Verifying a Harmonic Function via Laplace Equation

In this problem, we are asked to show that a given multivariable function is harmonic. A function is harmonic if it satisfies Laplace's equation.

First, let's write down the definition provided. Laplace's equation states that the sum of the second partial derivatives with respect to x and y must be zero.

The function we need to test is u of x comma y equals x squared minus y squared minus y.

To check if the equation holds, we first need to compute the partial derivatives with respect to x. Let's start with the first partial derivative.

Differentiating with respect to x, we treat y as a constant. The derivative of x squared is two x, and the other terms drop to zero.

Now, we find the second partial derivative with respect to x by differentiating two x. This gives us two.

Now, let's perform the same steps with respect to y. We rewrite our function first.