Chapter 14: Q 1 (page 1153)

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

Short Answer

Expert verified

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

Step by step solution

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

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

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

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Find a potential function for the vector field
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