Chapter 16.3, Problem 16E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Find a function *f* such that **F** = ∇* f* and (b) use part (a) to evaluate ∫_{C}**F** · *d***r** along the given curve *C*.

**16. F**(*x*, *y*, *z*) = (*y*^{2}*z* + 2*xz*^{2}) **i** + 2*xyz* **j**+ (*xy*^{2} + 2*x*^{2}*z*) **k**,

*C*:* x* =
*y* = *t* + 1, *z* = *t*^{2}, 0 ⩽ *t* ⩽ 1

**(a)**

To determine

**To find**: The potential function *f* such that

Explanation

**Given data:**

Vector field is

Consider

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

Substitute

Compare the equation

Integrate equation (1) with respect to *x*.

Apply partial differentiation with respect to *y* on both sides of equation (4).

Compare the equations (2) and (5)

**(b)**

To determine

The value of
*C*.

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