「数列の和としての積分　入門」

Transcription

1 7

2 I = 5. introduction () ( ) g(t) g(t) ( ) II 97 6 ()

3 6. ( ) (Talor ) ( ) ( ) ( ) 8 () 7 9 ( ) ( ) (,, ) () III

4 I

5 5 =. introduction ) r (B.C. ) Newton Leibnitz (7 ) Newton r ) () 8, 8 / 8 8

6 = 6 n n, r S = (r) r = r, = = S =, =, = n, = n,, k = k n,, n = n n = ` ` k k k n n ` n n n n n n n k [ k, k ]. δ k δ k = n f( k ) = ( ) k n Sn S n = f( )( ) + f( )( ) + + f( n )( n n ) n = f( k )δ k = k= n k= ( ) k n n = n n k = k= (n + )(n + ) 6n ;δ k ) S S = lim n S n = lim n (n + )(n + ) 6n = n = 5 = n = = n = ` 5 ` 5 ` = ` ` 8 ` 8 9 CG web site. ) d D

7 = 7 = ` ` n n = ` n n n n n n,, n f( k )δ k < S < k= n f( k )δ k k= n n (k ) < S < n n k= (n )(n ) 6n < S < k= k (n + )(n + ) 6n n n (n )(n ) 6n, (n + )(n + ) 6n S = ( ) = e =, e. ( ) ) a < b, = f() a b a b a = < < < < n < n = b { k } n δ k = k k (k =,,, n), δ k, η k k η k k (k =,,,, n) b a f()d = lim {f(η )( ) + f(η )( ) + + f(η n )( n n } n = lim f(η k )( k k ) k= = lim k= n f(η k )δ k ( ) )

8 = 8 η η = f() n n f() ( ) { k }, f() n k= f(η k)δ k ( ) {η k } ( ) η k f() ( ) f() a b ; f(η k )δ k d δ Leibnitz.) f(ηk ) δ k f() d b a f()d δ k f(η k )δ k () f(η k )δ k d ( ) ( ) ) )

9 = 9. {a n } a k = F (k + ) F (k) F (k) (k =,,, ) n n n a k = {F (k + ) F (k)} = {F (k) F (k + )} k= k= k= = [{F () F ()} + {F () F ()} + {F () F ()} + + {F (n) F (n + )}] = {F () F (n + )} = F (n + ) F () ( ) n k= k(k + ) = = n k= { k k + ( ) + = n + = } ( ) + n n + ( ) ( + + n ) n + F (n) = n a k = k(k + ) = k ( k + = ) ( ) = F (k + ) F (k) k + k n k= k(k + ) = n k= a k = n {F (k + ) F (k)} = F (n + ) F () = n + + k= () k+ k = ( ) k = k k = (k+ k )., n k = k= n ( k+ k ) = k= n ( k k+ ) k= = { ( ) + ( ) + ( ) + + ( n n+ ) } = + n+ = (n ) F (k) = k a k = k = F (k + ) F (k) n n k = {F (k + ) F (k)} = F (n + ) F (n) = (n+ n ) k= k=

10 = S n = a( rn ) r = a(rn ) r r k r k = (r )r k, r=\ r k = rk r rk r F (k) = a rk r ark = F (k + ) F (k). n ar k = k= n {F (k + ) F (k)} = F (n + ) F () = arn r k= 5 ( n k= k ) ar r = a(rn ) r k =,,, (k + ) k = k + k + (k + ) k = k + (k + ) k = + 6 (k + ) (k + ) + (k + ) 6 { k k + k } = k + k + 6 k = k F (k) = k k + k 6, k = F (k + ) F (k) n k = k= n {F (k + ) F (k)} = F (n + ) F () k= (n + ) (n + ) = + n + { = n + { (n + ) (n + ) + } 6 n(n + )(n + ) = 6 } S n = n(a+an)

11 =... ( ) b a f() d def = lim k= ( ) a b F () b a n f(η k )δ k f() = F () def = lim h F ( + h) F () h f() d = lim k= n f(η k )δ k = F (b) F (a) (a = < < < < n = b, k η k k, δ k = k k, = ma{δ k }) δ δ δ δ n a = n n = b η η η η n F () = f(), k =,,,, n F ( k ) F ( k ) = F (η k )( k k ) = f(η k )( k k ) = f(η k )δ k R = F () (η k, [ k, k ] ) Q k =,,,, n F ( ) F ( ) = f(η )δ F ( ) F ( ) = f(η )δ k η k k F ( ) F ( ) = f(η )δ // QR.. + ) F ( n ) F ( n ) = f(η n )δ n n F ( n ) F ( ) = f(η k )δ k k= = a, n = b n f(η k )δ k = F (b) F (a) k= ( )

12 = nx f(η k )δ k k= nx f(η k )δ k k= = f() = f() = f() V k η η η η η η k k η k η k η k, [ k, k ] η k, [ k, k ] ( f() {η k } {η k }.) η k, [ k, k ] n n n f(η k )δ k f(η k )δ k = {f(η k ) f(η k )}δ k k= k= k= [ k, k ] f() V k, V k (k =,,,, n) V f(η k ) f(η k ) V, α + β α + β n n {f(η k ) f(η k )}δ k f(η k ) f(η k ) δ k V k= k= k= k= k= n ( k k ) = V (b a) ( ) V 5) n n n f(η k )δ k f(η k )δ k = {f(η k ) f(η k )}δ k ( ) ( ) ( ) k= b a f() d = lim k= n f(η k )δ k = F (b) F (a) () a k = F (k + ) F (k) = f() = F () = lim h F ( + h) F () h = n a k = F (n + ) F () k= b a f()d def = lim k= n f(η k )δ k = F (b) F (a) a(k) = F (k + )-F (k) F (k), 5) V ( )

13 = 6 f() =, F () = [, ] { k } n = < < < < n = = n. F ( k + ) F ( k ) = ( k + ) ( k ) = ( k ) + k ( ) + ( ) ( ) ( ) ( ) ( ) k k = n n + + = k + k + n n n n F k = F ( k + ) F ( k ) = F (η k ) = f(η k ) k n < η k < k+ n F k = k + k + n = n (η k) k + k +. η k = n η k [ k, k+ ] n n f(η k ) = (η k ) = k= k= n k= k + k + n n = n = = F () F () n ( ) η k [ k, k+ ], [ k, k+ ] η k n I n. ( ) I n n n I = f(η k ) = (η k ) k= n ( ) I n = k= k < η k < k+ ( k n ) < η k < ( k+ n k= n { k } + k + n (η k ) ). k ( ) ( ) + k + k + n < k + k + n n (η k ) < k + k + k n n k + n < k + k + n (η k ) < k + n n k= n k + n n < ( ) I n < k= n n k + n n η k I n ( ) ( ) Comment ( ) ( ) (.)

14 = a = < < < < n = b { k } [a, b] n F () = f() F k = f( k ) + ɛ ( F () = lim n n n F k = f( k ) + ɛ k= k= ( ) ɛ E n n F k f( k ) = k= k= n n ɛ k= k= n k= F k = F (b) F (a), E 6) F (b) F (a) = lim k= ) F k = f( k) lim ɛ = k= n f( k ) = n ɛ E b a k= f()d = E(b a) 7) ɛ 8) 7 f() =, F () =,a = < < < < n = b { k }, F ( k+ ) F ( k ) = ( k + ) ( k ) = ( k ) + k ( ) + ( ) n n n {F ( k+ ) F ( k )} = ( k ) + ɛ k (ɛ k = k + ( ) ) k= k= n k= {F ( k+) F ( k )} = F ( n ) F ( ) = F (b) F (a), a k b ɛ k = ( k + )) ( k + ) ( ma( a, b ) + (b a)) k= E = ( ma( a, b ) + (b a)) n {F (b) F (a)} n ( k ) = n n ɛ k ɛ k E = E(b a) k= k= k= k= n lim ( k ) = F (b) F (a). k= b a d = F (b) F (a) 6) k E ( ). 7) [ k, k+ ] f() 8) α α, β lim β = α β β.

15 = 5 = f() = n = 5 n = V V V V V V k (k+) [ k, k+ ] f(), V j = ma{v k } (=\ ) f() f() = ( = ) f() = [, ] k = k (k =,,,, n) n =. n f() < V k = f( k ) f( k+ ) = n k (, ] = k n = n lim V k = lim n k n 5 5 n k + = n ` k + = n 5 n k(k + ), V k (n )V k k = 5 V = f( ) f() = = n V k, V n ( ) f() V f() f () lim V = f () ) f () a b [a, b] n { k } [ k, k+ ] V k, V k V lim V =, + h [ k, k+ ], f( + h) f() = hf (η) (η + h ) f () f () [a, b] M f( + h) f() = h f (η) h M M, + h [ k, k+ ], f( + h) f() M V k M. lim V k = Q.E.D. V k M M M k ( ) M f()

16 = 6 n 6 J n = f(η k ) n =, n =, n = η k (k = k=,,, n) J n F () F () f() = e, [, ] { k } n k = k n (k =,,,, n), = k+ k n n η k (k =,,,, n) [ k, k ] J n = f( k ), I n = f(η k ), I n J n (e ) lim I n J n = k= k= F () = f() F (), n k= f(η k)δ k = F (b) F (a) ( ( )) ( ( ))

17 = 7.. (,, g() = e d = lim k= e d = lim k= n f(η k )δ k = lim n n k= n g(η k )δ k = lim n k= ( ) k n n e k n ) =, (e ) = e, f() = [ ] n = = [ n = e ] = e e sin d f() F () = f() F () f() [a, b] (a < < b) F () = f() F () [a, ] n N, M n N a < n (N + ), n M < (M + ) n N, M, { k } = a, = n (N + ), = n (N + ),, M N = n M, M N+ = [ k, k ] (k =,,,, M N + ) f() m k F n () = M N+ k= m k δ k = M N+ k= m k ( k k ) = M N+ (.) n k= m k (), ) = f() n = f() n a N n N+ M M+ n n n n N a n+ N + n+ M M + n+ n+ n+ ( ) F n () n (f() >.) F n ()

18 = 8 ) F n () (b a) ( [a, b] f() ) lim n F n() 9) M N+ F () = lim F M N+ n() = lim m k δ k = lim m k n n n k= F () F () F () = f() h > ( + h b), [, + h] f() m, M, [ M n, + h] ) n (n =,,, ) k= n f() m n, = f() = M m n h F n ( + h) F n () M h = m = m n h m n F n( + h) F n () h M n N a n N+ M n n n ) + h m F ( + h) F () h M m, M n h h + f() M f() m f(). lim h + F ( + h) F () h = f() h < f(), f() = F () = lim k= n f(η k )δ k = F (b) F (a) (a = < < < < n = b, k η k k, δ k = k k, = ma{δ k }) ( ) {η k } 9)., a n M ( ) a n a n+ (n =,,, ) lim an n ) [, + h] n. ) a n b n (n =,,, ) {a n}, {b n}, lim an lim n n bn

19 = 9 ( ) f() [a, b] a b F () = a f()d = lim k= n f(η k )δ k ( a = < < < < n =, k < η k k, δ k = k k, = ma{δ k }), ( ) {η k } F () F () = f() ).5 () f(), a k = F (k + ) F (k) = f() = F () = lim h F ( + h) F () h S n = n a k = F (n + ) F () k= = lim k= n a k = a n+ = S n+ S n k= F () = def f()d = a n k= n f(η k )δ k = F (b) F (a) f(η k )δ k = F () = lim h F ( + h) F () h = f() ) ( ) f() ( ) {η k } ( ( )) S() Z S() def = f()d a S () = f() ( ) = f() =, t = a, t = S() S () = f() ( ) (i) : F () F () = f(). (ii) F () = f(), ( ) {η k },. (.) (ˆ ˆ ;),,

20 = S n = n a k = a n+ = S n+ S n k= a n S n S n = n k= k k = S k S k (k =,,, ), S n ( ) F () = f() F () a k+ a k = F k F k f()d n k= a k ) b a f()d n k= a k 8 I = sin sin + cos d, J = () t = I = J () I (.) () t =, cos sin + cos d dt = d, t I = = sin sin + cos d = cos t cos t + sin t dt = sin ( t) sin ( t) + cos ( cos sin + cos d = J t) ( dt) () I + J = () I = J sin sin + cos d + cos sin + cos d = sin + cos sin + cos d = d = I = J = ( ) I J sin cos =.( ) F () = f() F () a k+ a k = F k F k f()d n k= a k ( ) ) k, k a k+ a k = F k F k

21 = t =, = sin() dt = d, t = cos(), f(sin )d = = f ( ( )) sin t ( dt) f(cos t)dt = f(cos )d. sin cos = 5 S n = a + a + a + + a n S n = n n {a n } S n n 6 F () = (t ) sin t dt F (). 7 8 n =,,, I = I = n k= + ( ) d k k + (n k + )

22 g() = u g (). ( ) g() = u f(g())g ()d = f(u)du f(u) F (u) F (u) = f(u) df (g()) d = df (u) d = F (u) du d = f(u) du d = f(g())g () f(g())g () F (g()) f(g())g ()d = F (g()) + C = F (u) + C = f(u)du () ( A tpe) g() = u, g(a) = α, g(b) = β β α f(g())g ()d = b a f(u)du α β u a b f(g())g ()d = F (u) + C = F (g()) + C ( ), β α f(g())g ()d = [ ] β F (g()) = F (g(β)) F (g(α)) = F (b) F (a) = α b a f(u)du () g() = u g () = du d. g ()d = du. f(g()) g ()d = f(u) du g ()d du

23 9 ( A tpe) () log d () e e log d () log d = log (log ) d log = u d = du. log d = u du = u + C = (log ) + C du u = log, d ) =. d = du () () log d = (log ) + C e e log [ (log ) ] e d = = (log e ) e (log e) [ = u ] = = udu, log = u d = du. e e u log e log e e e log d = [ u u du = ] = = ( ) (A tpe), f(g())g () g() ( A tpe) () sin cos d () + d () d ) j d (log ) ff = log (log ) = log d

24 () sin cos d = sin (sin ) d sin = u cos d = du. sin cos d = u du = u + C = sin + C () + d = + ( + ) d, + = u ( )d = du., + d = u du = log u + C = log( + ) + C () d = ( ) ( ) d = u d = du. d = u d = du = ( u) + C = ( ) + C ) (, u) (t, ) ) ( B tpe) = g(t), a = g(α), b = g(β) b f()d = β a α f(g(t))g (t)dt a b t α β ( B tpe) () d () + d () = sin θ( θ ), d = cos θ, d = ). 8 < : = 6 6 θ 6 ( sin 6 θ) cos θdθ = [ ( + cos θ)dθ = θ +. sin θ ( d d {f()}n = n{f()} n f () (n=\ ) d d log f() = f () f() R {f()} n f ()d = {f()}n+ + C (n=\ ) n+ d = log f() + C R f () f() (),() ). ] 6 cos θdθ = +

25 5 () = tan θ( < θ < ), d = cos θ, θ. + d = tan θ + cos θ dθ = dθ = [ ] θ = g()g () (.) sin θ = cos θ, + tan θ = cos θ, cos θ = tan θ. a = a sin θ ( θ ) a +, a + = a tan θ ( < θ < ) a = a cos θ ( θ <, < θ ) ) Tpe A Tpe B Tpe B Tpe A a d () 9 () a a d () a a d () () sin d ( + ) + d 5 (.) ) = a cos θ, = a θ θ sin θ sin θ,tan θ = a sin θ ( θ ) = a sin θ ( θ < )

26 6. ( ) () = g(t), a = g(α), b = g(β) b f()d = β a α f(g(t))g (t)dt a b t α β.. g(t) g(t) t α = t t t t k η k µ k t k χ k k k (a) n (b) = f() ( ) b a f()d = lim k= n f(χ k )( k k ) ( ), a = < < < < n < n = b, χ k k χ k k, δ k = k k. i = g(t i ), χ i = g(η i ) g(t), k k = g(t k ) g(t k ) = g (µ k )(t k t k ) µ k (t k < µ k < t k ) b β = t n t = g(t) a f()d = lim k= n f(g(η k ))g (µ k )(t k t k ) f() t = ma{t k t k } t. 5) g(t) f(g(t)) [t k, t k ] η k, t n f(g(η k ))g (µ k )(t k t k ) k= n f(g(η k ))g (η k )(t k t k ) k= ( ) b a f()d = lim k= = lim = t k= β α n f(g(η k ))g (µ k )(t k t k ) n f(g(η k ))g (η k )(t k t k ) f(g(t))g (t)dt 5) ( ).

