Measuring the slope of multiple functions

We want to get really complicated, though, and measure the slopes of multiple functions at the same time. All we'll end up with is a matrix of gradients along the rows. In the following formula, we can see the solution that we just solved from the previous example:

In the next formula, we have introduced this new function, called g. We see the gradient for function g, with each position corresponding to the partial derivative with respect to the variables x and y:

When we stack these together into a matrix, what we get is a Jacobian. You don't need to solve this, but you should understand that what we're doing is taking the slope of a multi-dimensional surface. You can treat it as a bit of a black box as long as you understand that. This is exactly how we're computing the gradient and the Jacobian: