## The Total Derivative

**Point of Post: **In this post we discuss the notion of the total derivative for mappings which generalizes the notions of derivatives where or as is discussed in usual calculus.

*Motivation*

The basic notion of multivariable analysis begins by abstractly defining what the derivative for a function means. Intuitively, the derivative should be a ‘best approximation’ near a point. The question then is how to define this abstractly. For functions what we got when we took the derivative and evaluated it at a point was a number. But, let’s look a little closer at the definition of the derivative for functions . Namely, a function is differentiable at the point if the limit

exists. In other words, if there exists a number denoted such that the above limit evaluates to it. But, let’s rephrase this a little bit. Namely, we could write the fact that this above limit equals could be restated as

In other words, the derivative is really the linear transformation . Thus, it can easily be shown that a function is differentiable at if and only if there exists a linear transformation such that

This is precisely how we seek to extend the notion of derivatives to higher dimensions.

*The Total Derivative*

Let be a function where is an open subset of and let . We say that is *differentiable at *if there exists a linear transformation such that the limit

(note that the norm on top is the norm in and the norm on bottom is the one on ). We say that is t*he derivative (or total derivative) of at *. If is differentiable at every point of some region we say that is *differentiable on . *Note that it’s important that our set is open, so that there is some open ball and thus for we have that and so .

The justification for the ‘the’ follows from the following theorem:

**Theorem: ***Let be differentiable at and ‘two’ derivatives of at . Then, .*

**Proof: **We know that and so it suffices to show that agree on . To do this let and note that

and so

but since evidently we can easily deduce that

and so . Since was arbitrary the conclusion follows.

Given a function which is differentiable at some point we denote the derivative of at by . We define the *Jacbobian matrix *of at to be the matrix representation of with respect to the usual ordered basis. We denote the Jacobian of at as .

**References:**

1. Spivak, Michael. *Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus.* New York: W.A. Benjamin, 196

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