Chapter 16.9, Problem 1E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

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

**1. F**(*x*, *y, z*) *=* 3*x* **i** + *xy* **j** + 2*xz* **k**, *E* is the cube bounded by the planes *x* =0, *x* = 1, *y* = 0, *y =* 1, *z* = 0, and *z* = 1

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

Apply the limits of *x*, *y*, and *z*, and rewrite the expression as follows.

Simplify the expression as follows.

**Calculation of**
**:**

As the cube is bounded by the planes

Calculate the flux for each plane separately and add all the individual results to find the value of

Write the expression to find the value of

**For the plane**
**:**

Consider the surface for the plane

The normal vector for the plane

Substitute 0 for *x* in the vector field

Substitute

**For the plane**
**:**

Consider the surface for the plane

The normal vector for the plane

Substitute 1 for *x* in the vector field

Substitute

**For the plane**
**:**

Consider the surface for the plane

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