Chapter 16.3, Problem 29E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Show that if the vector field **F** = *P* **i +** *Q* **j** + *R* **k** is conservative and *P*, *Q*, *R* have continuous first-order partial derivatives, then

To determine

**To show**: For a conservative vector field

Explanation

**Given data:**

Conservative vector field is

When a vector field is conservative

Consider

Write the relation between the potential function *f* and conservative vector field

Substitute

Compare the equation

Apply partial differentiation with respect to *y* on both sides of equation (1).

Apply partial differentiation with respect to *z* on both sides of equation (1).

Apply partial differentiation with respect to *x* on both sides of equation (2).

Apply Clairautâ€™s theorem

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