Flux integrals Compute the outward flux of the following vector fields across the given surfaces S. You should decide which integral of the Divergence Theorem to use. F = <<x sin y, -cos y, z sin y>> , S is the boundary of the region bounded by the planes x = 1, y = 0, y = pi/2, z = 0, and z = x.

coexpennan 2021-01-31 Answered
Flux integrals Compute the outward flux of the following vector fields across the given surfaces S. You should decide which integral of the Divergence Theorem to use.
F=xsiny,cosy,zsiny , S is the boundary of the region bounded by the planes x = 1, y = 0, y=π2,z=0, and z = x.
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Expert Answer

odgovoreh
Answered 2021-02-01 Author has 107 answers
Step 1
Given:
F=xsiny,cosy,zzsiny, the surface S is the boundary of the region bounded by the planes x = 1, y=π2,z=0, and z = x.
Step 2
Using Divergence theorem
DFdv=SFnds
Compute The divergence of the field
F=x(xsiny)+y(cosy)+z(zsiny)
=sinyx(x)+y(cosy)+sinyz(z)
=siny+siny+siny
=3siny
Step 3
Therefore, the outward flux is
DFdv=0π2010x3sinydzdxdy
=30π201siny[z]0xdxdy
=30π201xsinydxdy
=30π2[x22]01sinydy
=320π2[sinydy
=320π2[sinydy
=32[cosy]0π2
=32(cos(π2)(cos(0)))
=32(0(
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