Chapter 14: Q 7 (page 1153)

Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it.
$F (x, y) = <y^{3} e^{x y^{2}}, (1 + 2 x y^{2}) e^{x y^{2}}>$

Short Answer

Expert verified

The vector field is non conservative

Step by step solution

Step 1:Given information

The given expression is $F (x, y) = <y^{3} e^{x y^{2}}, (1 + 2 x y^{2}) e^{x y^{2}}>$

Step 2:Simplification

Consider the vector field,

$F (x, y) = <y^{3} e^{x y^{2}}, (1 + 2 x y^{2}) e^{x y^{2}}>$

$A vector field F (x, y) = P (x, y) i + Q (x, y) j is conservative if and only if \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$

$Comparing F (x, y) = <y^{3} e^{x y^{2}}, (1 + 2 x y^{2}) e^{x y^{2}}> with F (x, y) = P (x, y) i + Q (x, y) j$

then,

$P = y^{3} e^{x y^{2}}$

$Q = (1 + 2 x y^{2}) e^{y^{2}}$

$Now calculating \frac{\partial P}{\partial y}, \frac{\partial Q}{\partial x}$

$\frac{\partial P}{\partial y} = 3 y^{2} e^{x^{2}} + 2 x y^{4} e^{x^{2}}$

and

$\frac{\partial Q}{\partial x} = e^{x y^{2}} + 2 y^{2} e^{x y^{2}} + 2 x y^{2} e^{y^{2}}$

Since,

$\frac{\partial P}{\partial y} \neq \frac{\partial Q}{\partial x}$

The vector field is non conservative

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it.
$F (x, y) = <y^{3} e^{x y^{2}}, (1 + 2 x y^{2}) e^{x y^{2}}>$

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Statistics

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it. F(x,y)=y3exy2,1+2xy2exy2

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Statistics

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it.
$F (x, y) = <y^{3} e^{x y^{2}}, (1 + 2 x y^{2}) e^{x y^{2}}>$