Chapter 16.8, Problem 14E

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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.*

**14**. **F**(*x*, *y*_{,}* z*) = -*2yz* **i** + *y* **j** *+* 3*x* **k**,

*S* is the part of the paraboloid *z* = 5 - *x*^{2} - *y*^{2} that lies above the plane *z* = 1, oriented upward

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 that is part parabolid above the plane is,

**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 parabolid intersects a plane

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

Consider the parametric equations since the orientation of surface is upward.

Write the expression for vector function

Substitute
*x*,
*y*, 1 for *z*,

Differentiate the expression with respect to *t*.

Find the value of

Find the value of

Write the expression for

Substitute

Find the value of

The expression for surface *S*,

Here,

Hence the equation is in the form of

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