Problem #6:Use the divergence theorem to find the outward fluxF.ndSoftheFtan¹(5y +82) i + ²+3 cosxj+field²+y² + z² k₂where S is the surface of the region bounded by the graphsof = √²+² and x² +3²+² = 64.vector

Given vector field is and surface S is the region bounded by the graphs Now So using Divergence…

Answered: Problem #6: Use the divergence theorem…

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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Question

Problem #6:
Use the divergence theorem to find the outward flux
F.ndS
of
the
Ftan¹(5y +82) i + ²+3 cosxj+
field
²+y² + z² k₂
where S is the surface of the region bounded by the graphs
of = √²+² and x² +3²+² = 64.
vector

Expert Solution

Step 1: Application of Divergence theorem

Given vector field is

$F equals tan to the power of negative 1 end exponent open parentheses 5 y plus 8 z close parentheses i plus e to the power of z squared plus 3 cos left parenthesis x right parenthesis end exponent j plus square root of x squared plus y squared plus z squared end root k$ and surface S is the region bounded by the graphs $z equals square root of x squared plus y squared end root space a n d space x squared plus y squared plus z squared equals 64$

Now $nabla. F equals fraction numerator z over denominator square root of left parenthesis x squared plus y squared plus z squared right parenthesis end root end fraction$

So using Divergence theorem ,

$double integral subscript S F. n d S equals triple integral subscript V nabla. F d V equals integral subscript x equals negative 4 square root of 2 end subscript superscript 4 square root of 2 end superscript integral subscript y equals negative square root of open parentheses 32 minus x squared close parentheses end root end subscript superscript square root of open parentheses 32 minus x squared close parentheses end root end superscript integral subscript z equals square root of open parentheses x squared plus y squared close parentheses end root end subscript superscript square root of open parentheses 64 minus x squared minus y squared close parentheses end root end superscript fraction numerator z over denominator square root of x squared plus y squared plus z squared end root end fraction d z d y d x$