Vector fields

We define a vector field as a function , and it can only be differentiated if the following applies:

Here,  is the derivative of F. 

We can think of M as a matrix that maps one vector to another, and we can now express F as follows:

Here,  for all , and therefore, the derivative of F is this:

Earlier on in single and multivariable calculus, we learned the importance of the chain rule, so it should be no surprise that we have it in vector calculus as well. And it goes as follows. 

Suppose we have  and  and the coordinates are , , and . Then, the chain rule gives us the following:

We can rewrite this in matrix form, as follows: