Question

Question#1: Verify stokes theorem for A(x, y, z) = (y^2, x^2, z^2) for the surface z = x^2 - y^2 over the disc x^2 + y^2 leq 4 Question#2 Verify stokes theorem for A(x, y, z) = (y^2, x^2, z^2) for the region y^2 + x^2 + z^2 leq 9 with z geq 0

I need someone to show me how to solve these two problems in very clear steps, thanks

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Answer 1

Consider the vector filed F(x, y, z) = (y^2) I + (x^2) j + (z^2) k. evaluate integral integral_s curl F. dS. Here, s is the part of the parabolic z = x^2 - y^2 that lies insider the cylinder x^2 + y^2 lessthanorequalto 4. Rewrite the place equation as z = x^2 - y^2 let z = g(x, y) = x^2 - y^2 g_x (x, y) = 2x g_y (x, y) = 2y curl F = Vertical-bar i partial differential/partial differential x y^2 j partial differential/partial differential y x^2 k partial differential/partial differential z z^2 Vertical-bar = (partial differential/partial differential y (z^2) - partial differential/partial differential z(x^2))i - (partial differential/partial differential x (z^2) - partial differential/partial differential z(y^2) j + (partial differential/partial differential x (x^2) - partial differential/partial differential y (y^2)) k = (0)I - (0)j + (2x = 2y)k (curl F). (- g_xi- g_y j + k) = ((0)i - (0)j +)