27 7 k = g (µ k ) t k d f (t)dt ( ) (.) f(ηk ) δ k f() d.. g(t) = f() g(t) α t γ γ t β g(α) = a, g(γ) = c, g(β) = b, a < b [α, γ] l α = t < t < t < < t l = γ g(t ) =, g(t ) =,, g(t l ) = c α = t t t l l (c) (a) n (b) γ α f(g(t))g (t)dt = lim t k= l f(g(η k ))g (η k )(t k t k ) (η k, [t k, t k ] ) t l g(t) γ = t l k k = g (η k )(t k t k ) δ = t m β = t n t lim t k= = g(t) η k [t k, t k ] g(t) l f(g(η k ))g (η k )(t k t k ) = lim k= l f(χ k )( k k ) (χ k [ k, k ] ) α t γ g(t), k k < (k =,,, l) γ α f(g(t))g (t)dt = lim k= l f(χ k ))( k k ) = γ t β g(t) a c f()d ( = c a ) f()d β γ f(g(t))g (t)dt = b f()d = a f()d + b c c a f()d, β α f(g(t))g (t)dt = a ( a f()d + f()d + b f()d c c a a ) = b f()d β α f(g(t))g (t)dt = b f()d a

28 8 Comment = g(t) t k = t k+ t k > k > ( g (t k ) > k > g (t k ) < k < ) g (t) < a c t α γ, γ α f(g(t))g (t)dt = c a f()d t α β a c a b t n f(g(t k ))g (t k )δt k = k= } {{ } α t β l m f(g(t k ))g (t k )δt k + f(g(t k ))g (t k )δt k + k= } {{ } α t γ l m f( k )δ k + k= } {{ } δ k n f( k )δ k k=m } {{ } a b k=l k=l } {{ } γ t δ n + f( k )δ k f( k )δ k } {{ } δ k k=m } {{ } δ k n f(g(t k ))g (t k )δt k k=m } {{ } δ t β. δt k β α f(g(t))g (t)dt = b a f()d, t α β a b (g(t), ) a b, a b,. ) g(t)

29 9. ( ) = g(t) δt k = t k+ t k > k+ k >. k+ = g(t k+ ), k = g(t k ) ) f() > f()δ k = f()( k+ k ) = k, = k+, = f(), = δs δi = f()δ k = f()( k+ k ) { { f() > (i) k+ k >, (iv) f() < δi = δs. k+ k < { { f() > (ii) k+ k <, (iii) f() < δi = δs k+ k > j (i) f() > k+ k > j (ii) f() > k+ k < j (iii) f() < k+ k > j (iv) f() < k+ k < k k+ k+ k = f() = f() k k+ k+ k = f() = f() { f() > (i) k+ k > f( k ) = f(g(t k ))( k = g(t k )). δ f (t k )δt k δ k : δt k = g (t k ) : = f(g(t))g (t), : g (t k ), g (t k ) : ( ) f( k )δ k f(g(t k ))g (t k )δt k δt, δ, = b f()d = β a α f(g(t))g (t)dt f() < g (t) < = f(g(t))g (t) = f() = f(g(t)) g (t) f(g(t k ))g (t)δt k f( k )δ k f(g(t k ))δt k k t k+ t k t k+ t k t k+ g (t k ) g (t k ) t

30 I = d, J = e et dt = t d = dt, t I = J h (h > ) t δt h δt = ( + h) = h + h h e t δt e h = e δ I = J = e = e t e e t + h ( + h) δ δt I = d, J = e et dt = t d = dt, t I = J h (h > ) t δt δt h. e t δt e h = e δ. e t δt = e δ. δt, δ I,J I = J < < δ = h >, δt = h < e t δt < e δ <. ( ) I = e, =, =, = ( ) J = e t, =, =, = ( ) =. = e = e t e δ + h t ( + h) δt = h

31 I = ( )e d = t ( )d = dt, t J = e t dt I = J = h (h > ) t δt ( )d = dt, δt = { ( + h) ( + h) } ( ) ( )h ( )e δ = ( )e h e ( )h = et δt I J I = J < < <, δt = ( ) < ( )e δ < et δt <. ( ) ( ) < < >, δt = ( ) > ( )e δ > et δt >. ( ) = et, =, t =, t = S, I = ( S) + } {{ } }{{} S = = ( )e ( )e = e + h + h g() g( + h) g() e t ( )e ( )h ( )h t t t = t = + h + h g() g( + h) g( + h) g() < < < <

32 CG δi = ( )e δ, δj = et δt, δs δi = δj = δs δi = δj = δs ( )e δ et δt t t t t t t

33 Tpe B.Tpe A 5 I = d = sin θ ( θ ) d = cos θdθ, θ sin θ cos θ dθ = cos θ dθ I = J θ δθ (δθ > ) J = δ cos θdθ = d, δ cos θδθ < θ < δ >, cos θ >, δi = δ cos θ cos θdθ δj = cos θδθ I I = J =. cos θ = cos θ θ δθ cos θ sin θ + δ θ θ + δθ θ δ = cos θδθ Comment = sin θ θ θ = tan θ θ I = sin d, J = sin tdt = f() = sin = sin t I = J I = + d = g(θ) = tan θ = f() = + θ = f(g(θ)) = f(g(θ))g (θ)

34 . d = sin θ = θ θ = g(t) g (t) g(α) = a, g(β) = b = sin θ ( θ ) d = cos θdθ.. θ 6, 6 θ = sin θ θ 5 6 d = = = 6 6 ( sin θ) cos θdθ [ sin θ + θ cos 6 θdθ = ] 6 = (cos θ + )dθ + d = = = = ( sin 6 θ) cos θdθ = 5 cos 6 θdθ + ( cos θ)dθ (cos θ + )dθ + [ sin θ + θ ] 5 6 [ + sin θ θ cos θ cos θdθ ( cos θ )dθ ] 5 6 = + ( 5 ) ( ) = +, cos θ = cos θ = { cos θ ( θ ) cos θ ( θ 5 6 ) θ 6, θ f(g(θ))g (θ)dθ = cos θ cos θdθ 5 6 f(g(θ))g (θ)dθ = 5 6 cos θ cos θdθ 6 f(g(θ))g (θ)dθ = θ 6

35 5 = g(θ) = sin θ g (θ) = cos θ = g(θ) = tan θ g (t) = cos θ = ±, ±, ± 5, : I = + d = tan θ ( < θ < ) d = cos θ dθ. θ I = + d = + tan θ cos θ dθ = dθ = g ( 5 ) = θ 5 I = 5 + tan θ 5 cos θ dθ = dθ = 5,! = tan θ = dθ = θ k θ k > d = k k > ( g(θ k ) = tan θ k = k ) I = I = 5 f(g(θ))g (θ)dθ = f(g(θ))g (θ)dθ = 5 dθ = dθ = [ ] θ [ θ ] 5 = > = > I +I = =\ g (θ) dθ = θ k θ k > d = k k < I I 6) = + = tan θ 5 θ 6) tan θ θ = R +tan θ cos θdθ, R R +tan θ cos θdθ = lim α R α +tan θ cos θdθ = lim α α dθ = lim α `α = g(t)?

36 6. f() > S = b a f()d f() < S = b { f()} d, a, a b f() g(), f() g(), = f(), = g(), = a, = b S f() g() = f() = g() S = b a def = lim {f() g()} d k= n {f(η k ) g(η k )} ( k k ) ( a = < < < < n = b, k < η k < k, = ma{ k k }) a Q b = f() = g() S ( ) {f() g()} (Q) ) (a) k k n (b) = f() = g() f(),g() f() g() = f() = f() = g() = α { f(α) = g(α) f (α) = g (α) ) = g() f() g() = ( α) h()(h() ) f() g() = = α. α ) (?) :Q δ k Qδ k ( δ k = k k ) ) f() g() = F ().(i) F (α) = F (α) = F () ( α) F () ( α) Q(), a + b F () = ( α) Q() + a + b F () = ( α)q() + ( α) Q () + a ( )

37 7 6 C : = f() = + a + b l : = g() = m + n = = f(), = g(), =, = S a, b, m, n C : = + a + b = f() = g() = f() = g() = f() g() = + a + b (m + n) = ( ) l : = m + n S = = {f() g()} d [ ( ) ( ) = ] = 6 Comment = ( ) = ( ) ( a, b, m, n ) f(), g() = α f() = g() ) { f(α) = g(α) = f() = g() = α f (α) = g (α) C : = f() = + a + b l : = g() = p + q + r =, = Q = f() = g() S a, b, p, q, r. < f() = sin, g() = a cos = f() = g() a = f(), = g() S a. ( ) F (α) = F (α) = j aα + b = a = a = b = F () ( α) F () = ( α) h() ( ), F (α) = F (α) = ). Q.E.D.

38 8. a C : j = (t) = (t) b C C : { = (t) = (t) d d dt dt t t (t = f () ) = f() = (t) (t) >, a < b = a, = b, C S 7 S = b a d = β α d dt, (α) = a, (β) = b ( ) dt () { = θ sin θ ( θ ) = cos θ S ( k, k ) ( k, k ) (, ) (, ) ( n, n) (, ) = k k = θ sin θ, d = ( cos θ)dθ. θ θ k θ k θ Comment = θ sin θ S = = = [ = d = ( cos θ) ( cos θ)dθ ( cos θ + cos θ ) dθ ( cos θ + θ sin θ + sin θ + cos θ θ [, ] = θ < θ < θ < < θ n = n θ k, k, k, k k = (η k )(θ k θ k ) (θ k < η k < θ k ) S = lim k= n k ( k k ) = lim θ k= n (θ k ) (η k )(θ k θ k ) = ] = d dθ dθ ) dθ

39 9 θ δθ (δθ > ) δ ( cos θ)dθ = d, δ ( cos θ)δθ δ >, cos θ, δi = δ ( cos θ) ( cos θ)dθ δj = ( cos θ) δθ δ S = lim δi = lim θ δj = ( cos θ) dθ cos θ ( cos θ) = ( cos θ) + δ δ = ( cos θ)δθ θ sin θ θ δθ θ δ ( cos θ) δθ, ( δθ ) t t t t t t

40 d dt (, ) ( d dt >, d dt <. ) 8 C : { = t + = t + t + C S t= 5 t = 5 t = S S = f () 5 t = ( t ) d dt = t, d dt = t + t d dt d ( dt ) ( ) + + ( ) + ) ( ( 5 9 ) 5 C t C = f (), t C = f (), S, S S = 5 f ()d, S = f ()d = f () = f () (, ), 5 t S 5 = f () (, ), t S = S S S = S S = Comment d dt dt d dt dt = d dt dt = ( t + t + ) tdt = 9 t δt (δt > ) δ dt = d, δ tδt < t <, δ <,, δi = δ ( t +t+) tδt δj = ( t +t+) tδt ( ) ( ) < t <, δ >,, δi δj ( ), δi t S, δi t S lim δ δi = ( t +t+) tdt S

41 t = t + t + t( t + t + ) = t( t + t + ) g(t + δt) g(t) 5 g(t + δt) g(t) δ = tδt δ = tδt = g(t) = t + t + δt t t t + δt t( t + t + ) t δi = δ δj = ( t + t + )tδt ( ) (t) >, t (, ) C : = (t), = (t) S t = α a b t = β S = β α d dt, (α) = a, (β) = b, α < β ( ) dt ( S).

42 9 C : { =sin θ cos θ =sin θ + cos θ + ( θ ) S d dθ = cos θ + sin θ = ( sin θ + ), d dθ = cos θ ( sin θ = sin θ θ 6 6 d dθ d dθ «««! «! + C θ, S = = = = = f ()d d dθ dθ d dθ dθ + d dθ dθ ) θ = f (), θ 7 = f () f ()d + d dθ dθ + d dθ dθ f ()d d dθ dθ d dθ dθ (sin θ + cos θ + )(sin θ + cos θ)dθ θ =, θ = 7 = f () θ = 6 { = sin θ + cos θ + ( } + ) sin θ cos θ + sin θ + cos θ dθ ( cos θ = + + cos θ + + ) sin θ + sin θ + cos θ dθ [ + = θ + sin θ + ] cos θ cos θ + sin θ = ( + ) S θ = = f () θ = C ( ) C C : = f(t), = g(t) (α t β) t α β (f(t), g(t)) C α t γ d d dt, γ t δ dt, δ t β d dt, C : S CG web site. t = α, β t = γ S t = δ a b c

43 ? = g(t) t δt(δt > ) δ, δi = δ. = g(t) d = g (t)dt. δt δ g (t)δt δi = δ g (t)δt (i) α < t < γ. g (t) > δ >. δi (ii) γ < t < δ. g (t) < δ <. δi ( ) (iii) δ < t < β. g (t) > (i) δi δi = δ t = α t = β t = γ t = γ t = δ δi = δ t = δ k k n m k k n t = α t. t k t k. t n = γ t n = γ t n+. t k t k t = f(t). t m = δ t = f(t) t m = δ. t l = β t = f(t) t = α t = α t = γ t = γ S t = δ t = δ S S

44 S, S, S l g (t k )δt k = k= n m g (t k )δt k + g (t k )δt k + k= k= k=n n m δ k + δ k + k=n S + ( S ) + S l δ k k=m l g (t k )δt k k=m δt = J = β α g (t)dt = β α d dt dt = S + ( S ) + S = S δi = δ t = α, β t = α. t k t k. X X + h δi = δ ( ) t i t i. β t = f(t) (, ) C (, ) [X, X + h] (h > ), t (, ) g (t)δt = δ > δj = g (t)δt = δ > δj ( ) g (t)δt = δ < δj = g (t)δt = δ < δj ( ) ( ) t k t t k t i t t i δj C = X, = X + h δs t α β δj δt J J = β α d dt dt = β α g (t) dt = lim g (t) δt = lim δj = lim δs = S t t t

45 5 δi = δ ( ) t = α, β t = α X δi = δ X + h t k t k t i t i β = f(t) t (, ) C C S [X, X + h] (h > ) t (, ), g (t)δt = δ < δj = g (t)δt = δ < δj ( ) ( ) g (t)δt = δ > δj = g (t)δt = δ > δj ( ). ( ). t k t t k t i t t i δj, C = X, = X + h δs ( ) t α β δj δt J J = β α d dt dt = β α g (t) dt = lim g (t) δt = lim t S = J = β α t d dt dt δj = lim ( δs) = S t, > < (.)

