Chapter 16.8, Problem 15E

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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*.

**15. F**(*x*, *y*, *z*) *= y* **i** + z **j** + *x* **k**,

*S* is the hemisphere *x*^{2} + *y*^{2} + *z*^{2} = 1, *y* ≥ 0, oriented in the direction of the positive *y*-axis

To determine

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

Explanation

**Given data:**

The field is

Consider the expression for the hemisphere surface S,

**Formula Used:**

Write the expression for curl of

Write the expression for the Stokesâ€™ theorem.

Here,

*S* is surface,

The hemisphere surface is oriented in the positive direction of *y*-axis. Hence consider

Substitute 0 for *y* in equation (1).

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

Write the expression for vector function

Substitute
*x*, 0 for *y*,
*z*,

Differentiate the expression with respect to *t*.

Find the value of

Find the value of

Write the expression for

Substitute

Apply limit values and simplify the equation.

Find the value of

Consider the parametric equations of hemisphere surface *S*,

Re-modify the equation (4) for spherical coordinate system.

Substitute
*x*,
*y*, and
*z*,

Write the expression to find

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