Chapter 16.3, Problem 9E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Determine whether or not **F** is a conservative vector field. If it is, find a function *f* such that **F** = ∇* f*.

**9. F**(*x*, *y*) = (*y*^{2} cos *x* + cos *y*) **i** + (2*y* sin *x* – *x* sin *y*) **j**

To determine

**To find:** Whether *f* such that

Explanation

**Given data:**

The vector field is

**Formula used:**

For a vector field

Where, *P*, and *Q*,

**Calculation:**

Compare the vector field

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

Apply partial differentiation with respect to *x* on both sides of equation (3).

Substitute

Hence

Consider

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

Substitute

Compare the equation

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