16.3 Summary
You know by now: half the success in mathematics is picking the right representations and notations. Although multivariable calculus can seem insanely complex, it’s a cakewalk if we have a good understanding of linear algebra. This is why we started our entire journey with vectors and matrices! Going from f(x1,…,xn) to f(x) is a big deal.
In this chapter, we have learned that differentiation in multiple dimensions is slightly more complicated than in the single-variable case. First, we have the partial derivatives defined by
where ei is the vector whose i-th component is one, while the others are zero. We can think about 
as the derivative of the single-variable function obtained by fixing all but the i-th variable of f. Together, the partial derivatives form the gradient:
However, the partial derivatives are not exactly the perfect analogue of the univariate derivatives. There, we learned that the derivative is the best local linear approximation, and...
