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4 ..1.1 D R. p : D R 3, : u, v) D, rankjp) u,v). Jp) u,v) Jacobi. p u, p v, Jp p u, p v ). rankjp) u,v) p u, p v.. D..3 pu, v), u, v : 1) u : u pu, v), ) v : v pu, v). p u u, p v v..4 : D : {u, v) π < u < π, π/ < v < π/}, x cos u cos v, y sin u cos v, z sin v. x + y + z 1... pu, v) uv,uv.,,,,,. 3

5 ..1.5 pu, v) pu, v) p u u, v) p v u, v) E, F, G. E : p u p u, F : p u p v, G : p v p v. pu, v) pu, v) pu + u, v + v) s u, v : s) pu + u, v + v) pu, v) p u u + p v v E u) + F u v + G v). s, u, v ds, du, dv,..6 ds Edu + F dudv + Gdv..7 [1], p66).... t t uv γ 1 t) u 1 t), v 1 t)) γ t) u t), v t)) pu, v) p γ 1 t) p γ t)...8 p γ 1 t) p γ t) θ, cos θ Eu 1 u + F u 1 v + v 1 u ) + Gv 1 v Eu1 + F u 1 v 1 + Gv 1 Eu + F u v + Gv..9 a:, a Ea 1 + F a 1 a + Ga. ) a pu, v) P pu, v ). a p u u, v ) p v u, v ). a a 1 p u u, v ) + a p v u, v ), a, a a a a 1 p u u, v ) + a p v u, v )) a 1 p u u, v ) + a p v u, v )) a 1p u u, v )) + a 1 a p u u, v ) p v u, v )) + a p v u, v )) a 1E + a 1 a F + a G.... )[.8 ] p γ 1 t) p γ t) t t, a 1 p u u 1 + p v v 1, a p u u + p v v 4

6 , θ.9, cos θ a 1 a a 1 a Eu 1 u + F u 1 v + v 1 u ) + Gv 1 v Eu1 + F u 1 v 1 + Gv 1 Eu + F u v + Gv..1 ) γ 1, γ pu, v) p γ 1, p γ E G, F.... ).8 E G, F, cos θ u 1 u + v 1 v u1 + v 1 v +,uv γ 1, γ.,e G, F. v [1], p6) pu, v) uv D. pu, v) D p u u, v) p v u, v) dudv. D.1 da : p u u, v) p v u, v) dudv. [1], p166, 1.7)..13 da EG F dudv... ) p u, p v θ. p u p v p u p v sin θ) p u p v 1 cos θ) p u p v 1 p u p v )/ p u p v ) ) ) p u p u )p v p v ) p u p v ) EG F., da p u p v dudv EG F dudv..14 uv D pu, v) EG F 1. 5

7 ... ).1 EG F 1,, p u u, v) p v u, v) dudv dudv D, D D. D D..4. uv γt) ut), vt)) a t b), p γt) put), vt)) pu, v)...15 p γt) )Lp γt)), Lp γt)) b a E u + F u v + G v dt.... )Lp γt)) b a d put), vt)) dt. p γt) chainrule, dt d dt put), vt)) p u u + p v v..9, Lp γt)) b a b a p u u + p v v dt E u + F u v + G v dt..16 D γt) a t b), p γt) E G 1, F.... ) [ ].15,E G 1, F. [ ] α,d u, v ) u α u + t cos α, v + t sin α) t c). uv c. pu + t cos α, v + t sin α) t c) c c d dt pu + t cos α, v + t sin α) dt,. c c E cos α + F cos α sin α + G sin α dt 6

8 c c u, v ) E cos α + F cos α sin α + G sin α 1. α, α E 1, α π/ G 1, F : , pu, v), 1. E G, F.. EG F E G 1, F.). 3. r,.,, [1], p99)., , x, y, z) u, v, x cos u cos v, y sin u cos v, z sin v π < u < π, π < v < π ). P uv,. 7

9 , E cos v, F, G 1. ds cos vdu + dv. du, dv E, G,, dv 1,..,,.,. 3.,,.,,.,.., ξ, η) ξ u, η tan v, ξη,., 8

10 ..,, ξ u, η log tan v + π 4 )) dv cos vdη η ηv), ξ, η) ξ u, η ηv),, E cos v, F, G cos v. ds cos vdξ + dη ),. η ηv),. η v dt cos t log tan v + π 4 )) η log tan v + π )). 4 cos ξ cosh η, sin ξ, tanh η). cosh η ξη. 9

11 E G, F,.,,., EG F 1,. lim v ±π/ η ±, 3.3.,, ζ sin v..., ξ u, ζ sin v, dζ cos vdv, ds cos vdu + dv cos vdξ + 1 cos v dζ 1 ζ )dξ +, 1 ζ cos ξ, 1 ζ sin ξ, ζ). ξζ,. 1 1 ζ dζ EG F 1,.., E G, F,. 1

