Chapter 16.9, Problem 2E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Verify that the Divergence Theorem is true for the vector field **F** on the region *E.*

**2. F**(*x*, *y*, *z*) *= y*^{2}*z*^{3} **i** + 2*yz* **j** + 4*z*^{2} **k**, *E* is the solid enclosed by the paraboloid *z* = *x*^{2} + y^{2} and the plane *z* = 9

To determine

**To Verify:** The divergence theorem for the vector field
*E*.

Explanation

**Given data:**

The vector field is

The region

**Formula used:**

Write the formula of divergence theorem.

Here,

Write the expression to find divergence of vector field

Write the expression for

Here,

*S*.

**Calculation of**
**:**

Substitute

**Calculation of flux**
**:**

Substitute

Write the equation of paraboloid as follows.

From the equation of paraboloid, the parameters are considered as follows.

Substitute

As the surface
*z*, *r*, and

Apply the limits and rewrite the expression in equation (4) as follows.

Simplify the expression as follows.

**Calculation of**
**:**

As the solid is enclosed by the paraboloid

Consider the surface

Consider the surface

Calculate the flux in each surface separately and add the individual results to find the value of

Write the expression to find the value of

**Calculation of**
**:**

As the surface

Substitute

As

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