When we find derivatives of functions with respect to vectors, we need to be a lot more diligent. And as we will see in Chapter 2, Linear Algebra, vectors and matrices are noncommutative and behave quite differently from scalars, and so we need to find a different way to differentiate them.
Vector calculus
Derivatives
Earlier, we saw that functions are differentiated by using the limit of the variable in the quotient. But vectors, as we know, are not like scalars in that we cannot divide by vectors, which creates the need for new definitions for vector-valued functions.
We can define a vector function as a function —that is, it takes in a scalar value as input and outputs a vector. So, the derivative of F is defined...