Further Properties of the Total Derivative (Pt. I)
Point of Post: In this post we finally prove the majority of the basic theorems regarding the total derivative (differentiable functions form a vector space, etc.)
So we now finish proving the ‘obvious’ facts one would hope that the total derivative would share with the normal derivative, such as the sum of differentiable functions is differentiable, the derivative of a sum is the sum of the derivatives, the product of two real valued differentiable functions is differentiable, the derivative of such a product is the product ‘rule’, the derivative of a vector valued function is differentiable if and only if each of its coordinate functions is, etc. A lot of the work is already done because of the corollaries of the total derivative of a multilinear function
Properties of the Total Derivative
We now show the claimed properties actually hold:
Theorem: Let , , with coordinate functions . Then, is differentiable at if and only if is differentiable at for . Moreover,
where the above makes sense since is a matrix or just a row vector.
Proof: Suppose first that is differentiable at . Then, each coordinate function is differentiable since the canonical projection map is differentiable everywhere and from where the differentiability follows from the chain rule.
Conversely, suppose that each is invertible and let be the linear operator whose matrix with respect to the canonical basis is
Note then that
and so taking the limit as on both sides (noting that each summand goes to zero) we may conclude that
and thus is differentiable at and .
Theorem: Let (where ) be differentiable at . Then, is differentiable at .
Proof: Consider the functions given by (where is the coordinate function of ) and given by
Clearly then is differentiable at since by the last theorem each is differentiable at and so each coordinate function of is differentiable. Moreover, it’s clear that is differentiable since it’s linear. Noting then that we may conclude from the chain rule that is differentiable at and . That said, we note from the previous theorem that
and since is linear we know that . Thus,
Since was arbitrary the conclusion follows.
1. Spivak, Michael. Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus. New York: W.A. Benjamin, 1965. Print.