Use Stokes' Theorem to find the line integral/ (r + 2y°)ï+ (y + z²)j+ (z + 2r*)K) • dřwhere C is the boundary of the triangle T with vertices (1,0,0), (0, 1,0), (0,0, 1) andoriented counter-clockwise when viewed from above.

Given that C is boundary of triangle with vertices 1,0,0,0,1,0,0,0,1 So, equation of the plane with…

Answered: Use Stokes' Theorem to find the line…

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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Use Stokes' Theorem to find the circulation of F = 3yi + 2zj + 2xk around the triangle obtained by tracing out the path (4, 0, 0) to (4, 0, 6), to (4, 6, 6) back to (4, 0, 0). Circulation = [ F . dr =
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Let F = (3z + 3x²) ¿+ (6y + 4z + 4 sin(y²)) 3+ (3x + 4y + 6e²²) k (a) Find curl F. curl F - (b) What does your answer to part (a) tell you about SF. dr where C' is the circle (x − 15)² + (y – 10)² = 1 in the xy-plane, oriented clockwise? ScF. dr = 0 (c) If C is any closed curve, what can you say about ScF. dr? SoF. dr = 0 (d) Now let C be the half circle - (x − 15)² + (y − 10)² = 1 in the xy-plane with y > 10, traversed from (16, 10) to (14, 10). Find SF. dr by using your result from (c) and considering C plus the line segment connecting the endpoints of C. -0 Lc F. dr =
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2. Compute the line integral of E from B(0,0,0) to A(2,1,3) if E = -4xya, – (4y + 2x2)a, + 8zaz. Use the path: a. Straight line segments: (0,0,0) to (2,0,0) to (2,1,0) to (2,1,3) b. Direct straight line
Suppose F(x, y) = (x² + 3y, 7x − 7y²). Use Green's Theorem to calculate the circulation of F around the perimeter of the triangle C oriented counter-clockwise with vertices (6, 0), (0, 3), and (-6, 0). Jo F. dr =
Assume that the second partial derivatives of the surface X = x(0, 4) are given by Xo4 = 0, X4 - e, + 2pes X00 = 0e, – 2pe,, And has the unit normal vector N = sin0e, + cos o ez. Then the second fundamental form coefficients L, M, and N of the surface are given by L-O sin -2o cos . M-0,N- sin ở + 20 cos o L = 0 sin 0, M = 0, N = sin 0 The above answer The above a nswer L= 0 sin 0, M = 0, N - 20 cos o I. - O sin 0. M- 0. N- sin 0 + 20 cos o The a bove ans wer The above a ns wer
Find the equation of the plane tangent to the surface S at (2, 3, 3) where S is parameterized by x = 2u²v², y = u +2v, z = 2u + v.
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Use Stokes' Theorem to find the line integral [((x² + 2y² ((x + 2y²)i + (y + 2²)j + (2 + 2x²)k) · dř where is the boundary of the triangle T with vertices (1,0,0), (0, 1,0), (0, 0, 1) and oriented counter-clockwise when viewed from above.