INSTRUCTOR: Find a potential function for capital F of x comma y comma z equal to angle bracket 12x squared comma cosine y times cosine of z comma 1 minus sine of y times sine of z close angle bracket. In finding a potential function, because we know that the gradient of our potential function, we're going to call it lowercase f. Because the gradient of lowercase F equals capital F, that includes partials, with respect to x, y, and z.

So we are going to find the antiderivative of each component, with respect to x, y, and Z, with something we'll look for at the end. So the antiderivative, with respect to x of 12x squared, is equal to 4x cubed. The antiderivative, with respect to y of cosine y cosine z is equal to sine y times cosine of z.

The antiderivative of 1 minus sine y sine z with respect to z is equal to z plus sine y cosine z. Now of course we need to add a constant here. But that's not our concern, because we're finding a potential function.

So we're going to say our constant of integration is equal to 0 for all of these. Now, notice that two of these terms have a sine y cosine z. So we can actually omit 1 and still have this still be the same potential function.

Lowercase f of x comma y comma z is equal to angle bracket. Actually, we don't even need that. It's equal to 4x cubed plus sine y cosine z plus z So if you want to check this, you can take the partial, this function with respect to x, y, and z. And see that you get capital F of x comma y comma z that we had at the first.