46 6 C > < ( = f() = g() ) t = α, β X X + h δi = δ ( ) δi = δ ( ) t = α. t k t k. t i t i. β t = f(t) (t) < C <, (, ) C. [X, X + h] (h > ) = X C,. =, =. δs C = X, = X + h, δ g (t)δt >, δ g (t)δt < δt = t k t k, δt = t i t i δt δt δj δj + δj = g (t k )δt + g (t i )δt δ + δ = h + ( h) = ( ) h ( ) δj + δj = δs δj t J J = β α d β dt dt = g (t) dt = lim g (t) δt = lim δj = lim δs = S α t t t ( ) δs > < ) ) < δj = g (t)δt = δ < δj ( ) ( ) δj = g (t)δt = δ > δj ( ) δt δt δj C = X, = X + h δs

47 7 t = α, β t = α, β δs a S δs b X X + h α α δt δt δt δt β β t t (, ) C [X, X + h] (h > ) C = X,,,,,,,. [X, X + h] t δt, δt, δt, δt, C S C = X, = X + h δs δs a, δs b, δ g (t)δt >, δ g (t)δt < δt δt δj = g (t)δt δj + δj δ + ( δ) = ( ) δ = h δs a δt δt δj δj + δj δ + ( δ) = ( ) δ = h δs b, δt, δt, δt, δt δj δs a + δs b = δs δj t J J = β α d β dt dt = g (t) dt = lim g (t) δt = lim δj = lim δs = S α t t t C. )

48 8 ( ) C ( ) C C : = f(t), = g(t) (α t β) t α β (f(t), g(t)) C, C S t = α, β S S = β α d dt dt ( ) C ( ) C C : = f(t), = g(t) (α t β) t α β (f(t), g(t)) C, C S t = α, β S β S = d α dt dt C 8, 9 C 5) C t β t γ ((t), (t) = (h(t), ) C (h(t) h(β) = f(β), h(γ) = f(α) ) β t γ S = γ α d β dt dt = d γ α dt dt + d β β dt dt = d γ α dt dt + d β β dt dt = d α dt dt 8 < = f(t) C : = g(t) : α t β 8 < = f(t) C : = g(t) : α t γ t = α t = β t = α, γ {z } t = β C : = l : = + t (t=\ ) () t () =, = S { = sin θ + cos θ 5 C : ( θ ) = sin θ + cos θ 5) = f(), = g(), = a, = b S

49 9. (r, θ), r θ (, ) r θ (, ) = r cos θ, = r sin θ, r = + S C, C : r = f(θ) C θ, α θ β S α β S = β α {r(θ)} dθ r θ r θ S r S = r θ S = δθ k S k α θ β, α = θ < θ < θ < < θ n = β [α, β] n θ θ k θ θ k S k θ k η k θ k S k = {r(η k)} δθ k (δθ k = θ k θ k ) η k (θ k < η k < θ k ) ( ) S = n S k = lim k= n θ k= {r(η k)} δθ k = lim n θ k= β {r(η k)} (θ k θ k ) = α {r(θ)} dθ ()

50 5 { = e θ cos θ C : ( θ ) C = e θ S sin θ j = e θ cos θ C : = e θ sin θ ( θ ) (, ) S θ { = r cos θ = r sin θ r(θ) = e θ S = r dθ = (eθ ) dθ = [ eθ dθ = eθ] = (e ) C S = = = = d dθ dθ = e θ (sin θ sin θ cos θ)dθ = eθ dθ [e θ] [ e θ sin θ e θ sin θ(e θ cos θ) dθ = e θ (cos θ + sin θ)dθ ] = (e ) θ cos θ sin θ e dθ e θ sin θ(e θ cos θ e θ sin θ)dθ, (.) 6 C : { = θ(sin θ + cos θ) = θ(sin θ cos θ) ( θ ) = S

51 5. S() : = a, = b ( a b) V V = b a S()d S() a b V = lim k= n S(η k )( k k ) = lim k= n S(η k )δ = b a S()d S(η k )δ ( a = < < < < n = b, k < η k k, δ k = k k, = ma{δ k }) a k η k k b = f() = a, = b (a < b) f() = f() S() = f() = {f()} V a S() b V = b {f()} d = b a a ( = f()) d = f() f()

52 5 = g() = a, = b (a < b) g() S() d S() = g() = {g()} S() = g() V V = d {g()} d = d c c ( = g()) d c, ( ) = sin ( ) = + Q Q R QR (, ) QR R z z t R Q = + = sin S() = QR V = S()d = Q d = ( + sin ) d { = ( + ) ( + ) sin + sin } z d { ( + ) = ( + ) + } ( = + ) ). ) Z Z Z ( + ) sin d = ( + )( cos ) d = ( + )( cos ) ( + ) ( cos )d = ( + ) cos + sin + C Z ( + ) d = ( + ) + C, Z Z sin d = cos d = sin + C

53 5 r V + = r = ± r. : r ( r r) r V = r = r r d = r r (r )d (r )d = [r ] r = r r r = r r = f(), = g(),., : + ( ) V ( ) +( ) = = ± + ( ) ( ) ) V = = = 8 ( + ) d ( ) d { ( + ) ( ) } d d = 8 = ( ) = + = + ( ) = ) ( ), R = sin θ.k..

54 5 = f() = g() a b f() g() = f(), = g(), = a, = b V = f() f()(a b) V f()(a b) V = g() V = = b a b a {f()} d b a {g()} d [ {f()} {g()} ] d a b = f() = g() a b = g() = f() S() f() g(), S() S() = {f()} {g()} = [ {f()} {g()} ] V = b a S()d = b a [ {f()} {g()} ] d, V = b a [f() g()] d () = f() = g() (f() g().) <

55 55 = = + V = = + = = ( ) = +. = + = =, = = + V = ( + ) d + [ 5 = 5 + ] ( + ) d + [ ( + ) = = ] ( ) d [ ] = + 7 = sin = cos

56 56. 5 ( ) { = θ sin θ ( θ ) = cos θ V V = d = θ sin θ d = ( cos θ)dθ. = cos θ. θ. (, ) C V = = d = ( cos θ) dθ = ( cos θ) ( cos θ)dθ ( cos θ + cos θ cos θ)dθ cos θdθ = ( sin θ)(sin θ) dθ, sin θ = u, cos θdθ = du, cos θdθ =. ( = cos θ ) V = { cos θ + } [ ( + cos θ) dθ = θ sin θ + ( θ + sin θ ) ] θ u. = 5 Comment θ δθ (δθ > ) δ ( cos θ)dθ = d, cos θ δ ( cos θ)δθ δ >, θ δθ δv, + δ δ = ( cos θ)δθ θ sin θ δv δ = ( cos θ) ( cos θ)δθ δθ δθ V = lim δv = ( cos θ) dθ θ θ = θ sin θ

57 57 6 ( = cos ) = cos ( ), V = cos V = = cos = cos (cos cos ) cos = cos d = sin d,. V =, V = = Comment sin d = (sin ) d = d d = ( sin )d = ( cos ) d = [ sin ] [ ] ( cos ) sin d = V = ( ) sin d ( cos )d = cos d = t, = cos t V = d. = cos t = t δ(δ > ) δ, δv, d = sin d, δ sin δ δ <, δv δ ( sin )δ = sin δ δ V = lim δv = lim sin δ = sin d

58 58 7 ( ) C : { =sin θ cos θ =sin θ + cos θ + ( θ ) V ( ) C C θ, 7 θ = f (), θ 7 = f () V = = = = = = {f ()} d d dθ dθ d dθ dθ + d dθ dθ 7 7 {f ()} d + d dθ dθ + d dθ dθ + 7 (sin θ + cos θ + ) (sin θ + cos θ)dθ 7 d dθ dθ d dθ dθ {f ()} d θ =, θ = 7 = f () θ = 6 { sin θ + sin θ cos θ + cos θ + 6 sin θ + 6 } cos θ (sin θ + cos θ)dθ θ = = f () θ = (, ) sin θdθ = sin θdθ = cos θdθ = cos θdθ =. sin θ cos θdθ = ( cos θ) cos θdθ =, sin θ cos θdθ = ( sin θ) sin θdθ =. V = ( 6 sin θ + 6 ) cos θ dθ = { ( cos θ) + } ( + cos θ) dθ = 6( + ) Comment ( ),

59 59. t δt (δt > ) V δv δ = (cos θ + sin θ)δθ > V (δv > ), δ = (cos θ + sin θ)δθ < V (δv < ), δv > δv < V. z z z z z

60 6 t = α, β δj = h t = α, β X X + h δj ( ) h t = α. t k t k X X + h δj ( ) ( h) ( ) t = α. t k t k δj = ( ) ( h) ( ).. t i t i t i t i.. β t = f(t) β t = f(t) C = f(t), = g(t) (α t β) ( ) t α β (f(t), g(t)) C. (, ) [X, X + h] (h > ) C V, = X, = X + h δv, t (, ) g (t)δt = δ > δv = δ = g (t)δt g (t)δt = δ < δv = ( δ) = g (t)δt. δj = g (t)δt,,,, δt = t k t k, δt = t i t i δt δt δj δj = δj + δj = ( ) g (t k )δt + ( ) g (t i )δt ( ) h + ( ) ( h) = ( )h = δv J = β α g (t) dt = β d α dt dt, β J = g (t) dt = lim g (t) δt = lim δj = lim δv = V α t δt t C < () ) (*) C ) ( ) C < ( ) ( ) <.

61 6 ( ) C ( ) C C : = f(t), = g(t) (α t β) t α β (f(t), g(t)) C C, C V t = α, β β V = d α dt dt ( ) C ( ) C C : = f(t), = g(t) (α t β) t α β (f(t), g(t)) C C, C V t = α, β β V = d α dt dt C = f() C C V β V = d α dt dt t = α, γ t = α, γ t = β {z } 8 = f() = sin ( ) V 9 C : r = a( + cos θ) ( θ ) t = β

62 6. 8 () z (cos θ, sin θ, ), Q(cos ( ) ( ) θ +, sin θ +, ) θ Q, z = z = V z. z. z z. z. z Q A(,, ) z = Q S θ z θ t R Q -t z = t SR = ( t) S + t SQ ( t) + t z = t ( t ) Q z = t Q R R z t R : RQ = R z z : Q z R z = t : ( t) ( z, z ) z = t z S(, t) = t cos θ sin θ + t sin θ cos θ = ( t) cos θ t sin θ ( t) sin θ + t cos θ t t z = t S(t) S(t) = SR = { ( t) + t } V V = S(t)dt = { ( t) + t } dt = [ (t ) + t ] =

63 6, 9 ( ) z A z A(,, ),B(,, ),C(,, ) ABC z V C B z = t ( t ) z, AB Q, AC R, QR H, z = t S(t) AQ : QB = (A Q z ) : (Q B z ) = ( t) : t z A Q = t A + ( t) 6 t B = t + t = t ( t) + t t ( R = Q, R = A = C = ) R = t A + ( t) C = t + t = ( t) + t t 6 t t Q t B R C QR H (i) Q = t t H QR H R S(t) = (R H ) R H = HR S(t) = HR = (R ) = H Q R

64 6 (ii) Q = t t H QR Q R S(t) = (R Q ) R = H + HR, Q = H + HQ R Q = (H + HR ) (H + HQ ) = HR HQ { ( ) } t S(t) = {(R ) (Q ) } = H Q ( ) = t t R V = S(t)dt + S(t)dt = dt + ( ) t t dt = Comment (ii) S(t) = (HR HQ ) Q,R z Q,R S(t) Q R = t (. Q Q R R, A,B,C z A,B,C ABC A B C z = k ABC z V A B C z V V = ( ) ( ) = = = V z A A z A C B C B C B

65 65 A,B,C z A,B,C ABC A B C z = k ABC z V A B C z V ( z =, z =.),.

66 第章 66 体積 例 (立体図形の回転体) の直円錐を 頂点を原点に固定して 平面上を図のようにすべることなく転がす 半径 高さ 円錐の中心軸が一周して元の位置に戻るとき 円錐が通過する領域の体積 V を求めよ 類題 東京電気大) z. z z z z.5. z 原則は 廻してから切るのではなく 切ってから廻せ です 円錐を 平面 z = t で切った断面を考えま す (下の図は 交線が見やすいように底面を抜いてあります ) 図からもわかりますが 平面と円錐の 交線は双曲線になります (数 C の範囲)

67 67 z z Q R S M Q S R M Q t θ M z = t z Q, R S, RS M z = t S(t) S(t) = (R Q ), R R = (,, t) R = 9 + ( ) =. R = R = t z., tan θ = Q = = t S(t) = { ( t ) ( ) } t = ( ) t t t V = S(t)dt = ( t ) dt = 8 Comment z, z = t, QS ( QS QS S(t) ) QS t ( ).

68 68... ( ) z A(,, ), D D : = D z E V z z = t D R D Q H A Q R Q H Q z = t ( < t < ) Q,,Q =, Q, z = t z R(,, t), R Q H, S(t) S(t) S(t) = (RQ RH ) = HQ = R S(t) D = D, D z E z = t E E, E E V = ( ) = = H = A AH = t. S(t) = H = ( t ) V = S(t)dt = ( t )dt = [ ] ( t )dt = t t = Comment (6 ) D Q Q Q Q Q Q D Q D D D Q

69 69 ( ) z A(,, ), D D : = D z E V z = Q z = A Q Q R Q R Q z = t u = H Q = M = D,Q,R,H, S(t),Q =, Q H Q, {( S(t) = (RQ R ) = RQ + QQ ) ( R + )} = (RQ R ) D = D, D z E E E =, z = u u z D (, ), V = (u ) + z = u = ± z + RQ R = ( + t ) ( t ) = t S(t)dt = ( RQ R ) dt = t dt = = ( ) Q M HQ H = (HQ + H)(HQ H) = HM Q = t ( ) ( ) Comment Q = Q = D H HQ = Q, ) ). V = =

70 7 ( ) z A(,, ), D D : z = + D z E V z z R Q H Q R Q R Q z = t A H Q D C,Q,R,H,,Q, S(t) S(t) S(t) = (RQ RH ) = HQ = R S(t),D = D, D z E E E z 5,Q R = AH Q = Q z, C D, C 5) D, C z V = ( ) = = H = A AH = ( t ) = t. V = Comment S(t)dt = ( t )dt = ( t )dt = [t t] = D z ABCD A(,, ), B(,, ), C(,, ), D(,, ) z E V 5).

71 7 () z D D : ( ) + + z, D z E V z z z = t t R(,, t) Q t A H Q R T T T Q H A S(t) H Q Q = t z = t ( < t < ) D z = t,q,r,t R Q H, S(t) S(t) { S(t) = (R RT ) = + ( t ) ( t ) } = t V = S(t)dt = t dt = = Comment QT =, Q T Q T z E E E z Q,T Q T z E E E, (.) z ABCD A(,, ), B(,, ), C(,, ), D(,, ) z E V

72 7. S(t) S(t) () (,?) 5 ( ) ( ) V.5 z. z.5. z = k z = k( k ) z k k k (,, k) k + z, = + z, = z = k C z + z C (,, z) C : + z ( ) C z + z C : + z ( ) D (,, z) { + z + z z = k, () { { + k + k k k k k

73 7 z = k ( k ) S(k) S(k) = ( k ) = ( k ) V = S(k)dk = ( k )dk = 8 ( k )dk = 6 C z = k k.) C z = k k ) (.) + (5 ) z

74 7 6 ( ) z ) D, A (,, ) B ( ),, D E V z z = t : z = + B z t + R R Q B S Q S A A t t D +. : z = +. E (,, z) { + z + = t ( t ) = t { + t z t + { t t z t + = t S(t) S(t) = S Q = ( t t + ) V = S(t)dt = t ( t + ) dt = t t dt + t dt t = u, tdt = du, t t dt = t dt t dt = + V = ( ) 8 t u ( u ) [ du = ] u = + = = 8 Y Y = t t

75 75 S(t) = S Q = ( t t + ) () z B S t z : z = + R S A Q t A B t t + Comment (), = t CG

76 76 7 ( ) z, z + + ( ),, +,, z V ( 98 ) = k = k, z = k = k z z = ( k) + k + k + S(k) k k + (, k, ) = k( < k < ) = k ( ) z ( k) + k + k +, k k + ( k) + k + k + = f() = k z = f(k) f(k) = k + >, f(k + ) = k > k k + f() >. = k S(k) S(k) = k+ k { ( k) + k + k + } d = {(k + ) + ( k)} = k + V = S(k)dk = k + dk = 7 = k ( = k )

77 77 5 = e =, e (7 ) [, ] n s n n n s n = f( k )δ k = e k n n = { + e n n + e n + + e n n k= k= } = n (e n ) n e n h = n n, h +. S Comment S = lim n s n = lim n n e e n = lim h + e e h h = e ( ) e d = [ ] e = e S n = n k= f( k)δ k = n k= e k n n S n = n(a+an) ( ) ( ) { (k + ) k = k + (k + ) k = (k + ) (k + ) k k = k f(k) = k k (k =,,, ), k = f(k + ) f(k) n k = k= n n {f(k + ) f(k)} = {f(k) f(k + )} k= k= = {f() f() + f() f() + f() f() + + f(n + )} = f(n + ) f() = (n + ) (n + ) = n(n + ) a n = a + (n )d n a k = k= n n n n {a + (k )d} = a + d (k ) = a k= = a n + d n(n ) k= k= k= = n {a + (n )d} = n(a + a n ) n + d k= k

78 5 78 n 6 J n = f(η k ) n =, n =, n = η k (k =,,, n) k= J n F () F () (6 ) 6, [, ] f() =, F () = n n J n = f(η k ) = f (η k ) n, η k + k + k = n. (i) n = k= k= n J = f(η k ) n = f(η ) = f k= (ii) n = n ( J = f(η k ) n = f(η ) + f(η ) = f k= = (iii) n = Comment + 7 = ( ) ( ) = = = F () F () ) = = F () F () n J = f(η k ) n = f(η ) + f(η ) + f(η ) k= ( ) ( ) ( ) = f f f 7 = = ( 7 + f ) = = F () F () n =, =, F () F () = jf ( ff ) F () + jf ( ) F ( ff ) + jf () F ( ff ) = F (η ) + F (η ) + F (η ) = f(η ) + f(η ) + f(η ) (η ) F () F ( ) = f(),, =, = [ k, k+ ] = f(),, = k, = k+, f(η k ) δ k {η k } = = = J q q 7 q 7 q 7 7 q 9 7

