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Chapter 14: Q. 3 (page 1153)

Potential Functions: Find a potential function for each vector field.

G(x,y,z)=xzey2i+2xyzey2j+xey2k

Short Answer

Expert verified

Potential function isgx,y,z=x2zey22

Step by step solution

01

Step 1. Given Information.

The Vector field is G(x,y,z)=xzey2i+2xyzey2j+xey2k

02

Step 2. Calculation.

Potential function is given by

gx,y,z=鈭�xdx+B+C

Here, B is integral with respect to y of the terms in which the x does not appear, and C is integral with respect to z of the terms in which the x and y does not appear.

So,

鈭�xzey2dx=x2zey22+C.

AndB=0,C=0

Potential function isrole="math" localid="1650887740924" gx,y,z=x2zey22

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Most popular questions from this chapter

ComputethedivergenceofthevectorfieldsinExercises17鈥�22.G(x,y,z)=(x2+y鈭�z)i鈭�2ycoszj+ex2+y2+z2k

fx,y,z=1xlnz+2y, where Sis the surface given by ru,v=2ui+u+vj+2u-2vkfor 3鈮�u鈮�7and2鈮�v鈮�10.

Use the same vector field as in Exercise 13, and compute the k-component of the curl of F(x, y).

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.

Use what you know about averages to propose a formula for the average rate of flux of a vector field F(x, y ,z) through a smooth surface S in the direction of n.
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