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Chapter 14: Q 1 (page 1153)

Find a potential function for the vector field

Fx,y=(3x2y2,2x3y)

Short Answer

Expert verified

The potential function isx3y2

Step by step solution

01

Step 1:Given information

The given expression isFx,y=(3x2y2,2x3y)

02

Step 2:Simplification

Consider the vector field

Fx,y=(3x2y2,2x3y)

To find potential function

let F(x,y)=鈭�f鈭�xi+鈭�f鈭�yj

=fx(x,y)i+fy(x,y)j

=3x2y2i+2x3yj

now integrating fx(x,y)=3x2y2w.r.t x

鈬�fx,y=鈭�3x2y2dx

=x3y2+B(y)

Here B(y) is integral w.r.t y

Differentiating f(x,y)partially w.r t y

鈬�fyx,y=2x3y+B'(y)

On comparing with

fyx,y=2x3yB'(y)=0

On integrating it

鈬�B(y)=0+b

鈬�f(x,y)=x3y2+bas constant is 0

Therefore f(x,y)=x3y2

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Most popular questions from this chapter

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

ComputethedivergenceofthevectorfieldsinExercises17鈥�22.G(x,y)=xcos(xy),ycos(xy)

In what way is Green鈥檚 Theorem a special case of Stokes鈥� Theorem?

ComputethedivergenceofthevectorfieldsinExercises17鈥�22.F(x,y,z)=sin-1(xy)i+ln(y+z)j+12x+3y+5z+1k

What is the difference between the graphs of

G(x,y)=xi+yjandF(x,y)=鈭�xi+(-y)j
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