79 5 79 f() = e, [, ] { k } n k = k n (k =,,,, n), = k+ k n n η k (k =,,,, n) [ k, k ] J n = f( k ), I n = f(η k ), I n J n (e ) k= k= lim I n J n = (6 ) f() [, ] () f( k ) f(η k ) f( k ) f( k ) (k =,,,, n) n n {f( k ) f(η k )} {f( k ) f( k )} k= k= n n {f( k ) f( k )} = {f( k ) f( k )} = {f( n ) f( )} = {f() f()} k= k= J n I n = n k= {f( k) f( ηk )},, J n I n = k= n n {f( k ) f(η k )} {f( k ) f( k )} {f() f()} = (e ) k= I n J n (e ) (e ), lim I n J n =. = f() = e f( k ) f( k ) f( k) f(η k ) e Comment k k η k η η η η η 5 J n I n ( n = 5 ) f() δ k = k k J n I n = n n n {f( k ) f(η k )}δ k {f( k ) f(η k )}δ k {f( k ) f( k )} k= lim I n J n = k= k=

80 5 8 5 S n = a + a + a + + a n S n = n n {a n } S n n (( )) n a n = S n S n = ( n n ) { n (n ) } = n (n ) n = a n = n (n ) (n =,,, ) ( ) a n n =,,, n a n 7 5 S n n 8 a n > ( ) S n S n = a n a n > S n > S n, a n < S n < S n. S > S > S > > S 6 S 6 < S 7 < S 8 < S n n n = 6 ( ) 6 F () = (t ) sin t dt F ().( ) d F d = d d (t ) sin t dt = ( ) sin F () + F () F () = ( ) Comment

81 5 8 7 ( ) t =, dt = d, I = + ( ) d t. J = I = ( ) +( ) ( t) ( t) + t ( dt) = d I = J I + J = ( t) t + ( t) dt = + ( ) + ( ) d = ( ) + ( ) d d = I = ( ) 8 n =,,, ( ) I = n k= k k + (n k + ) I = + n + + (n ) + + (n ) + + (n ) (n ) + + (n ) (n ) + + n n + J = n k= (n k+) (n k+) +k J = n n + + (n ) (n ) + + (n ) (n ) (n ) + + (n ) + I + J = I = J n k= k k + (n k + ) + n k= (n k + ) n (n k + ) + k = k= + n k + (n k + ) n (n k + ) + k = = n k= I = J = n ( ) Comment

82 5 8 9 () a a d () a a d () () sin d ( + ) + d (5 ) () = a = a, ( a. a ( a ) a d = = a 8 ) + = ( a ( ) ),, I ` a = a, a = a ( ) a a = a sin θ, d = a cos θdθ, a θ I = = a 8 a ( a ) a ( a sin θ cos θdθ = [ θ + ( + cos θ)dθ = a 8 sin θ ) ] = a 8 cos θdθ a. () a a d = a a t a. a a d = (a ) a d t = a dt = d, t ( ) dt = a a [ ] a tdt = t = a ( ) () sin d = sin sin d = ( cos ) ( cos ) d cos = t sin d = dt, t. sin d = sin sin d = ( t )( dt) = ( t )dt = [ ] t t = ( ) Comment Z sin d =, cos. Z Z sin sin h d = (cos ) ( cos ) d = cos cos i =

83 5 8 I n = R sin n d (n =,,, ) I n+ = Z sin n+ d = Z Z sin n+ ( cos ) d = Z h i sin n+ ( cos ) Z + (sin n+ ) cos d = (n + ) sin n cos d = (n + ) sin n ( sin )d ( Z Z ) = (n + ) sin n d sin n+ d = (n + )(I n I n+) = (n + )I n (n + )I n+ (n + )I n+ = (n + )I n. I n+ = n + n + In ( ) () Z sin d = I = I = Z sin d = = () ( + + ) d = = { + d ( + ) + d + } d ( + ) d = tan θ d = cos θ dθ, θ. ( + ) d = = = ( + tan θ) cos θdθ = [θ + ] sin θ cos θ dθ = ( + cos θ)dθ = 8 + cos θ cos θ dθ + d = ( + ) [ ] + d = log( + ) = log, [ d = ] =, ( + ) + d = ( + log ) = 8 + log + 7 ( ) Comment + d = tan θ d, d, = t, + = t + d

84 5 8 I = sin d, J = sin tdt = f() = sin = sin t I = J ( ) sin = sin = sin t = sin t + h t = t = h sin t = h [, ] n = < < < < n =, = n = h g() = t k = g( k ) = k (k =,,,, n) = t < t < t < < t n = {t k } t [, ] n t = = h I n, J n n n I n = sin k = sin k h, J n = k= n k= k= n sin t k t = k= sin k (h) I n = J n h I n I, J n J Comment I = lim h I n = lim h J n = J g() = I n = J n g() I n J n I n J n I n J n

85 5 85 I = + d = g(θ) = tan θ = f() = + θ = f(g(θ)) = f(g(θ))g (θ) ( ) cos θ k = θ k θ = + = cos θ k tan θ k tan(θ k + θ) θ k θ θ k cos θ θ f(g(θ)) = f(tan θ) = +tan θ = cos θ θ [, ] n = θ < θ < θ < < θ n = = cos θ, g (θ) = cos θ, f(g(θ))g (θ) = cos θ cos θ =., θ = n k = g(θ k ) { k } [, ] n k = k k cos θ θ (k =,,, n). δi k δj k { δik cos θ θ δi k δj k (k =,,, n) n n δi k δj k k= k= δj k = θ = θ n k= + ( k ) k n θ θ = k= lim θ k= n δi k = lim θ k= n δj k + d = dθ ( = )

86 5 86 C : = f() = + a + b l : = g() = p + q + r =, = Q = f() = g() S a, b, p, q, r.(7 ) f() g() =, = ( ), =, f() g() = + a + b (p + q + r) = ( ) ( ) < < ( ) ( ) < S = = f() g() d = [ ( ) ( ) ] + { ( ) ( ) } { } ( ) d = ( )d ( ) d = [( ) ] = 8 ( ) < f() = sin, g() = a cos = f() = g() a = f(), = g() S a.(7 ) = t { { f(t) = g(t) sin t = a cos t f (t) = g (t) cos t = sin t = a cos tan t =. sin t = ± 5, cos t = ± 5 ( ). α α + = sin ( a = sin t + cos t = ± ) ( + ± ) = ± a > 5 α a = 5 ( ) tan t = sin t > t, t < [, ] t α, cos α = 5, sin α = 5. a = 5 f() g() = sin + cos 5 = { 5 sin 5 + cos 5 } 5 = 5 {sin sin α + cos cos α} 5 = 5 {cos( α) } < f() g() = α, α + α+ { } [ ] α+ S = 5 cos sin d = 5 sin + cos α α { } = 5(α + ) sin(α + ) + cos(α + ) ( 5α sin α + cos α) = 5 ( )

87 5 87 C : = l : = + t (t=\ ) () t () =, = S (8 ) 6 t + t t = t = t t S S = + 6 = + (t + t) () l C = + = + t ( + t) = = t + t = t + t = + t ( t ) t = + t ( t + t, t ) t ( ) () = t + t = t t + = t =, t = 6 t t = 6. = t + t d = ( ) t dt, t 6. Comment S = = 6 d = 6 ( t t + t ( t ) ( t t ) [ t dt = ) dt 8 log t ] 6 t = 9 log ( ) a b = (, t) l : = ± b a + t t S = d, = cos θ Z d = log + p + C, Z p d = p log + p + C (..)

88 5 88 C : { 5 = sin θ + cos θ = sin θ + cos θ ( θ ) (8 ) (θ) = sin θ + cos θ, (θ) = sin θ + cos θ { (θ + ) = sin(θ + ) + cos(θ + ) = ( sin θ + cos θ) = (θ) (θ + ) = sin(θ + ) + cos(θ + ) = (sin θ + cos θ) = (θ) ((θ), (θ)) ((θ + ), (θ + )) θ d dθ = cos θ sin θ, d dθ = cos θ sin θ. tan α = α, C θ = α θ α d dθ + d ( dθ + ) ( ) ( ) ( 5 ) ( 5 ) θ = θ = ((θ), (θ)) C θ = S = = = d dθ dθ = (sin θ + cos θ)(sin θ + cos θ)dθ { sin θ + cos θ + sin θ cos θ } { dθ = sin θ + cos θ } dθ { } ( cos θ) + ( + cos θ) dθ = [ θ + ] sin θ = ( ) ( ) ( ) ( ) ( ) ( ) sin θ + cos θ cos θ = = sin θ + cos θ sin θ Q(cos θ, sin θ) A = ( ) A = ( ) a b c d A ad bc C S = ( ) () = ( )

89 5 89 B(, ) Q A (, ) θ Q A(, ) A= B (, ) Comment M = ( a c d b ) M ad bc A(, ), B(, ) M A,B A (a, c), B (b, d). A, B S S = ad bc A B, : ad bc ad bc (Q.E.D.) B (b, d) B A a b c d A (a, c) C : j =sin θ cos θ =sin θ + cos θ + ( θ ) S ( ) C : j =sin θ cos θ =sin θ + cos θ ( θ ) ««««sin θ cos θ = sin θ + = cos θ cos θ sin θ S = ( ) () = ( + ) ( ) C C

90 5 9 C : { 6 = θ(sin θ + cos θ) = θ(sin θ cos θ) ( θ ) = S (5 ) = θ(sin θ cos θ) = θ sin ( θ = θ ( sin θ + cos θ ). cos ) = θ ( sin θ sin + cos θ cos ) = θ ( cos θ ) { = θ cos ( ) { θ = θ sin ( ) θ = r cos φ = r sin φ, r = θ, φ = θ r, δφ δs δs r δφ, dφ = dθ, φ θ. S = r dφ = θ dθ = [ θ ] = ( ) = C (, ) θ θ Comment

91 5 9 7 = sin = cos (55 ) = sin = cos sin = cos =, 5 = sin, = cos V,. = { V = = sin d [ sin ] V cos d } = [ + sin ] = ( cos )d ) ( + ) ( ( + cos )d = + ( V = + ) = + ( )

92 5 9 8 = f() = sin ( ) V (6 ) = g () = g () = g () ( ) (.) [, ] = g (), [, ] = g (), [, ] = g () V = {g ()} d + {g ()} d {g ()} d + {g ()} d = sin d = cos d, = g (), = g (),, = g () V = = = (cos d) + (cos d) cos d + (cos d) (cos d) + cos d (cos d) + (cos d) + (cos d) (cos d) = = f() cos d = (sin ) d = sin sin d sin d = ( cos ) d = ( cos ) ( cos )d = cos + sin + C cos d = sin ( cos + sin ) + C = sin + cos sin + C V = [ sin + cos sin ] + [ sin + cos sin ] = + 6 = 8 ( )

93 5 9 [, ] n δ (δt > ), [, ], [, ],,,, δ, δ, δ, δ [, ],[, ] f(), [, ] f() δ f ( ) δ, δ f ( )( δ ), δ f ( )( δ ), δ f ( ) δ δ δv, δv {( ) ( ) }δ ( ) f ( )( δ ) ( ) f ( )δ = ( ( ) f ( )δ + ( ) f ( )δ ) = ( ( ) cos δ + ( ) cos δ ) δv {( ) ( ) }δ ( ) f ( )δ ( ) f ( )( δ ) = ( ( ) f ( )δ + ( ) f ( )δ ) = ( ( ) cos δ + ( ) cos δ ) δv δv ( ( ) cos δ + ( ) cos δ ) δ δv V, δv ( ( ) cos δ + ( ) cos δ ) δ V = cos d + cos d () V = sin d + ( sin )d = 8 cos d cos d

94 5 9 9 C : r = a( + cos θ) ( θ ) (6 ) r = a( + cos θ) (, ) θ a r(θ) = a( + cos θ) r( θ) = a( + cos( θ)) = a( + cos θ) = r(θ) C a = a V a = (, ), θ { = r cos θ = ( + cos θ) cos θ = r sin θ = ( + cos θ) sin θ d dθ = sin θ( + cos θ) θ (, ) C V V a = d dθ dθ = = ( + cos θ) sin θ sin θ( + cos θ)dθ ( + cos θ) ( cos θ) ( + cos θ)( cos θ) dθ cos θ = t sin θdθ = dt, θ t. V a = = ( + t) ( t )( + t)( dt) = ( 5t + t + ) dt = [ t 5 + t + t ( t 5 5t t + t + t + ) dt ] = ( + ) + = 8 V = a 8 = 8 a ( ) Comment C () ) ),

95 5 95 z ABCD A(,, ), B(,, ), C(,, ), D(,, ) z E V (7 ) z z = t ( t ) z = t ( t ) A D Q B Q C H R Q Q = H Q R Q = z = t ( t ) AB, CD z,q,r,,q =, Q, R Q H A,B,C,D = H H(,, t). H D z ABCD t = H HQ z = t S(t) (i) t HQ H ( S(t) = (RQ RH ) = HQ (= (RQ ) ) = t ) = ( t) (ii) t H HQ S(t) = (R RH ) = H (= (R ) ) =, V = S(t)dt = ( t) dt + dt = 8 + = ( ) z = t R H A D H = Q z = t z = A z D = H Q z = t t B C B C u ABCD = A,B,C,D = A,B,C,D, S(t) = (RQ ), S(t) = (R ) A B C D z E S(t) E V E V. V = V = ( ) 7 + ( ) = ( )

96 5 96 z ABCD A(,, ), B(,, ), C(,, ), D(,, ) z E V (7 ) z z -t B S A Q t D S R z = t A S B z = C z = t Q z z = t ( t ) AC, AD, AB, z,q,s,r z = t E S(t) z = t z R = RQ, RS { S(t) = (RQ) (RS) } A B z z = (, S(,, z) z = S z z = t t = = t + ( ) t +. S,, t A z, Q z t Q AD -t : t Q = t A + ( t) + t t D = t ( t) + t + t = 6 t t t,, { { ( S(t) = (RQ) (RS) } ) ( ) ( ) } { 6 t t + t (6 t) = + = } t Comment V = S(t)dt = { (6 t) } t = 6 ( ) S

97 II

98 98, δ k = k+ k, = ma(δ k ) =ma(δ k )=δ k = k+ k, > < ( = g(u) g (u) ) )?) ( ) ( ) ) ( )

99 99 6 () 6. ). ) β α, β lim α = β α β = o(α) = k ( k ) β α β lim α β α β = (α) ) lim = sin θ. lim θ θ ==\ sin θ cos θ θ lim θ θ = cos θ θ θ = cos θ θ cos θ lim θ θ cos θ = lim θ θ = o(), sin θ = (θ), cos θ = (θ ), cos θ = o(θ) 6. = f() = a = f( + ) f() ), lim = { sin (=\ ) f() = sin θ sin ( =) lim f() ==f(), f() = lim + = lim f(h) f() h + h = lim h + h sin h h = lim h + h sin h lim ) ) t θ. ) β α / o o,,order( ) (order) = o( ). DNA DNA= o( ). = ( ). = ( ). β α lim = k(=\ ) lim α β = (=\ ) α β β α k ) lim a f() =f(a) ) g() = sin, {a n} =,,,, n, g(a n ) = n sin n =. {b n } = n o q 5, 9,,, n+, g(b n ) = n + sin `n + q = n + n. n a n, b n sin.

