Just like in one variable, we’re interested in local maxima and minima of a function , where is an open region in . Again, we say that has a local minimum at a point if there is some neighborhood of so that for all . A maximum is similarly...

When I talked about David Bowie last week, I of course had to mention one of his recurring characters, Major Tom. Interestingly enough, Bowie was not the only one to use this character. In 1983, German singer Peter Schilling wrote “Major Tom (Coming...

Okay, today we’re going to prove the implicit function theorem. We’re going to think of our function as taking an -dimensional vector and a -dimensional vector and giving back an -dimensional vector . In essence, what we want to do is see how this...

Let’s consider the function . The collection of points so that defines a curve in the plane: the unit circle. Unfortunately, this relation is not a function. Neither is defined as a function of , nor is defined as a function of by this curve. ...

At last we come to the theorem that I promised. Let be continuously differentiable on an open region , and . If the Jacobian determinant at some point , then there is a uniquely determined function and two open sets and so that , and is...