## The Total Derivative of a Multilinear Function (Pt. II)

**Point of Post: **This post is a continuation of this one.

**Theorem: ***Let . Then, is differentiable everywhere on and for any (where it’s understood to think of this -tuple as a tuple where each represents clumped together numbers) and *

**Proof: **We merely note that if then

But, by definition of multilinear functions we may rewrite this as

where is if and if . The thing to notice is that this may be rewritten as

where (if (such as in the case of an honest-to-god linear transformation we take the sum to be zero) . Thus,

is equal to

and so clearly if we can show that

for every we can apply the triangle inequality to the definition of the derivative to conclude. To do this we note that since there exists (unless is empty in which case we’re done) such that . We prove this (for notational convenience) only for when we can choose (i.e. there are ‘s in the first two slots) since the method extends effortlessly to the general case. To do this we note that if for any then we’re done since then , so assume not. Then,

where we used the lemma to obtain the upper bound . The conclusion now follows from previous remarks.

*Corollaries*

Since a lot of functions are multilinear it makes sense that we get from the above a lot of corollaries. Some are:

**Corollary 1: ***Let , then is differentiable everywhere ant . In particular, the function is differentiable everywhere for *

**Corollary 2: ***The inner product is differentiable and for every .*

**Corollary 3: ***The function is differentiable for every and has derivative*

**Corollary: ***If one identifies with then is differentiable and*

*where .*

**References:**

1. Spivak, Michael. *Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus.* New York: W.A. Benjamin, 1965. Print.

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