PRESENTER: I want to make sure that you understand the partial derivatives of the u and v velocity with respect to x and y because we will soon be using these terms a lot. Let's start with the partial derivative u with respect to x. Consider a constantly increasing x so that the change in x is positive. As x increases, u becomes initially less negative, hence a positive change; then becomes positive, another positive change; and then becomes more positive, another positive change. Since the change in u and the change in x are both always positive, the partial derivative is positive, greater than 0.

Look at the case where a partial derivative is less than 0, or negative. As x increases, u becomes less positive hence, a negative change. Then becomes negative, another negative change, then becomes more negative, another negative change. Since the change in u is always negative with a positive change in x, the partial derivative is always negative.

Same logic applies to the partial derivative of v with respect to y. Up is positive for y, so you should look at how v changes as y becomes more positive. Look at the case of the change in u with respect to y. It does not matter that u is in the x direction perpendicular to y because we are interested in how u changes as a function of y.

Let's look at what happens as y becomes more positive. On the left, u becomes less negative, a positive change in u, then positive, and more positive. Thus the partial derivative is a positive change in u over a positive change in y and therefore is positive, or greater than 0. The change in u with respect to y is always positive in this case.

Using the same logic on the right, we see that the change in u with respect to y is always negative. And because a change in y is positive, the partial derivative is negative, or less than 0. The same logic applies to the partial derivative of v with respect to x. To the right is positive for x. So you can determine how v changes as x becomes more positive to see whether the partial derivative is positive or negative.