100 6 () 6. ( ) = f() = a f(a + h) f(a) lim h h = lim a f() f(a) a = f (a) f (a)=\ f (a) = = ( ) f = f() = g() F = f(a) = a = g() F () =f() g() F () F (a) lim a a = lim a f() {f(a)+f (a)( a)} a { f() f(a) = lim a a a } f (a) = F f() = a 5) 6. (Talor ) F () =f() f(a) f (a)( a) ( a) f (a) f (a)=\ ( a) f (a) = ( a) f (a)=\ f (a) =, f (a)=\ () = f() f = f() = g() F = g() a a = F () =f() g() k( a) = F () =f() g() k( a) j f (a)=\ F () =f() f (a)( a) k( a) (k=\ ) f (a) =,f (a)=\ F () =f() f (a)( a) k( a) (k=\ ) a 5)

101 6 () f() =cos f () = sin, f () = cos, f () =sin. f () = =\, f ` ` =,f =. =F () = F () = =cos = + () f(),g() b g(a)=\ g(b).a<<b {f ()} + {g ()} =\ B c. f(a) f(b) g(a) g(b) = f (c) g (c), a<c<b A Γ = g(t), = f(t) t = a, t = b, t = c A,B, AB, f (c) g Γ (c) f(a) f(b) g(a) g(b) Γ A B AB ( ) {f ()} + {g ()} =\ f(),g() = a g ()=\ ( = a g (a) = ) f() f(a) lim a g() g(a) = lim f () a g () 6) f ()=\ >a a <η< η f() f(a) =f (η)=\ c a c a lim a+ g() g(a) f() f(a) = g (c) f (c) (a<c<) g() g(a) f() f(a) = lim c a+ g (c) f (c) = lim a+ g () f () <a lim a g() g(a) f() f(a) = lim g () a f () lim a g() g(a) f() f(a) = lim a g () f () (Q.E.D.) 6) g (a)=\ f() f(a) lim a g() g(a) = lim a f() f(a) a g() g(a) a = f (a) g (a)

102 6 () ) sin lim ( sin ) cos = lim = lim ( ) ( cos ) sin = lim = lim ( ) 6 = 6 f() = a F () =f() f (a)( a), g() =( a) F () F (a) lim a g() g(a) lim a f() f(a) f f() f (a)( a) f () f (a) = lim = lim a ( a) a ( a) f (a) (a)( a) ( a) ( a) n f() f (a)( a) a ( a) = lim f() =f(a)+f (a)( a)+ f (a) = f (a) o f (a) = ( a) + o(( a) ) f (a)=\ F ( ) f() = a, G() =f() f (a)( a)! f (a)( a) g() =( a), G() G(a) lim a g() g(a) = lim a f() f (a)( a)! f (a)( a) ( a) f () f (a) f (a)( a) f () f (a) = lim = lim a ( a) a 6( a) = 6 f (a) f() f(a) f (a)( a) f (a) ( a) f (a) ( a) = o(( a) ) 6 f (a) =, f (a)=\ F ( ) f() = a n, Talor f() =f(a)+( a)f (a)+( a) f (a)! +( a) f (a)! + +( a) n f (n) (a) n! f() =e f() = f () = f () = f () = = f n () =, a = e =+ +! +! + + n n! + o(( a) n ) + o( n ) sin = ( ) n n+! 5! 7! (n +)! + o(n+ ) cos = ( ) n n!! 6! (n)! + o(n ) log( + ) = + + n +( )n n + o(n ) = n + o( n ) 5,, >, 5 =\ n = n+ = n + n+

103 6 () 6.5 ( ) f() = a lim a f() f(a) a = f (a) lim = f f (a) (a) lim f (a) = o( ) = f (a) + o( ) = A + o( ) A A = ɛ 7) lim = lim A + ɛ f() = a f (a) =A = f() = a lim A f (a) 8) = A lim = A = A + o( ) f A ( ) = df d = lim, f A = df d = lim f = A f = A = ɛ = ( ( ;) ) f A f f() = f =( + ) = +( ).( ) f () =. 6.6 f(x) > = f(x) X = a,x = ( >a), = S() f(x) [, + ] m, M, S = S( + ) S() m S M m S M Y = f(x) m f(),m f() ds d = lim S = f() f() S a + X 7) e error. 8) f (a)

104 6 () f() S, ( E) S f() ( E ) Y = f(x) f() f (), () ( ) ( ) f() f() = f (),lim = E S f() ( ) ds d = f() a + X C (, ) = r(θ)cosθ, = r(θ)sinθ C θ = α, ( θ>α) S(θ) r [θ, θ + θ] m, M, S = S(θ + θ) S(θ) m θ S M θ m S θ M θ m r(θ),m r(θ) ds dθ = lim θ S θ = r θ θ S r θ S r θ, r ( E) S r θ r θ r r(θ) r r (θ) θ, r θ ( ) r θ r ( θ) ( ) E θ. θ r r θ θ r(θ) θ r,lim θ r θ r θ = T θ S = {r(θ)} θ + o( θ) ds dθ = {r(θ)}

105 6 () 5 = f(x)( f (X) ) X = a X = ( >a) l() l = l( + ) l(), 9) ) + l ( ) +( ) = + ( ) = f ()+ɛ ( ɛ ) ( ) = +(f ()+ɛ) = +({f ()} +ɛf ()+ɛ ) = +{f ()} + ɛf ()+ɛ +{f ()} = +{f ()} ( + ɛf ()+ɛ ( + {f ()} ) = +{f ()} + (ɛ) l = +{f ()} + o( ). ) + o(ɛ) = f(x) dl d = +{f ()} l ( ) +( ) ( ) = a X +. lim lim =. ) S V f() S V ( θ) ) ( ) f () f() (?) 9). ) lim + ( + ) = lim ( + ) `+ ` + + `+ = lim + + «= o() ) f() = p lim h + f(h) h =. = + + `+

106 6 () = f(x),x = a, X =, = S() S f () () S f () ()

107 6 () ( ) [a, b] n, = a, = a + b a n, = a + (b a) n,, n = b, = k+ k ( ), η k [ k, k ] ) b a f()d = lim k= n f(η k ) F () =f() F k = F ( k+ ) F ( k ) F k = F (η k ) = f(η k ) η k [ k, k+ ] ) n n F k = {F ( k+ ) F ( k )} = {F ( ) F ( )} + {F ( ) F ( )} + + {F ( n ) F ( n )} k= k= = F ( n ) F ( )=F (b) F (a) n n F (b) F (a) = F k = f(η k ) k= k= nx f(η k ) k k= nx f(η k ) k k= = f() = f() = f() V k η η η η η η k k η k η k η k η k f() V k [ k, k+ ] ; α + β α + β lim k= n f(η k ) lim k= n f(η k ) = lim k= lim k= V k ma(v k ). ) lim k= n f(η k ) = lim ma(v k ) lim k= n (f(η k ) f(η k )) n (f(η k ) f(η k )) k= n f(η k ) n =ma(v k )(b a) ), ) k V k ( ).

108 6 () 8 n F (b) F (a) = F k = lim k= k= n f(η k ) = lim k= n f(η k ) = b a f()d F () =f() (F () f() ) ( ). F () =f() F (b) F (a) = n k= f(η k), f() ) f() =, F () = [, ] { k } n = < < < < n = = n. F ( k + ) F ( k )=( k + ) ( k ) =( k ) + k ( ) +( ) ( ) ( ) ( ) ( ) k k = n n + + = k +k + n n n n F k = F ( k + ) F ( k )=F (η k ) = f(η k ) k n < η k < k+ n F k = k +k + n = n (η k) k +k +. η k = n η k [ k, k+ ] n n f(η k ) = (η k ) = k= k= n k= k +k + n n = n ==F () F () ( ) n η k [ k, k+ ], [ k, k+ ] η k n I n. ( ) I n n n I = f(η k ) = (η k ) k= n ( ) I n = k= k <η k < k+ ( k n ) <η k < ( k+ n k= n { k } +k + n (η k ) ). k ( ) ( ) +k + k + n < k +k + n n (η k ) < k +k + k n n k + n < k +k + n (η k ) < n k + n ) f(), F () =f() F ().

109 6 () 9 n k= n n k + n < ( ) I n < k= n k + n n η k I n ( ) 5) F () =f() ( ) F k = f( k ) + ɛ ɛ lim ɛ = n n n F k = f( k ) + ɛ k= ( ) ɛ E n n n F k f( k ) = n n ɛ ɛ E = E(b a) k= k= k= k= k= n k= F k = F (b) F (a), E 6) F (b) F (a) = lim k= k= k= n f( k ) = b a f()d 7) ɛ f() =,F() =,a = < < < < n = b { k }, k= F ( k+ ) F ( k )=( k + ) ( k ) =( k ) + k ( ) +( ) n n n {F ( k+ ) F ( k )} = ( k ) + ɛ k (ɛ k = k +( ) ) k= n k= {F ( k+) F ( k )} = F ( n ) F ( )=F (b) F (a), a k b ɛ k = ( k + )) ( k + ) ( ma( a, b )+(b a)) k= E =(ma( a, b )+(b a)) n {F (b) F (a)} n ( k ) = n ɛ k lim k= k= k= k= n ( k ) = F (b) F (a). n ɛ k E b a k= = E(b a) d = F (b) F (a) 5) F () =f()! 6) k E ( ). 7) [ k, k+ ] f()

110 7 ( ) Γ :( ) + =, B, A(, ) Γ (, A.) Γ Q ABQ t Γ S ( ) Q B t A(, ) t t t + t Q Q = AQ = t t t Q, S S Q Q S = Q t + o( t) ( ) t t B A o( t) t ds dt = Q = t., S = t dt = [ t ] = 6 (.) BQ t Q BQ ( 9 ) t BQ = (cos t, sin t), Q = t(cos(t 9 ), sin(t 9 )) = t(sin t, cos t) = B + BQ + Q = (, ) + (cos t, sin t)+t(sin t, cos t) =(+cost + t sin t, sin t t cos t) (, ) j =+cost + t sin t =sint t cos t t,, S S = = Z d dt = Z Z t sin tdt + (sin t t cos t)t cos tdt = Z t ( + cos t)dt ( ) Z t sin t cos tdt + Z t cos tdt

111 7 ( ) (( ) ) f() = f() Q Q = f() Q Q Q Q Q θ o( ) R H Q Q H θ R Q Q H θ R Q (a, f(a)) () Q = a +,Q Q Q S,Q Q H Q Q Q R, Q = r,q Q Q r + r, QR r r Q r r θ () r + r θ + Q, Q H () (r r) θ S Q H (r + r)+ (r + r) θ r + r r r r r. Q Q lim Q H = Q H=o( ) ( ) θ (r r) S Q H (r + r)+ ds d = dθ r d S = r θ + o( ) f () > θ (r + r) θ dθ d = dθ d(tan θ) d(tan θ) d =cos θ df () d =cos θ f ()=\ θ S = r θ + o( θ) ds dθ = r ( ) ( ) f() Q () Comment Q H ( ) Q R ( ) (Q Q.)

112 7 ( ) Γ : = f() = e +e ( log( + )) Q Q Q =( ) Q Γ A(, ) B(log( + ), ) Q Q(t, f(t)) Q Q f (t) Q = (,f (t)) = (, et e t ) = e +e ( ) Q et = e + t ( ) et + e = t = et + e t Q θ(t), t t θ θ t t Q S Q B θ S = Q θ + o( θ) ds dθ = Q A t =log(+ ) θ = t log( + t + ) ( e t + e t ) dθ S = Q dθ = θ tan θ θ tan θ ( ) dθ tan θ = u du = cos θ dθ =(+tan θ)dθ =(+u )dθ =(+{f (t)} e )dθ = t +e t dθ = ( e t +e t ) du, θ u S = ( e t + e t ) dθ = ( e t + e t ) ( e t +e t ) du = du = ( )

113 7 ( ), θ θ S = Q Q Q Q Q Q=Q Q =, et e t = ««Q + Q = t, et + e t +, e e t =(t +,e t ) = e +e = e = e S, α = log( + ) e α =+ Z α e + e S = d + + jz α+ ff e d + h e e i α = + + h e i α+ = eα e α + + (eα ) = ( + ) α α + = = Comment Q Q Q = Q Q Q S S S = Q dθ = Q dθ = Q = S S S

114 7 ( ) Γ : = f() = e +e ( log( + )) Q Q Q =( ) Q Γ A(, ) B(log( + ), ) Q Q(t, f(t)) (t >) Q f (t) ( Q = k, ) f (k ) (t) Q,k = f (t). ( e Q = (f t e t ) (t), ) =, = e +e ( ) et Q = e + t ( ) et + e = t = et + e t A Q B θ Q θ(t), t t Q, Q, θ θ Q S, Q s. Q Q Q R θ S Q, s ( QR Q ) Q θ ( Q R ) ) S Q s + Q θ θ ) ( ) Q s t Q log( + ) Q Γ θ [, ] n θ k(k =,,,,,n) θ = θ k+ θ k, n n S Q s + k= k= n k= θ Q θ ( ) θ θ Γ L R S = L Qds + Q dθ ds = ( ) et +{f (t)} e dt = + t = et + e t dt ) ( ) S = Q s + Q θ + o( θ)

115 7 ( ) 5 L log(+ ) e t + e t Qds = et + e t dt = = [ ] log(+ ) e t e t +t = 8 8 +log(+ ) = log(+ ) { ( + ) Q Q dθ = e t ++e t dt ( + ) +log(+ ) },, +log(+ ) + S = ( ) Comment ( ) S Q s + Q θ θ ) ( ) θ ds dθ = Q ds dθ + Q S = Z Q ds dθ dθ + Z Q dθ, R L Qds S = Z L Qds + Z Q dθ ( ) Q Q R S = f() = g() S = Q + o( ) s = + o( )., f () > θ S = Q s + o( θ) Q s Q Q R S θ S = Q R θ + o( θ) = Q θ + o( θ) R t t + t S = S + S = Q s + Q θ + o( θ) Q

116 7 ( ) 6 ( ) Q = (f e (t), ) = t e t, = ««Q + Q = t, et + e t e t e t +, = t + et e t, et + e t α = log( + ) S «Z α e + e S = d + Z α «e t +( + e t ) d dt dt Z α e + e = d + Z α e t + e t «e t + e t «+ dt = he e i α + he t t e ti α 8 = (eα e α )+ `eα α e α 8 = + «+ + ( ( + ) log( + ff! ) 8 + =+ n + log( + ) ( o ) 8 = ++log(+ ) Comment (?) ) S, ) F () B = e +e F (b) F (a) F, A Q = f() = a, = b, = t α α + S = f() +o( ) f() first step

117 7 8 () f() = sin = f() V V = f()d ( 989 ) = f(x) f() + X V V S + S = ( + ) = + ( ) [, + ] f() M, m m S V M S m( + ( ) ) V M( + ( ) ) m( + ) V M( + ), M f(),m f() dv d =f() V lim =f() V = f()d = sin d Q.E.D. = t d = dt, t. V = t sin tdt = [ ] t cos t ( cos t)dt = ( )

118 8 () 8 = f() (, f()) f() α [,α] > >, [α, ] > < Z α Z α d d d = f()( α) V, d d d = f()(α ) V = f(x) V V = V V = = Z Z α d d d Z α cos( )+sin( ) d d d = d Z d d d = t ( ) α X z. z. z (!) r l(r) V V = b a r l(r)dr l(r) r r + r S = (r + ) r =r r + ( r) r r ( r ) ( ) a r b r + r r r + r V, S, l(r) V l(r) S r l(r) ( r ) V = b a r l(r)dr Q.E.D. ( ) r S =( ) () S

119 8 () 9 V = d V = d 5 = cos ( ), V V = d = [ ] cos d = sin = V = d ( =cos ) =cos d = sin d, V = ( ) sin d = [ ] sin d = ( cos ) [ ( cos )d = sin. 6 C : { =e cos θ =sin θ ( θ ) V ] = d dθ = sin θecos θ, d dθ =cosθ θ d dθ d ( dθ + + ) ( ) ( ) ( ) e e C θ = θ = e e θ = C C : = (θ), = (θ) ( ) θ, C : = (θ), = (θ) ( θ ) V C = V C =

120 8 () ( ) θ = θ = e e θ = θ = e e θ = V = V V V = d = d dθ dθ V = d = d dθ dθ = d dθ dθ V = d dθ ( ) dθ d dθ dθ = d dθ dθ ( ) ( ). Comment V = e cos θ sin θ cos θdθ = (e cos θ ) cos θdθ [ ] = e cos θ cos θ e cos θ ( sin θ)dθ = ( e ) + [e cos θ] = ( e ) + (e ) = e V = V V = d d Z e V = d = e = Z Z d dθ = dθ sin θ( sin θ) e cos θ dθ = Z Z d dθ dθ sin θe cos θ dθ ( ) ( 6 ) θ cos θ = u, sin θ = du, u. V = Z ( cos θ)sinθe cos θ dθ = Z ( u )e u du ( )

121 8 () C t t α β C ((t),(t)), C V ( ) Z V slice = d = Z β α d dt dt ( 6 ). C ( ), V baum = Z β α d t = α, β. Z β V slice = d h dt dt = i Z β β + ( ) dt α α = (α) (α) (β) (β) + = = Z β α Z β α d dt dt d = V baum α Z β α d dt dt (α) =(β),(α) =(β) C : Z β α d dt dt = Z β α d t = α, β C, V V = Z β α d dt dt = Z β α d ( < ) C, t = γ t = α, β V = Z β α d dt dt = Z β α d t = δ ((t),(t)) d dt ((t),(t)) d dt Z β α d dt dt = Z γ α < R γ d α d dt dt + Z δ γ d dt dt + dt dt + R β Z β δ > R δ d dt = V γ dt, d dt = V dt δ d dt dt = V V = V ( ) ( t R γ d dt α dt )

