How to Find Potential Functions? A 10 Minute Introduction

[Greg Bernhardt]

Definition/Summary
Given a vector field $\vec F(x,y,z)$ that has a potential function, how do you find it?
Equations
$$\nabla \phi(x,y,z) = \vec F(x,y,z)$$ $$\nabla \times \vec F(x,y,z) = \vec 0$$
Extended explanation
Suppose we are given a vector field $\vec F(x,y,z)=\langle f(x,y,z),g(x,y,z),h(x,y,z)\rangle$ that has a potential function $\phi$ and we wish to recover the potential function. We know that we must have$\nabla \phi =\vec F$, so $\phi_x = f,\, \phi_y=g,\, \phi_z = h$. This means we can recover $\phi$ by integrating the components of $\vec F$.
To look at a particular example, consider $$\vec F =\langle 2xz^3+e^z,-z\sin(yz),3x^2z^2-y\sin(yz)+xe^z\rangle$$Our unknown potential function $\phi$ must satisfy$$\phi_x=2xz^3+e^z,\, \phi_y=-z\sin(yz),\,\phi_z=3x^2z^2-y\sin(yz)+xe^z$$Students often solve this type of problem by taking the anti-partial derivative of each equation:$$\phi = \int 2xz^3+e^z\,\partial x = x^2z^3+xe^z$$ $$\phi = \int -z\sin(yz)\...

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[fresh_42]

See
https://www.physicsforums.com/threads/intermediate-math-challenge-august-2018.952511/ (problem #4)
https://www.physicsforums.com/threads/basic-math-challenge-august-2018.952503/ (problem #4)
https://www.physicsforums.com/threads/intermediate-math-challenge-july-2018.950690/ (problem #3)
for more examples.