Chapter 16.3, Problem 18E

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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*.

**18. F**(*x*, *y*, z) = sin *y* **i** + (*x* cos *y* + cos *z*) **j** − *y* sin *z* **k**,

*C*: **r**(*t*) = sin *t* **i** + *t* **j** + 2*t* **k**, 0 ⩽ *t* ⩽ *π*/2

**(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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