Chapter 16.8, Problem 13E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Verify that Stokes’ Theorem is true for the given vector field **F** and surface *S.*

**13. F**(*x*, *y _{,} z*) =

*S* is the cone *z*^{2} = *x*^{2} + *y*^{2}, 0 ≤ *z* ≤ 4, oriented downward

To determine

**To verify**: Whether Stokes’ theorem is true for given vector filed
*S*.

Explanation

**Given data:**

The field is

Consider the expression for surface of cone,

**Formula Used:**

Write the expression for curl of

Write the expression for the Stokesâ€™ theorem.

Here,

*S* is surface.

Consider surface *S*,
*S*.

Here,

*A* is area.

The boundary value of surface cone is a circle. The maximum value of *z* is 4.

Substitute 4 for *z* in equation (1),

Consider the parametric equations since the orientation of surface is downward as,

Write the expression for vector function

Substitute
*x*,
*y*, 4 for *z*,

Differentiate the expression with respect to *t*,

Find the value of

Find the value of

Write the expression for

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