Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

Q 1

Expert-verified

Calculus

Found in: Page 1153

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

Find a potential function for the vector field
$F (x, y) = (3 x^{2} y^{2}, 2 x^{3} y)$

The potential function is $x^{3} y^{2}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:Given information

The given expression is $F (x, y) = (3 x^{2} y^{2}, 2 x^{3} y)$

Step 2:Simplification

Consider the vector field

$F (x, y) = (3 x^{2} y^{2}, 2 x^{3} y)$

To find potential function

let $F (x, y) = \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j$

$= f_{x} (x, y) i + f_{y} (x, y) j$

$= (3 x^{2} y^{2}) i + (2 x^{3} y) j$

now integrating $f_{x} (x, y) = 3 x^{2} y^{2}$ w.r.t x

$\Rightarrow f (x, y) = \int 3 x^{2} y^{2} d x$

$= x^{3} y^{2} + B (y)$

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

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

$\Rightarrow f_{y} (x, y) = 2 x^{3} y + B' (y)$

On comparing with

$f_{y} (x, y) = 2 x^{3} y B' (y) = 0$

On integrating it

$\Rightarrow B (y) = 0 + b$

$\Rightarrow f (x, y) = x^{3} y^{2} + b$ as constant is 0

Therefore $f (x, y) = x^{3} y^{2}$

Most popular questions for Math Textbooks

$Compute the curl of the vector fields in Exercises 23 - 28 . F (x, y, z) = {xe}^{yz} i + {ye}^{xz} j + {ze}^{xy} k$

Problem Zero: Read the section and make your own summary of the material.

Evaluate the integrals in Exercises 43–46 directly or using Green’s Theorem.

\iint_{R} (3 x y - 4 x^{2} y) d A

, where R is the unit disk.

What are the outputs of a vector field in $ℝ^{3}$ ?

Find the areas of the given surfaces in Exercises 21–26.

S is the portion of the surface determined by $x = 9 - y^{2} - z^{2}$ that lies on the positive side of the yz plane (i.e., where $x \geq 0$ )

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free