TEST TOMMOROW PLEASE SHOW YOUR WORK AND AVOID SPHERICAL COORDINATES IF POSSIBLE

Question

TEST TOMMOROW PLEASE SHOW YOUR WORK AND AVOID SPHERICAL COORDINATES IF POSSIBLE

Will give lifesaver to most accurate answer as I need to be able to understand how to do these flux integrals for my exam.

Explanation / Answer

Let f(x, y, z) = x² z + y² z ?? f(x, y, z) dS = ?? f(r(u,v)) * ||r? x r?|| dA ? . . . . . . . . . . . . ? where r(u,v) = is parameterization of S. Parameterization of sphere is given by: r(?,f) = < ? sinf cos?, ? sinf sin?, ? cosf > r? = < -? sinf sin?, ? sinf cos?, 0 > r? = < ? cosf cos?, ? cosf sin?, -? sinf > r? x r? = < -?² sin²f cos?, -?² sin²f sin?, -?² sinf cosf sin²? - ?² sinf cosf cos²? > r? x r? = < -?² sin²f cos?, -?² sin²f sin?, -?² sinf cosf > ||r? x r?||² = ?4 sin4f cos²? + ?4 sin4f sin²? + ?4 sin²f cos²f . . . . . . . . = ?4 sin4f (cos²? + sin²?) + ?4 sin²f cos²f . . . . . . . . = ?4 sin4f + ?4 sin²f cos²f . . . . . . . . = ?4 sin²f (sin²f + cos²f) . . . . . . . . = ?4 sin²f ||r? x r?|| = ?² sinf Since we are working on the upper half of a sphere with radius ? = 2, then we get ||r? x r?|| = 4 sinf and limits are 0 = ? = 2p and 0 = f = p/2 f(x, y, z) = x² z + y² z f(r(?,f)) = f(2 sinf cos?, 2 sinf sin?, 2 cosf) . . . . . . . = 8 sin²(f) cos²(?) cos(f) + 8 sin²(f) sin²(?) cos(f) . . . . . . . = 8 sin²(f) cos(f) (cos²(?) + sin²(?)) . . . . . . . = 8 sin²(f) cos(f) ?? f(x, y, z) dS = ?? f(r(?,f)) * ||r? x r?|| dA ?? (x² z + y² z) dS = ?0²???0??/² (8 sin²(f) cos(f) * 4 sinf) dfd? = ?0²???0??/² (32 sin³(f) cos(f)) dfd? = ?0²?? (8 sin4(f) |0??/²) d? = ?0²?? (8 (sin4(p/2) - sin4(0))) d? = ?0²?? (8 (1 - 0) d? = ?0²?? 8 d? = 8? |0²?? = 16p

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