Answered: Given F = (M,N,P) is continously…

Given F = (M,N,P) is continously differentiable on R^3. Let E be a regular surface with a regular boundary such that E: z = f(x,y) on domain E(xy). How would I prove Stokes theorem?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Given F = (M,N,P) is continously differentiable on R^3. Let E be a regular surface with a regular boundary such that E: z = f(x,y) on domain E(xy). How would I prove Stokes theorem?

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Step 1: we have given,

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Step 2: To compute line integral of F over c,

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Step 3: To find Curl of F,

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How to solve this question
Let z = g(x, y) = f(3 cos(xy), y + e#») provided that f(3, 5) = 5, f1(3, 5) = 2. f2(3, 5) = 5. i) Find g1 (0, 4). i) Find 92 (0, 4). iii) Find the equation of the tangent plane to the surface z = f(3 cos(xy), y + e=") at the point (0, 4). i) 20, ii) 5, ii) 20x + 5y - z = 15 i) 20, ii) 5, iii) 20x - 5y - z = 5 i) 20, ii) 15, ii) 20x + 15y + z = 55 O i) 60, ii) 5, ii 60x - 5y - z = -45 O i) -40, ii) 15, ii) -40x + 15y + z = -25 O i) 60, ii) -10, i) 60x -10y - z = -25 i) -20, ii) -10, ii)-20x -10y - z = 15 i) -40, ii) 20, ii) -40x + 20y - z = 15
Let z = g(r, y) = f(3 cos(ry), y + e") provided that f(3, 5) = 7, fi(3, 5) = 2, f2(3, 5) = 5. i) Find g, (0, 4). i) Find 92 (0, 4). iii) Find the equation of the tangent plane to the surface z = f(3 cos(ry), y + ev) at the point (0, 4). Türkçe: f(3, 5) = 7, fi(3,5) = 2, f2(3, 5) = 5 olmak üzere z = g(x, y) = f(3 cos(zy), y + e") olsun. i) 91 (0, 4) değerini bulunuz. in 92 (0, 4) değerini bulunuz. im) z = f(3 cos(ry), y + e*") yüzeyine (0, 4) noktasında teğet düzlemin denklemini bulunuz. O i) 20, ii) 5, ii 20x + 5y - z = 13 O i) 20, i) 5, ii) 20x - 5y - z = -1 O i) 20, i) 15, ii) 20x + 15y + z = 53 O i) 60, i) 5, ii) 60x - 5y - z = -47 O ) -40, i) 15, ii) -40x + 15y + z = -27
I Let S be the surface parametrized by r(u, v) = (u cos v, u sin v, 9 - u- -2), for 0