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

**Point of Post: **In this post we prove that a multilinear form is differentiable everywhere and compute its derivative.

*Motivation*

Now that we have the definition of the derivative for mappings it’s time to get our hands a little dirty and compute something. In particular we aim at proving that the wide sweeping class of multilinear function on spaces of the form are everywhere differentiable and compute their derivative. From this we will be able to recover as a corollary a lot of particulary (and important) functions are differentiable, in particular linear trnasofrmations, the functions of the form given by and , the usual inner product on , and the determinant.

*Total Derivative of a Multilinear Form*

We begin with a small lemma that will allow us to bound a multilinear function on the boundary of a polydisc. Namely:

**Lemma: ***Let be a multilinear function from to . Then, is bounded on . *

**Proof: **We’d be done if we knew was continuous (which we don’t, in fact it will be a corollary of the following theorem). So, we have to do this old school. Namely, let then we know that we can write

where is the element of the canonical ordered basis on . But, by definition of a multilinear function we then have that

and thus clearly (applying the triangle inequality) since for all

and since was arbitrary the conclusion follows.

So, with this in mind:

**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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