122 9 ( ) ) 7, C : = l : = A A V ( 98) (t, t t), l H,H = X,H = Y V = Y dx A C Q,Q(t, t) Q = t (t t) = t +t X Q(t, t) H Y (t, t t) t l 5 { X = Q HQ = ( ) Q = t ( t +t) = t Y = Q = ( t +t) X = t dx = tdt, X t. V = Comment Y dx = { } ( t +t) tdt= ( t 5 t +t ) dt = 8 5 X Y ( ) : V = R β Y dx dt X Y ) α dt (, ) ( 5 ) (, ) = «««cos( 5 ) sin( 5 ) = sin( 5 ) cos( 5 = ««) Y X (, ) ( = ( + ) = (t +(t t)) = t = ( + ) = ( t +(t t)) = (t t) 5 (, ) )? ), (, ) θ (, ) ««cos θ sin θ = sin θ cos θ «

123 第9章 回転体の体積 (斜回転体) 9. 傘型分割 このように パラメータ表示された曲線の回転体の体積の応用 と考えると 単純計算の問題になりま す しかし 輪切り分割 でなく 傘形分割 すれば 計算が ちょっとだけ 簡単になります l l A Q Q Q(t, t) S = l = rl l l の周りに S = rl H r l 回転する (t, t t) r t t t + t V 今 t が t + t に変化したときの,Q をそれぞれ, Q とし 四角形 Q Q を l の周りに回転させ た立体を考えると 中央上図のような傘型になる 一般に半径 r, 母線が l の円錐の側面積は S = rl で あるから 上の傘型は, 傘の面積が H Q で, 厚み が t の傘型と考えてよい よって V H Q t = ( t + t) ( t + t) t (誤差は t より高位の微小量) ( ) V は V を集めたものだから 8 ( t + t) t = V = lim ( t + t) dt = (t t + t )dt = t 5 先の計算と比べると ほとんど同じですが dx dt がない分だけ簡単になっています 参考 傘型が集まって立体を作っていく様子

124 9 ( )? V =( ) ( ) QQ V =( ) QQ V =( ) t dv dt =( ) ( t ) dx dt. V =( ) ( ), r h, l V = r h r, h, l r, h, l r, h, l V V = (r + r) (h + h) r h = (r h +rh r + r h +r r h + h( r) +( r) h) r h (rh r + r h) ( r ) = ( rl h l r + rl r ) l h θ, d, h h l θ d h r r =cosθ r = d, l h =sinθ h = d l V (rl d + rl d) =rl d ( d ) r r dx dt l θ( <θ<9 ). = f() l, = a, = b(a <b) l V V V =( ) ( ) =Q H = Q cos θ l H Q Q = f() V =cosθ b a Q d θ a θ b + Q = f() (tan θ), θ θ (cos θ cos θ.)

125 9 ( ) 5 8 C : = f() = l : = V ( ) = θ, tanθ = cos θ = 5. C (, f()), l Q(, ) l H Q l Q, H, V V ( ) ( ) =Q H = Q cos θ = 5 V =cosθ = 5 5 = 5 = 5 5 [ 5 { } ( ) d ( ) 5 d ( ) d = ] 5 θ H Q 5 = 9 C : + =l : = l V ( 985) + = =( ). = θ, cosθ = cos 5 =. Q Q l Q, H, V V ( ) ( ) =Q H = Q cos θ V =cosθ Q d = { ( ) } d = ( +)d = [ 8 + ] = 8 =( ) Q H θ + = Q

126 9 ( ) 6, =sin( ) (, ), (, ) l : = V ( 976) <t< = t = f() =sin = A :sin ( ) V B : + ( ) V = = ( ) Q = = Q sin t = 5 V = (cos 5 ) V = (cos 5 ) Q d = Q d = { ( )} d = ( ) d = 6 ( sin ) d = ( sin +sin )d [ sin d = ( cos ) sin d = ] ( cos )d = [ ] ( cos )d = + sin [ ] sin = = V = ( ) = V = V V =

127 9 ( ) 7 ( ) =sin (, ) ( 5 ) (X, Y ), V = Y dx ( ) ( )( ) X cos( 5 ) sin( 5 ) = Y sin( 5 ) cos( 5 ) ( )( ) ( ) = = + + (t, sin t) ( t ) (, ) 5 (X, Y ) { X = ( + ) = (t +sint) Y = ( + ) = ( t +sint) = X = (t +sint) dx = ( + cos t)dt, V = Y dx = ( t +sint) X t. ( + cos t)dt () dx +cost =+( t+sint) dt I = R ( t +sint) ( + cos t)dt I = = = = Z Z Z Z ( t +sint) +( t +sint) dt V = Z ( t +sint) dt + ( t +sint) ( t +sint) dt h ( t +sint) ( t +sint) i dt + ( t +sint) dt I = Z ( t +sint) dt ( θ) (t, f(t)) (X, Y ), j X = cos θ + sin θ = t cos θ + f(t)sinθ Y = sin θ + cos θ = t sin θ + f(t)cosθ 6 dx dt dx dt sin θ cos θ dy dt =tanθ dy dt + cos θ =(cosθ + f (t)sinθ) sin θ cos θ ( sin θ + f (t)cosθ) = cos θ.

128 8 ( ). C = l l C Q,R QR = = V () (,t)( t ) QR D t S(t) () V Q D t C l R C : = R l : = + t 5 (,t) +t Q Q Q t t C R l R () l = + t. C = + t t = =± +t QR = Q = ( + +t) ( +t) = ( + t) S(t) =( ( + t)) =(+t) () t t, D t V, D t, t V = S(t) t + o( t) ( )

129 ( ) 9 t [, ] t V V V = V = lim S(t) t = t S(t)dt = [ V = S(t)dt! ] ( + t) dt =6 S(t) S(t) V = S(t)dt V f(t) t ( t ) V = β V = lim f(t) t = f(t) dt t α S(t) t

130 ( ) C ( ) + = C C V C =+cosθ, =sinθ z θ θ A θ θ = cos θ ( θ ) ) θ θ, V, ( ) V =( ) + o( θ) ( ) r, θ r θ () ( ) = θ + o( θ) V θ = θ = ( cos θ) θ θ V = lim cos θ θ = cos θdθ =8 cos θdθ =8 (sin θ) ( sin θ)dθ sin θ = t cos θdθ = dt, θ t. V =8 [ ( t )dt =8 t t ] = 6 ) A r =R cos(θ α) (R,α A )

131 ( ) Comment ( ) [t, t + t] m, M, S θ m S V M S m S θ V θ M S θ V θ m,m lim θ S = lim. θ V S lim θ = V = S + o( θ) Q.E.D z. z. z z. z. z

132 ( ) C = e C C l Q Q () (t, ) t C l, S(t) S(t) () A(, ), B(,e) C A B V A Q () f (t) = e t l e t = e t ( t) = e t + t e t + et = Q(t + e t, ). H S(t) = t e d + QH = et + et e t = et + e t () t t,q,q V Q S S S(t) () ( ) S S (t) t = et + e t t t V Q = (e t ) +(e t ) = e t e t + C V S, Q V Q S = e t ( ) e t + et + e t t S Q Q t V = e t ( ) e t + et + e t dt = ) e t +( et + e t dt e t +=u e t +=u, e t dt =udu, t u e +. Comment V = { } ( u (u ) + du =(e +) e + ) 5 5 e + S S(t) S.

133 ( ) ( ) = h C Z t+h S = e d + H Q HQ t = he i t+h + et+h e (t+h) et e t t (t, e t ) = e t+h e t + (e(t+h) e t ) = e t (e h ) + et (e h ) H H Q(t + e t ) Q e h lim = e h +h ( h ). h h V = e t h + et h + o( h) = e t + «et = e t + «et + o( ) h S =( ) Q + Q θ + o( t) ( θ Q Q ) θ ) S Q + QQ. Q = ( e t,e t ), Q =(h e t,e t+h ) Q = e t (h e t ) ( e t )e t+h ) = et (e h ) + h et QQ = (e(t+h) + h e t )e t+h = et+h (e h ) + h et+h S Q + QQ = et (e h ) + h et + et+h (e h ) + h et+h e h h( h ) e h h, e t+h = e t e h e t ( + h), e t+h = e t e h e t ( + h) h S et h + h et + et ( + h) h + h et ( + h) = e t + «et h + e t + «et h e t + «et h h S(t)? S(t) f() =e e e

134 ( ). ( ) ( ) ( ) C : z = z K, 5 H K H ( 98 ) z z z z (X,,z) H (,, z) H H ` + = h H H : z = ) = k, = k (z = k.) H : z = + h K K (,, z), H, z z (X,,z), H X = H = H = +. C z = X. z = ( + ). K K : z = ( + ) : z = + h, + h = ( + ) ( ) + = h h> K (,, z), ) K h () 5 cos 5 = K S(h), S proj (h) S(h) = S proj (h) S(h) = S proj (h) = ( h) ) z = CG z =. ) Q(,, z) H(,, )

135 ( ) 5 z = + h, z = +(h + h) h, K V V V = V = S(h) S(h)dh = h + o( h) ( h)dh = h h z h z = + z = + h Comment z =, z z = f() z, ( ) z = f + z z = +, =( z ) : + = A(, ), z :z = ( z ) z z = + z z : +(z ) = z z +(z ) =, = ( + )+(z ) = + +(z ) = (,, ) z (), f(,, z) =,g(,, z) =, f(,, z) = g(,, z) = z z, h = ( ) + = ) + = K, : z = + (

136 ( ) 6 + +(z ) = z = + z + +( ) = + = z = + ( ). + (, ) ( ) z = + z = H z = + = + ( ) = +, = ) f(,, z) = g(,, z) = z F (, ) =, 5) β β H (,, z) H(,, ) α β θ α S β S H S =cosθ S α α β l, QRS α Q l, S//l { Q =cosθ Q, S = S Q R R S S β S = Q S =cosθ Q S =cosθ S θ Q ). 5) G(, z) = z, G(z, ) = z

137 ( ) 7 l cos θ l cos θ, 6) K K : z = ( + ) z = k = k z = k H: z = = k z = k z = k z = k + z = k +. z = k ( ) = k k + > k <k< α, β(α <β), 6 S(k) = β α ( α)( β)d = 6 (β α) = 6 = k ( k + k + ) V = S(k)dk = ( k + k + ) k + k + dk ( ) k + k + = (k ) + k = t V = ( t ) t dt () z = k z = k ), 6) (..) ( )

138 8 (,, ). θ 5 z 5 A(,, ), B(,, ), C(,, ), D(,, ), (,, ) ABCD + V ( 998 ) + =, z + = z + z k T z = k S Q θ R k k z z = k = k, = k, z = k z z = z = k = k, = k ( z = k.) z = k θ( θ ) S(θ) S(θ) = QR ( QRS ) ( QST ) QS = QR =cosθ, RS =sinθ. ( QRS ) =QR RS = cos θ sin θ θ ( QST ) = QR RQS = ( θ ) { S(θ) = QR sin θ cos θ ( θ ) } = cos θ sinθ cos θ +θ QR = k =cosθ, k k = cosθ

139 (,, ) 9 θ θ k k, k dk dθ θ =sinθ θ. z = k z = k + k V V S(θ) k = S(θ) sinθ θ θ [, ], θ V V V = S(θ) sin θdθ = ( cos θ sinθ cos θ +θ ) sin θdθ cos θ sin θdθ = sin θ cos θdθ = θ sin θdθ = sin θdθ = [ θ( cos θ) [ ] cos θ cos θ( cos θ) dθ = [ sin sin θ(sin θ) θ dθ = ] = [ cos θ ] ( cos θ)dθ = ] = 8 + = ( ) ( ) V = ( ) ( ) =+ Comment V S(θ) k = S(θ) sinθ θ = V = S(θ) sin θdθ k = cosθ dk =sinθdθ, ( k θ = k ). V = k S(θ)dk = S(θ) sin θdθ k k θ. θ sin θ, cos θ 99 5 = k, = k

140 (,, ) 6 z +,z =, (,, ), ( ) +,z =, ( ) +,z =. (, ) D D ( ) ) z θ Q z = k k θ B R z z=k z = k ( <k<) C,C, z C k C (,,k), C B, B C D, Q,R, Q R = θ( <θ< ) Q = θ S =( Q )+( BQ ) ( BQ ) = ( k) θ + ( θ) sin( θ) B Q H H= Bcosθ =cosθ. Q= H=cosθ. C k, B Q Q=cosθ = k. k = cosθ, S =θ cos θ +( θ) sin θ, V, V = Sdk, dk =sinθdθ, k θ V = = Sdk = ( θ cos θ +( θ) sin θ ) sinθdθ ( 8θ sin θ cos θ + sin θ θ sin θ sin θ cos θ ) dθ ) z = cosθ θ. ().

141 (,, ) θ sin θ cos θ dθ = = [ θ = = ( θ cos θ ( cos θ )] ) dθ (sin θ) ( sin θ)dθ [ sin θ sin θ ] ( ) cos θ dθ = 9 θ sin θdθ = θ( cos θ) dθ = [ sin sin ] θ θ cos θdθ = = [ ] θ( cos θ) ( cos θ)dθ = V =8 9 + = 9 ( )

142 (,, ). 7 z + + z, ( ) + (,, z) V () ( ) + : ( ) + z z = k ( k ) ) θ r S r r θ Q R r z = k z = k ( k ) z = k + k ( k ), ( ) + ( ) r = k z = k (,Q ) θ ( ) <θ< SQ = θ QS = θ. z = k S(θ) S(θ) =( RS )+( QS ) ( QS ) = r θ + ( θ) r sin θ QS Q S H H = cos θ. r = S = H = cos θ { S(θ) = ( cos θ) θ + ( θ) r = k } cosθ sin θ =θ cos θ sinθ cos θ + θ cosθ = k k = cos θ k =sin θ k =sinθ dk = cos θdθ, k θ V = S(θ)dk = (θ cos θ sinθ cos θ + θ) cosθdθ ( ) ) k k =sinθ

143 (,, ) cos θdθ = (sin θ) ( sin θ)dθ =sinθ sin θ Comment θ cos θdθ = = [ θ ( θ sin θ sin θ ( sin θ sin θ = [ ] + cos θ = = 7 9 sin θ cos θdθ = θ cos θdθ = cos θdθ = [ sin θ ) ] ) dθ [ cos θ cos θ (cos θ) cos θdθ = θ(sin θ) dθ = ] = [ ] θ sin θ ( ) sin θ sin θ dθ (cos θ) ( cos θ)dθ ] [ cos θ ] = sin θdθ = ( V =6 7 ) 8 ( ) = 6 6 9

144 (,, ). (). z z z V M(r) f(r, θ) m(r) θ v θ θ r R θ R f() z = f(),z =, =, = R z θ( ) : v ( ) z v = R f()d θ v = θ v = θ R f()d = θ R f()d ( ) % θ V, r θ h (> ) θ h = f(r, θ) f(r, θ) (r ) θ [θ, θ + θ] h = f(r, θ) M(r), m(r) z = m() z θ z = M() z θ, ( ) θ r r m(r)dr V θ r r M(r)dr f(r, θ) θ M(r) f(r, θ), m(r) f(r, θ). r r r f(r, θ)dr r M(r)dr r r r m(r)dr = r f(r, θ)dr = r r r {f(r, θ) m(r)} dr r {M(r) f(r, θ)} dr θ r r m(r)dr r r f(r, θ)dr, r r M(r)dr r r f(r, θ)dr, ) V lim θ θ = ). r r f(r, θ)dr V = θ r r f(r, θ)dr + o( θ) ( )

145 (,, ) 5 ( ) % V r r f(r, θ)dr, θ, ( ) z θ z = f(r, θ) f(r, θ) f(r, θ) > ) V V V θ r r f(r, θ)dr ( ) ( θ. θ. ) θ f(r, θ) θ r R H H H R θ R θ R h(r, θ) =h(r) θ ( ) e h(r) =H ( ) V = θ R r h(r)dr = θ R e h(r) =H H R r ( H R ) R R V = θ r h(r)dr = θ r (H HR ) [ r r H dr = θ e h(r) = H R r ( H R ) θ R [ r H ] R r Hdr = θ = R H θ = R θ Hr R ] R = R H 6 H θ R θ = R θ H V = θ R r h(r)dr = θ R r H R rdr = θ [ H R r ] R = R H θ e h(r) = R r V = θ [ = θ R r h(r)dr = θ R (R r ) r R R r dr = θ ] R = R θ θ = V = R (R r ) R r dr

