Problem #8: Use Divergence theorem to find the outward flux ofF = (6x², et², cos³ (√² + x))of the cube in R³ bounded by the planes x = 0, x = 3, y = 0, y = 3, z = 0, z = 3.

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Answered: Problem #8: Use Divergence theorem to…

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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#8: Use Divergence theorem to find the outward flux of F = (3x², e¹², cos³(y² + x)) of the cube in R³ bounded by the planes x=0, x= 1, y = 0, y = 1, 0,= - 1.
Let 3. f(2) = z² – 2iz be a complex valued function defined on a complex plane. Writing z = x + iy, 6. find real valued functions u(x, y) and v(x, y) such that f(2) = u(x, y) + iv(x, y). u(x, y) = 2x² + 2y? + x and v(x, y) = 4xy – Y u(x, y) = x² – y² + 2y and v(x, y) = 2xy – 2x u(x, y) = 2x ² – 2y + y and v(x, y) = 4xy- x u(x, y) = x2 +y2 +x and v(x, y) = 2xy- y
Verify that u(x, y) = xy - x +y is harmonic in the complex plane and find a harmonic conjugate of u(x, y).
Let F = 2xyzi + (x²z + 2z)j + (x²y + 2y + 2z)k. (a) Show that Curl(F) = 0 (note that Curl(F) may also be written as VXF). (b) Use the method of partial integration to find an f such that F = Vƒ.

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Problem #8: Use Divergence theorem to find the outward flux of F = (6x², e¹², cos³(y² + x)) of the cube in R³ bounded by the planes x = 0, x = 3, y = 0, y = 3, z = 0, z = 3.