Answer: Stokes' Theorem for evaluating surface integrals

Chapter 13, Problem 19E

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QUESTION:

Stokes' Theorem for evaluating surface integrals Evaluate the line integral in Stokes' Theorem to evaluate the surface integral \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\). Assume that n is in the positive z-direction.

\(\mathbf{F}=\langle 2 y,-z, x-y-z\rangle\); S is the cap of the sphere (excluding its base) \(x^{2}+y^{2}+z^{2}=25\), for \(3 \leq x \leq 5\)

Text Transcription:

iint_S (nabla x F) cdot n dS

F = langle 2y, -z, x - y - z rangle

x^2 + y^2 + z^2 = 25

3 leq x leq 5

Questions & Answers

QUESTION:

Stokes' Theorem for evaluating surface integrals Evaluate the line integral in Stokes' Theorem to evaluate the surface integral \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\). Assume that n is in the positive z-direction.

\(\mathbf{F}=\langle 2 y,-z, x-y-z\rangle\); S is the cap of the sphere (excluding its base) \(x^{2}+y^{2}+z^{2}=25\), for \(3 \leq x \leq 5\)

Text Transcription:

iint_S (nabla x F) cdot n dS

F = langle 2y, -z, x - y - z rangle

x^2 + y^2 + z^2 = 25

3 leq x leq 5

ANSWER:

Problem 19E (F)n = -32The val

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