Problem on finding the potential function of a vector field
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$\mathbf{F}(x,y) = 2 \mathbf{i} + 3 \mathbf{j}$ is a conservative vector field. Find a potential function for it.
Solution
Recall that
Conservative vector fields and potential functions
Because $\mathbf{F}(x,y)$ is conservative, it has a potential function. That is, there is some function $\phi(x,y)$ such that $ \mathbf{F}(x,y) = \nabla \phi(x,y)$.
Finding the scalar potential of a vector field
The function $\phi(x,y)$ can be found by integrating each component of $$\mathbf{F}(x,y) = \nabla \phi(x,y)$$ and combining the results into a single function $\phi$.
To find the potential function $\phi(x,y)$, we write out $$ \begin{align} \mathbf{F}(x,y) &= \nabla \phi(x,y) \\
2 \mathbf{i} + 3 \mathbf{j} &= \partial_x \phi(x,y) \ \mathbf{i} + \partial_y \phi(x,y) \ \mathbf{j}
\end{align} $$
Thus, $\partial_x \phi(x,y) = 2$ and $\partial_y \phi(x,y) = 3$.
Integrating, we get $\phi(x,y) = 2 x + g(y)$ and $\phi(x,y) = 3 y + h(x)$.
Combining, a possible potential function for $\mathbf{F}(x,y)$ is $\phi(x,y) =2x + 3y$.
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