146 (,, ) 6 z = f(r, θ) r a h = f(r, θ) (h>) θ V = θ b a r f(r)dr () θ ( θ ) θ = V = b a r f(r)dr () h = f(r, θ) θ V θ r f(r, θ)dr ( θ ) V θ V r f(r, θ)dr ) 8 : { = θ cos θ C : = θ sin θ D θ C ( ) θ C C z θ Q θ θ r = θ D θ Q Q = θ. D θ r z ( θ, ), r ) θ ( r θ ) ( ) ( θ + z θ =, z z = θ ) θ θ D θ V, θ ( θ V θ r r θ ) dr ) S()

147 (,, ) 7 r θ = θ sin t dr = θ cos tdt, r θ t. θ r ( θ r θ ) dr = = θ 8 = θ 8 θ = 6 θ (sin t +) θ cos t θ (cos t +cos t sin t)dt cos tdt ( + cos t)dt ( sin t cos t ) V 6 θ θ ( θ ) V = 6 θ dθ = [ θ 6 ] = 5 Comment V θ rf(r)dr θ ( ) V S() ( ). z z z

148 (,, ) 8 9 z + + z, ( ) + (,, z) V z z = r cosθ r θ cosθ r rz z θ ( θ ) : :( ) + =,z = = cos θ r z z = r, r cosθ θ θ + θ V cosθ V θ [ = θ r zdr = θ ( r ) cosθ ] cosθ z V, r cosθ r dr = θ ) = θ ( 8 sin θ + 8 ( r ) r dr = 8 ( sin θ) θ V = 8 ( sin θ)dθ = 6 ( sin θ)dθ dθ =, sin θdθ = (cos θ) ( cos θ)dθ = [ cos θ cos θ ] = V = 6 ( ) = 8 9 z = k

149 (,, ) 9, z +,z =, (,, ), ( ) +,z =, ( ) +,z =. (, ) D D ( ) r z A z = r Q θ R cosθ cosθ Q cosθ r z rz = (tan θ) z (θ) A(,, ), ( cos θ, sinθ, ) (θ) A. :( ) + = Q Q=cosθ (θ) D, Q z A QRA. θ θ V V θ cosθ r zdr = θ cosθ ] ( r( r)dr = θ [r r cosθ = cos θ 8 ) cos θ θ θ. D z V V = =8 ( cos θ 8 ) cos θ dθ cos θdθ 6 = ( + cos θ)dθ 6 = [θ + ] sin θ 6 = 6 = 9 cos θdθ (sin θ) ( sin θ)dθ [ sin θ sin θ ]

150 (,, ) 5 Comment V (θ) (θ + θ) S, R R z H θ S QQ = Q θ = cos θ θ θ, z V ( A-HRR )+( HRR -QQ ) = HRR HA+ QQ H = HA S +H S A H R R = cosθ S +( cosθ) S = ( ) cos θ S = ( ) cos θ cos θ θ θ Q Q Comment,, English Japanish,, sin θdθ, cos θdθ, θ sin θdθ, θ cos θdθ I n = sin n θdθ I n = n n I n (n =,,, ) (test )

151 5 ) () ( ) (). z (z ) : z = + + (z ) D. ( 976, ) ) : + =,z= (cos θ, sin θ, ) z Q(cos θ, sin θ, z) Q z z = + +=cosθ + sinθ + Q D, z cos θ + ( sinθ + sin θ + ) 6 θ θ θ Q S S Q ( ) =(cosθ + sinθ +) θ θ S = (cos θ + [ sinθ +)dθ = sin θ cosθ + θ ] B A = + θ Q Q θ ) ) : + z = CG.

152 5 Comment θ = A, θ = B ( A ) =θ +, Q=cosθ + sinθ +=sin ( θ + 6 ) + Y Y =sin`x 6 + Q A z A, AB X Q Y, Q(X, Y ) { X = θ + ( Y =sin ( ) Y =sin X ) + θ A B sin`θ + θ ( X ) X

153 5. z +, z= A(,, ) : z = C () ( cos θ, sinθ) A C Q AQ θ () C () Q(,, z) AQ=t A(t ) cosθ = z + t sinθ Q z Q z =, cos θ=\ θ t = t cos θ t = + cos θ (Q z )= ( t), t cos θ θ. AQ = t A = t = + cos θ

154 5 A z φ φ φ T Q Q E D Q Q θ C B B C θ D θ D θ B () θ θ Q,Q AQ T B(,, ), C(,, ), D(,, ), E(,, ) AE BA = φ. ) ( ) :( ) =AB:B=: φ = θ, T AQ φ = θ φ ( ) φ = +cosφ ( ) cos φ = φ cos φ φ AQ T T = = cos dφ = φ cos dφ = φ cos φ ( +tan φ ) (tan φ) dφ =[ tan φ + tan φ ] dφ cos φ A=, θ A,, ABD Comment Q T = 6 AQ = cos φ AC ( = cos φ cos φ = cos φ = cos φ sin φ = sinφ cos φ = = ±, = 6 ) φ...

155 55. ) A l R R r θ φ Q A φ l S Q Q θ r S A φ R φ = b θ = β R θ B φ = a θ = α B l r AQ S, Q S. φ : θ = r : l S = l φ = l θ r ( ) = l r θ l r = l r S ( ) θ, φ R, BA = φ. R R, BR = θ R AR = r l, φ = r l θ R φ = a, φ = b (a <b) S, R θ = α, θ = β S S = b a β AR dφ = α ( ) l r R r l dθ = l r β α (R ) dθ = l r S ( ) Q AQ Q R R () :( ) =l : r =( ) :( ) Q AQ Q α cos α = r l. ( ) ( ) A ) =( ) cos α (α ), α Q ) X H

156 56 S 5) S S = S cos α α ) α, R (: ) z z R S θ θ θ R S H H R cos θ θ R R sin θ z θ ( ) <θ< θ + θ z θ S, S S S cos θ, H,H H = R sin θ. HH (R sin θ) θ = R cos θ θ S H HH =R sin θ R cos θ θ =R sin θ cos θ θ S S cos θ =R cos θ sin θ θ 6) S = R cos θ sin θdθ =R cos θ( cos θ) dθ =R [ cos θ ] = R 5) S ( 6) θ.

157 57 z +, z= A(,, ) : z = C () C C () C D S z A = G H S () H, G z AG:G = := :. (,, z) AG= G z = + (z ) =( + ) :z = ( ) =( + ) += + = z = C C : = ( ) () D S S = ) ( d = 8 :, S S :. ( S = 8 ) = 6 Comment p z = f() z z = f + z = z, z = p + z = p + z = p + ( )

158 58. z +,z =, (,, ), ( ) +,z =, ( ) +,z =. (, ) () S () S z θ A B ` cos θ Q B θ θ z () z z = ( z ) z z = + (z ) = +, z A(,, ), B(,, ), (cos θ +, sin θ) z Q(,, z) l : cos θ + cos θ + = sin θ + t = sin θ z t Q l, (t ) = (cos θ +) +sin θ =(+cosθ) =cos θ (. t = cos θ ) z t ( Q = (Q z ) = cos θ ) ( BQ )=θ θ θ Q θ Q S = Q θ = ( cos θ ) [ dθ = θ sin θ ] =( )

159 59 D () D, D D D D S S = = :, S S :. S = S = Comment

160 6 z ( ) z (,, z) S z Q θ A B θ sin θ θ R A(,, ), B(,, ), (cos θ+, sin θ) z l, l Q R. l n =(,, ) l : = cos θ + cos θ + + t n = sin θ + t = sin θ z t ( (cos θ +) +sin θ + t = t =( cos θ) = sin θ ). t = ±sin θ n =, QR = n (t ) = sin θ ( BQ )=θ θ θ QR θ QR S = QR θ = sin θ dθ =8 sin θ [ dθ =6 cos θ ] =6 Comment

161 6 z A = {(,, z) + z =,,,z } B = { (,, z) z + z =,,,z} C C (,, ) C ( 99 ) z + z = n Q z θ R z : + z =,= (,, ), (, cos θ, sin θ) ( θ <) l, l B, (,, z). l n =(,, ) l : = t + t n = cos θ + t = cos θ z sin θ sin θ B t t sin θ +sin θ = t = sinθ ± sin θ = sin θ ± (θ cos θ =sin ± ) 6 t =sin ( ) θ + 6, t =sin ( ) θ 6 (,, z) Q, R B l { { Q = + t n, Q = t n, R = + t n R = t n =θ. X, n Y, Q R Y t,t, Y =sin ( ) ( ) X + 6,Y =sin X 6 ( )

162 6 Y Q Q Y =sin`x + 6 n z θ R X Y =sin`x 6 θ R Comment (i). (ii)qr Q R z A B S. ( QR = sin θ + ) ( sin θ ) = cos θ 6 6 S = cos θ = cos θdθ = ( ) (i) (,

163 III

164 6. () I = lim n () J = lim n n n n + k n k= n + k n + k n k= () I = lim n () J = lim n n n k= n e k n ( ( k + n k= ) ( ) ) k e ( k n ) n n ( I = lim sin n k= (k +) n sin k )( sin k ) n n n ( I = lim tan n k= (k +) n tan k ) n + ( tan k n ) 5 C : = + V.

165 65 6 = + A V (, 98 ) 7 z A(,, ) +,z= 8 z C : +, z= n =(,, ) l : = = z C l V ( 98 ) 9 z 6 V (988 ),, z, + + z, +z, z. a <a<, ( ) a ( ), ( 997 ),, 9 D. 5, E V. D. E D D 5

166 66 z C : +, C : + z D D S, D C r z + r, + z r, z + r ( 5 ) A(,, ), +,z =, + z = C. C. 5 A(,, ), +,z = + z = 6 z + z,z, z (,, z) V, k,z z = k S(k) () k =cosθ S(k) θ θ () V ( 99 ) 7 z ; z = z z =,z = D, ;,, z D. D D V.

167 67 8 z l; =,= z = m; =,+ z =,l m Q. l, Q C,z,z E.,E. 9 C +( ) = ( ) l l C Q Q ( ) l C V Q C = A A(, ), C A AQ C B(, ) V A Q +, A(, ) K K H H K H C : + = = C : + =

168 68. () I = lim n () J = lim n ) n n n + k n k= n + k n + k n k= () lim n n n + k n = [ +d= ( + ) k= () >, > ( + ) ( + ) =( ) + > ] = ( ) ( ) >, > < + < + =+ k n, = k n + kn < + kn + kn < + k n + k n k =,,,,n, () lim n n + kn n < + kn + kn n < + kn n + k n k= k= k= k= n + k n n < n + k n n + k n < n + k n n + n n k= n k= k= + k n = ( ). lim n lim n n n k= n k= k n n + k n + k n = ( ) k= n k= = lim n + n n k n = ) (988 )

169 69 = + = + n n n n n n n n I, J (n =5) () f() = + [, ] n { k } = < < < < < n = k n < k n + k n < k+ n η k = k n + k n, k <η k < k+ (k =,,,,n ). J = lim n k= n f(η k ) = f()d = +d ( ) Comment ( ) + kn + k n + k n k n lim n k n = lim n k n n =, n k= k n n. =( ) k n =, k n J ɛ k lim n δ k= k= ɛ k = n K = + kn kn n, L = + k n + k n k= I ( MuAD ) k= n I J K L

170 7 () I = lim n () J = lim n n n k= n e k n ( ( k + n k= () ( ) ) ( ) ) k e ( k n ) n I = lim n n n k= e k n = e d = [e ] = e ( ) () J = lim n = lim n n k= n ( ( k + { n k= ) ( ) ) k e ( n k n ) = lim n n n k= } k n e( k n ) + n e k n n k= k + n e ( k n ) () lim n n n k= e k n J = e d = u, d = du, u. J = e u du = e ( )

171 7 () u =. k = k n, k u u k u k = ( ) k. n k = n u u k u k = u k+ u k = ( ) k+ ( n k ). n ( n ( k + J = lim n k= n ) ( ) ) k e ( n k n ) = lim n k= e u k u k J = e u du ( ) = e u kn e u u k =( k+ n ) ( k n ) n n. k n k+ n... ( k n ) ( k+ n ) u k u k+ u u = Comment n J = lim n k= k n e( k n ) = e d J = e d = ( ) J ( n =) f(),g () lim k= n f(g( k )) (g( k+ ) g( k )) = lim e u du u k= ( u = g() ) n f(u k ) u k = β α f(u)du

172 7 n ( I = lim sin n k= (k +) n sin k )( sin k ) n n k = k n (k =,,,,,n ) = < < < < n = { k} [, ] n = k+ k = n. g() =sin, sin (k +) n sin k n =sin k+ sin k = g( k+ ) g( k )=g (c k )( k+ k ) = cos c k k <c k < k+ c k [ k, k+ ] I = lim n n k= ( sin (k +) n sin k )( sin k ) n = lim cos c k (sin k ) n n n k= ( ) I = lim n k= n cos c k (sin k ) = cos sin d = sin (sin ) d sin = u du =cosd, u. I = sin d(sin ) = [ u u du = ] = ( ) Comment ( ) n (sin k= sin (k+) ) n sin k k n n ( n =6). u =sin = g(), f(u) =u I = lim k= = lim k= n f(g( k )) (g( k+ ) g( k )) n f(g( k ))g (c k )( k+ k ) () I = lim u k=,, u = g() n f(u k ) u k sin k n n ṇ. k n (k+) n... sin (k+) n sin k n sin k n u k u k+ = u u sin (k+) n u b f(g())g ()d = β a α f(u)du u =sin

173 7 n ( I = lim tan n k= (k +) n tan k ) n + ( tan k n ) θ k = k n (k =,,,,,n ) =θ <θ <θ < <θ n = {θ k} [, ] n θ θ = θ k+ θ k = n. g(θ) =tanθ, tan (k +) n tan k n =tanθ k+ tan θ k = g(θ k+ ) g(θ k )=g (c k )(θ k+ θ k )= cos c k θ θ k <c k <θ k+ c k [θ k,θ k+ ] I = lim n n k= ( tan (k +) n tan k ) n + ( tan k n ) = lim n n k= cos c k + (tan θ k ) θ I = lim n n k= cos c k +tan θ = θ k cos θ +tan θ dθ = dθ = ( ) Comment = g(θ) =tanθ, k = g(θ k ), k+ k = I = lim n k= = lim n k= = lim n k= = lim n k= n (tan θ k+ tan θ k ) n (g(θ k+ ) g(θ k )) n ( k+ k ) n +( k ) +( k ) = + (tan θ k ) +g(θ k ) + d ( ) I = + d = +tan θ (tan θ) dθ n n. k n (k+) n. θ = + k k+ =tanθ + d( ) lim θ n k= f(g(θ k)) (g(θ k+ ) g(θ k )) = lim θ n k= f(g(θ k))g (c k )(θ k+ θ k ) = a sin θ a d

174 7 5 C : = + V. = k = k = + k = + + k = =± k k<, = + = k,q = g (), Q = g (), < g () =+, g () = V = = {g ()} d {g ()} d = [ d = ( ) ] { (g ()) (g ()) } d = 8 ( ) = g () Q = g () C = t, = t +t ((t),(t)) = g (), Q((t),(t)) = g (), V = Comment = {g ()} d {(t)} d dt = dt t, t Q t = t = t = {g ()} d = {(t)} d dt dt {(t)} d dt dt [ t ( t +)dt = t + t] = 8 ( ) I = (t) dt I = (t) dt + (t) dt. I,I I = <t< t d dt > <t< t d dt < (t) dt = (t) d dt >, dt I = (t) dt = (t) d dt dt < I = g () =,= V ( )I = g () =,= V I = I + I = V V = V C : = (t), = (t),α t β t ((t),(t)) C V = β d t α dt dt ()

175 75 ( ) V = f()d = [ = ( +) ] ( +)d = ( +)d = ( ) ( +)d ( +)d ( ) Comment ( )

176 76 6 = + A V (, 98 ) A ( ) z z Q k Q H (X,, ) A k k Q A = k A (,, z) H(,,), (X,, ) = X +, X = + z = ( + z )+. A, ) = ( + z )+ A = k ( <k<) = k = z k + z = k, A = k Q(k, t k +,t)( k t k ), A(k,, ) (AQ) =( t k +) + t =t +(8k 7)t +( k +) =s +(8k 7)s +( k +) s = t s k (AQ) (AQ) s =( Q ) s =( Q, Q ) ((AQ) ) =ma(( k +), k ) k > ( k +) ( k )=( k ) ( k )=( k )( k +) ((AQ) ) = { <k<, ( k +) k<, k = k z { } ( V = ( k +) dk + ( k )dk = + 7 ) ) = f(), = f( + z ).

