Q. 20 - Calculus

Question

In Exercises 17–24, find a potential function for the given vector field.

$\mathbf{G}(x,y)=\mathbf{i}-\mathbf{j}$

Step-by-Step Solution

Answer

A potential function for the given vector field is $g(x,y)=x-y$.

Step 1. Given Information

In given exercises we have to find a potential function for the given vector field. 

$\mathbf{G}(x,y)=\mathbf{i}-\mathbf{j}$

Step 2. Since G ( x , y ) = i − j

$$\begin{gathered} g(x,y)=\int dx+B \\ g(x,y)=x+\alpha +B \end{gathered}$$

where $\alpha$ is an arbitrary constant and B is the integral with respect to y of the terms in $G2(x,y)$ in which the factor x does not appear.

Step 3. In this case, that is all of G 2 ( x , y ) , so

$$\begin{gathered} B=-\int dy \\ B=-y+\beta \end{gathered}$$

where β is an arbitrary constant.

Step 4. Setting the constants equal to zero since they do not affect the gradient of g ( x , y )

We have, 

$g(x,y)=x-y$