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Chapter 14: Q. 21 (page 1095)

In Exercises 17鈥�24, find a potential function for the given vector field.

F(x,y,z)=yzi+xzj+xyk

Short Answer

Expert verified

A potential function for the given vector field is f(x,y,z)=xyz.

Step by step solution

01

Step 1. Given Information

In given exercises we have to find a potential function for the given vector field.

F(x,y,z)=yzi+xzj+xyk

02

Step 2. Since F(x,y,z)=yzi+xzj+xyk

F(x,y,z)=鈭�(yz)dx+B+CF(x,y,z)=yz鈭�dx+B+CF(x,y,z)=yzx+伪+B+CF(x,y,z)=xyz+伪+B+C

where 伪 is an arbitrary constant and B is the integral with respect to y of the terms in role="math" localid="1650473109511" F2(x,y,z) in which the factor x does not appear.

03

Step 3. In this case, that is all of F2(x,y,z), so

B=鈭�xzdyB=xz鈭�dyB=xzy+尾B=xyz+尾

where尾 is an arbitrary constant.

04

Step 4. Now finding C=∫xydz

C=xy鈭�dzC=xyz+纬

where纬 is an arbitrary constant.

Setting the constants equal to zero since they do not affect the gradient of role="math" localid="1650473089885" f(x,y,z)

We have,

f(x,y,z)=xyz

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Most popular questions from this chapter

What are the inputs of a vector field in 鈩�3?

Do the vectors in the range of F(x,y)=xi+yjpoint towards or away from the origin?

Give an example of a field with positive divergence at (1, 0, 蟺).

Q. Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A smooth surface with a smooth boundary.

(b) A surface that is not smooth, but that has a smooth boundary.

(c) A surface that is smooth, but does not have a smooth boundary

F(x,y,z)=鈭�xzi鈭�yzj+z2k, where S is the cone with equation z=x2+y2between z=2,4, with n pointing outwards.
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