12 3.4 ) N S z 1). P N P, P P π : P P. Px, y, z) X, Y, 1)., X x 1 z, Y y 1 z, x + y + z 1, z 1, x 4X X + Y + 4, y 4Y X + Y + 4, z X + Y 4 X + Y + 4. px, Y ) x, y, z),, I X + Y ) dx + dy ). XY E G, F,. 1: 9 6 : 9 6 ) 11

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14 4.. 1/6.,.. 4.1,., 1,. OACB, OA a, OB b., a a OA BC), b b OB AC). a P a, b P b, P a β arctan α + β π a b, P b α arctan α + β π b a. 3,. 13

15 a + b 1. O 1 C O. P, O, A B P. A B P., A B P, OP P,., a OP BC P.. a b, P P a, P a B C A B.. 14

16 P BC, P P t, b) t a). P OP k OPk R), OP 1,. P t b + t, b ) t a)) b + t b, P AC,P A C,. P a b + t, P B C. ft) t/ b + t, gt) b/ b + t, t ) t b)) b + t f t) bt/b + t ) 3, g t) b /b + t ) 3, B C a a a a b f t)) + g t)) dt b t + b 4 b + t ) 3 dt b b + t dt du 1 + u arctan a b. A C b a a + t dt arctan b a. P a B C A B arctan a b π π arctan a b, P a A C A B arctan b a π π arctan b a,. a, b, 1/. 15

17 4. 3 1, 3. 3 a, b, c, E O. a + b + c 1,. ab. P, ab OP CDEF P. P CDEF P x, y, c) x a, y b). P P P x x + y + c, y x + y + c, c ) x a, y b) x + y + c. ab P S ab. 1/8 π/ P ab S ab /π/). S ab.. S ab [,a] [,b] x P y P dxdy x P y P y + c xy,, xc x + y + c ) 3 x + y + c ) 3 x + y + c ) 3 ), xy, x + c yc, x + y + c ) 3 x + y + c ) 3 x + y + c ) 3 ), 16

18 ., x P y P,. 1 x + y + c ) xc, yc, c ) dx a + x ) 3 x a a + x, a x + 1) a + x + 1 dx arctan ax a + x + 1 c x + y + c ) 3 1 S ab [,a] [,b] a b a b c a a a c [ arctan x P y P dxdy c x + y + c ) 3 1/c dydx x/c) + y + 1) 3 1/c)y dydx x/c) + 1) x/c) + y + 1 ] b c dx 1/c)b/c) x/c) + 1) x/c) + b/c) + 1 dx b/c x + 1) x + b/c) + 1 dx b/c)a/c) a/c) + b/c) )... 1 ), P ab S ab π, arctan ab c a + b + c. arctan ab c a + b + c π P ab π arctan π arctan ab c a + b + c. ab c a + b + c, P bc π arctan bc a a + b + c, P ca π arctan ca b a + b + c. ab 1/. 17

19 4.3. a b c. ) P ab P bc P bc π arctan ,. c. ) P ab 1, P bc P ca. P ab 1, P bc P ca..p ab + P bc + P ca 1.) )x π/)p ab, y π/)p bc, z π/)p ca. tan x ab/c a + b + c ), tan y ab/a a + b + c ) x, y π/). tan x + y) tan x + tan y)/1 tan x tan y) b/ac) a + b + c.,arctan θ + arctan 1/θ) π/, x + y + z π/)p ab + π/)p bc + π/)p ca arctan b/ac) a + b + c ) + arctanca/b a + b + c )) π/., P ab + P bc + P ca ,,. a 1, b 1, c 1 ab ,.,. [3], p59.,.,,,,.. 18

20 [1], :,,. [] :, 9 C A. [3] :, 8 1, pp56 59,. [4] :, 9 1, pp6 66,. [Maple ] The Maple Maptools Package Maple. Maple Maptools example worksheet by: Vince Costanzo [update 9-jan-4]. worksheet The Maple Maptools Package C:\local\maptools, Coastal point databases Fine resolution 9883 points). >restart; >libname : libname, "C:\\local\\maptools"; >withmaptools); >load"c:\\local\\maptools\\fine.m"); >f : x,y) -> x, y); >mapplotf, lonlat[-18*deg..18*deg, -8*deg..8*deg], meridian_spacing15*deg); f. >f : x,y) -> x, lntany/ + Pi/4))); >f : x,y) -> x, siny)); >f : x,y) -> *cosx)*cosy))/1-siny)), *sinx)*cosy))/1-siny))); 19

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