177 77 7 z A(,, ) +,z= z z z z A A z = u + Q (,, z) Q C t C t R H u u H p + B R = t ( t 5 ) B Q R = t ( 5 t ) (,, z), H(,, ) u H A z = u +. u = H = +, z = + + = t ( t ) z = + t + (z ) =(t + ) (z ) t (t) = = t C t B(,t,) C t B (,, z) C t B = + z = (z ) t + z = 5 z z + t C t z C t = Q, z = R, BQ BR = u z z = u + Q(,t, t +). BQ =( t +) =( t) z = BR = = t, BQ BR =( t) ( t )=( t)( 5t) 5 t 5 BQ BR, 5 t BQ BR. = t S(t) { BQ =( t) S(t) = BR = ( t ) V = S(t)dt = 5 ( t) dt + 5 ( t )dt = 75 ( )

178 78 8 z C : +, z= n =(,, ) l : = = z C l V ( 98 ) n QR QR ( ) ) z l n =(,, ) Q t θ Q H θ m =(,, ) H R R l = t, l + =,z = Q R, QR H R (R) =(R) () = t n =, n m m =. H = θ cos θ =. H = cos θ = t. (H) =(H) () = t t = t C QR l ( ) S(t) { S(t) = (R) (H) } = ( t ) t = ( ) t H t. l z V = S(t)dt = ( ) [ t dt = t t] = 6 9 ( ) l } QR α, C α C S(t) = {(R) (H) = (RH) ) C C S(t) C l C l. ) )

179 79 α l l : = = z α, θ C α α ( =, z =) cos θ =, = l V Comment V =( ) = = 6 9 C l l C α C α C l ) (,, ) l.

180 8 9 z 6 V (988 ),, z, + + z, +z, z = t = t z = t z z = t z = t (t ) + t t t +,, t,, t =z = t S(t) t + t Y Y = t Y = t + t t Y = t t (i) t z = t S(t) = {( t)+( t)}(t +)= ( t +t +5) t t + ( t, t +) (ii) t t t z = t S(t) = {(t ) + ( t)}( t) = t + t + t ( t, t ) V = S(t)dt = ( t +t +5)dt + ( t +8)dt = 7 t t

181 8 Comment a + b + cz = d n=(a, b, c), 6 z = t t = t =. t =.7 t = t =.5 t =

182 8. a <a<, ( ) a ( ), ( 997 ) t ( ) = ± p s( s) a = ± p ( ) a a a s ( ) t = s( s) a a a α β s ( ) = s( s) a β α = ± p ( ) a s =,t= ( ) t s( s) a s t <a< a > s a > s = ± s( s) a s (.) t = s( s) a s = ± s( s) a s = ±s, ( ) ( ),, V = ± ( ) a = ( ) + a = = ± a α = a, β = + a = ± α, ± β. β β V = f()d = ( ) ad α α = s d = ds, α β s α β. β β ( V = s( s) ads= α α a s ) ds

183 8 α, β s( s) a = = s( s) a ( ) s + = a (, ), a (!) V = ( ) a ( ) = a ( ) Comment 989 = sin ( ) ( ) = f( ) ( ) V = f( )d = t d = dt V = f(t)dt t = f(t) t ( ) t = t sin t ( t ) t = ( ) a (a = 8 )

184 8,, 9 D. 5, E V. D.,,, z,,., (,, ) (,, ). z = t(t ) ( t, t, t) ( (, ). θ θ ), z = t(t ) S t =sinθ, z = t = sinθ ( ) () S = ( ) θ ( ) t(cos θ t) = θ sin θ(cos θ sin θ) V = ( θ +sin θ sin θ cos θ) dz = = = ( θ +sin θ sin θ cos θ) cos θdθ dz = cosθdθ, [ sin θ = cos θdθ ] [ θ sin θ +cosθ θ cos θdθ + ] + sin θ cos θdθ [ sin θ +cos θ ] z θ sin θ cos θdθ ( ) S D 5 θ ( t, t, t) E Comment 999 5) 5)

185 85 z C : +, C : + z D D S, D C z m() z m(θ) Q + z = θ R + z + + = S, + z = S + = C (cos θ, sin θ, ) z m(θ) cos θ cos θ m(θ) : = sin θ + t = sin θ (t ) z t m(θ) C + z = z cos θ + t = t = ± cos θ = ± sin θ Q z = sin m(θ) C Q(cos θ, sin θ, sin θ) R(cos θ, sin θ, sin θ) R z =sin C m() m() m(θ) θ. C D [ ] S = QRdθ = sinθ dθ = sinθdθ = cos θ =8 S =S =6 ( ) Comment ( ) C.G. r, S =6r

186 86 r z + r, + z r, z + r ( 5 ) + z r + r,, z r ( ) = k ( <k<) k + z r r = k ( <k<) ( ) z + z r + r D V r = V r z = k ( k ), { { k + z + k k k k z k z S k = k k = k D = k S θ Q = cos θ, QR = sin θ. R k θ Q k =cosθ k =sinθ = k S = Q ( QR) RS = cos θ sinθ cos θ ( ) θ S = cos θ ( θ) sinθ cos θ 5

187 87 k =sinθ dk =cosθdθ. θ θ V = =8 S(θ)cosθdθ = cos θdθ ( θ)cosθdθ 8 { cos θ ( θ) sinθ cos θ } cos θdθ sin θ cos θdθ cos θdθ = (sin θ) ( sin θ)dθ = ( θ)cosθdθ = [ = sin θ cos θdθ = ( θ)(sin θ) dθ = ] cos θ = ( cos θ) cos θdθ = [ sin θ sin θ ] [ ( θ)sinθ [ cos θ ] = 5 ] = ( ) sin θdθ ( V =8 5 ( ) 8 ) =8 V = r V = ( 8 ) r ( ) Comment, = k k θ : [ ], θ V = V = lim S(θ) k = lim S(θ)cosθ θ k θ

188 88 A(,, ), +,z =, + z = C. C. (, cos θ, sin θ) ( ) m(θ) m; = cos θ + t z sin θ z m Q ( + )=(z ) θ n (t +cos θ)=(sinθ ) t =sin θ sinθ +=(sinθ ) l(θ) l(θ) = t = sinθ n; =,z = m(θ) n θ, n, m(θ) n θ = θ. ( z ). S S = l(θ)dθ = ( sin θ )dθ = 9 ( ) C l(θ) Q m θ n Comment 998 9,

189 89 5 A(,, ), +,z = + z = z A V m z A(,, ) θ Q θ Q n h H θ (, cos θ, sin θ) ( ) m(θ) m,q,, Q = sinθ, Q, :cosθ+sinθz = A(,, ) h, h z cos θ+sinθz = A(,, ) h = sinθ cos θ +sin θ = sinθ (θ), m(θ) A(,, ),K,(θ) (θ + θ) V, θ, V Q Q, h V ( Q Q ) h = (Q θ) h = ( sin θ ) θ θ Comment V = ( sin θ ) dθ = 9 ( ) A A m r, Q = h

190 9 6 z + z,z, z (,, z) V, k,z z = k S(k) () k =cosθ S(k) θ θ () V ( 99 ) + = z z = z z 6) z = z z = z () k k R k θ k k k z = k Q. z = k z = k ( k ) z = k + z z + k ( ) k ( = k ) z = k θ k cos θ = k cos θ = k k θ z = k S(k) S(k) =( QR ) ( QR ) = k QR k sin θ k =cosθ S(k) = cos θ (θ) cos θ sin θ = θ cos θ sin θ cos θ k =cosθ dk = sin θdθ. k θ. V = S(k)dk = (θ cos θ sin θ cos θ)( sin θ)dθ = θ sin θ cos θdθ sin θ cos θdθ 6) z z = f() z z = f( p + )

191 9 I n = cos n θdθ I n+ = n+ n+ I n(n =,,, ). θ sin θ cos θdθ = = sin θ cos θdθ = [ θ ( θ cos θ ( cos θ ) ] ) dθ ( cos θ)cos θdθ = I n+ = n+ n+ I n(n =,,, ) ( cos θ ) dθ = cos θdθ I = I = =, I 5 = 5 I = 5 = 8 5 cos θ = I cos 5 θdθ = I I 5 V = I (I I 5 )=I 5 I = 8 5 = 5 ( ) Comment z z =cosθ ( ) n =,,, I n+ = =(n +) cos n+ θdθ = n =,,, cos n+ (sin θ) dθ = cos n θ sin θ sin θdθ =(n +) [ ] cos n+ θ sin θ (cos n+ θ) sin θdθ cos n θ( cos θ)dθ =(n +)I n (n +)I n+ I n+ =(n +)I n (n +)I n+ I n+ = n + n + I n ( ) sin n θdθ = cos n θdθ, sinn θdθ = sin n θdθ θ =

192 9 7 z ; z = z z =,z = D, ;,, z D. D D V. (,,k) θ k ( +k,,k) z = k z = k( k ) S(k). z = k, θ, k =tanθ, dk = cos θ dθ, k θ,, z = k { S = k + ( +k ) V = { tan θ ( + tan θ), I,I I = =, ( ) } ( ) θ = tan θ +tan θ θ } cos θ dθ + θ( + tan θ) cos θ dθ { tan θ ( + tan θ) [ tan θ tan θ } (tan θ) dθ ( tan θ + tan θ ) ] = I = = [ θ θ( + tan θ) (tan θ) dθ = ( tan θ + tan θ tan θdθ = I = = + ( + ) ) ] ( tan θ tan θdθ = [ log cos θ ] tan θdθ [ tan θ 6 ( θ tan θ + tan θ tan θ + tan θ ) dθ ) dθ ( tan θ + ) cos dθ θ ] tan θ (tan θ) dθ = 6 log

193 9, V =(I + I )= ( log ) ( ) Comment 8 z l; =,= z = m; =,+ z =,l m Q. l, Q C,z,z E.,E.,Q, = + t, Q = + s z Q m Q = Q = s t s X X Q =s t =, t =s X R Q Q X R,R Q X :( X) R = ( X) Q + X = X X(t s)+s (X )(s ) = X X(s ) + s (X )(s ) R, z, { = X(s ) + s z =(X )(s ) s (X,, z) = X C s z + = X z = X X + ( ) (X,, )

194 9 = X,K = X. = X E S(X) S(X) = X X + E V V = + d = ( ) + d = [ ] log( +) =log ( ) Comment Q = X R s C Q C.G. =

195 95 9 C +( ) = ( ) l l C Q Q ( ) l C V Q. Q θ V f(θ) θ v A(, ) cosθ θ sinθ Q( sin θ, cosθ) θ A θ v Q H Q θ Q Q G H θ θ A(, ), AQ = θ l C θ =, θ θ AQ=(sinθ, cos θ), Q Q(sinθ, cosθ) Q = (Q ) = ( cos θ) S(θ) S(θ) =Q =8( cos θ) θ θ,q,q Q Q H, Q C Q G v θ θ HQG = θ. ( QH = QG cos θ ) (θ < ) QG ( QQ ) = θ θ ( QH = QG cos θ ) ( cos θ ) θ = (cos θ +sinθ) θ θ θ V H V S(θ) H = 8( cos θ) (cos θ +sinθ) θ ( θ ) V = 8 (cos θ +sinθ)( cos θ) dθ =8 { cos θ( cos θ) +sinθ( cos θ) } dθ

196 96 () cos θ( cos θ) dθ = = = [ sin θ cos θ( cosθ +cos θ)dθ = { cos θ ( + cos θ)+( sin θ)(sin θ) } dθ (θ + ) ( sin θ + sin θ sin θ (cos θ cos θ +cos θ)dθ ) ] = + sin θ( cos θ) [ ( cos θ) dθ = ( cos θ) ( cos θ) dθ = ] = V =8 ( ) = 6 + C v θ () d C :((t),(t)) v, n, t t,,, d d (, ) = dt, d «t = t v dt ((t),(t)), ((t + t),(t + t)), H, H = φ φ d = H = cos φ = n t v =( n C v ) t ( ) v φ t d n v n = (, ), d v = dθ, d «=(cosθ, sinθ) dθ t v n H d d =( n v ) θ = (cos θ +sinθ) θ ( ) (t, ) l : = + t. C, +( + t ) = (t ) + t t = = t ± t +t + Q = t t + «t +t + = (t + p t +t +) S(t) =Q = (t + p t +t +) V = S(t) t V = Z + S(t) Z + dt = (t + p t +t +) dt ()

197 97 C = A A(, ), C A AQ C B(, ) V (t, t ) t t, Q,Q, A Q A, AQ A Q V V (A QQ ) ( t ) A A Q Q Q (A QQ ) (A A QQ ) (A A QQ ) A S, A V V (A QQ ) = S Q S Q = S Q C C = H A C ( A = H = t ) = t + = Q A(, ),(t, t ) ( A = t, t ), ( A = t + t, (t + t) ) A H S = ((t t + t) ) ( t ) (t + t) (t = + ) ( t + t( t) t + ) t t V ( t + ) ( t t + ) = ( t + ) t Comment V = ( t + ) dt = ( t + t + ) dt = V

198 98 +, A(, ) K K H H K H z h z = + h + h z = + h h h h z = θ = h + h h 5 z = K z :z = ( z ) z (,) K, z = + ( z ) H z = H : z = + h ( H h ). H + h = + + h = + ( + h ) =( + ), + h = h h K z = + h h. h h = h = ± h h h K S(h), S proj (h) 6 S proj (h) = 6 h ( h ) = ( + h) h 6 θ cos θ = 5 S proj (h) = cos θ S(h) = 5 S(h). S(h) = 5 S proj (h) = 5 6 ( + h) h z = + h, z = +(h + h) 5 h, K V V = S(h) 5 h + o( h)

199 99 V V = S(h)dh = ( + h) h dh = h dh + 6 h h dh h dh ( h ) ) h h dh = ( ) h dh = = ( h ) ( h ) dh = [ ] ( h ) = 8 V = 9 CG (.) K.5 z.5 z.5 z z z z

200 C : + = =, = f() C ( 5 ) C (, ) ( 5 ) Q(X, Y ), Q 5 ( ) ( )( ) ( )( ) cos 5 sin 5 X = = X sin 5 cos 5 Y Y C + =, (X Y ) + (X + Y ) (X Y ) (X + Y )= Y = X ( X) X + Q C ( 5 ) C Y = X =,. C X +=u dx = du, 8 V = = [ 8 ( u ) 8 u u (u 6) V = du = 8 ] 8 +logu = Y dx = X u 8. 8 X ( X) X + dx u +u 6u + du = u 8 ( ) 8 log 8 { (u 6) + u } du ( ) = C (, ) C 5 5 Q(X, Y ) Q(X, Y ) Comment + = C Y = ± X X X+ C C = C

201 C : + = = C (, ) θ + = S = X X X+ dx, C = r, θ = r cos θ, = r sin θ r (cos θ +sin θ) r (sin θ cos θ) = r=\ r(cos θ +sin θ) (sin θ cos θ) = r = sinθ cos θ sin θ +cos θ = S = r dθ = 8sin θ cos θ (sin θ +cos θ) = 8tan θ ( + tan θ) cos θ dθ tan θ = t cos θ dθ = dt, θ t. S = 8t ( + t dt ) +t = u t dt = du, S = 8 t u. u du = 8 [ ] u = ( )

202 Comment r = r r = r = = t C + t n t = ( + t ) o t = C = t t +t, «t +t r = = p +t = p +t t +t S = R r dθ, tanθ = t dt = dθ cos θ =(+t )dθ Z S = r dθ = Z r «dt = +t Z p+t «t +t dt +t 7) = f() = g()... ( = k, = k, + = k, = t ) ( k) = f() = k ( 5 ) ( ) ( + ) =X ( ) =Y X =tanθ θ 8) 7) = f() = g() + = F (, ) =. = f() = f() 8) = f(t), = g(t) = f(t) t

203 ( ) (). ( 5 III 9) ( ) ( ) TEIC ( ) CS TEIC ) 9) ) TEIC,,

204 ( ).. ( ) 6 II.