# Abstract Nonsense

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

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

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Theorem: Let $K\in\text{Mult}\left(\mathbb{R}^{n_1},\cdots,\mathbb{R}^{n_p};\mathbb{R}^m\right)$. Then, $K$ is differentiable everywhere on $\mathbb{R}^{n_1+\cdots+n_p}$ and for any $(a_1,\cdots,a_p)\in\mathbb{R}^{n_1+\cdots+n_p}$ (where it’s understood to think of this $p$-tuple as a $n_1+\cdots+n_p$ tuple where each $a_k$ represents $n_k$ clumped together numbers) and $(x_1,\cdots,x_p)\in\mathbb{R}^{n_1+\cdots+n_p}$

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$\displaystyle \left(D_K(a_1,\cdots,a_p)\right)(x_1,\cdots,x_n)=\sum_{j=1}^{p}K(a_1,\cdots,x_j,\cdots,a_p)$

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Proof: We merely note that if $(h_1,\cdots,h_p)$ then

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$K\left((a_1,\cdots,a_p)-(h_1,\cdots,h_p)\right)=K\left(a_1-h_1,\cdots,a_p-h_p\right)-K(a_1,\cdots,a_p)$

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

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$\displaystyle \sum_{j_1=1}^{2}\cdots\sum_{j_p=1}^{2}K\left(v_{1,j_1},\cdots,v_{n,j)n}\right)$

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where $v_{k,j_k}$ is $a_k$ if $j_k=1$ and $h_k$ if $k=2$. The thing to notice is that this may be rewritten as

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$\displaystyle K(a_1,\cdots,a_p)+\sum_{j=1}^{p}K(a_1,\cdots,h_j,\cdots,a_p)+\sum_{(s_1,\cdots,s_p)\in S} K((s_1,\cdots,s_p))$

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where $S=\left\{(s_1,\cdots,s_k)\in \{a_1,h_1\}\times\cdots\times\{a_p,h_p\}:s_k=h_k\text{ for at least two }k\right\}$ (if $S=\varnothing$ (such as in the case of an honest-to-god linear transformation we take the sum to be zero) . Thus,

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$\displaystyle K\left((a_1,\cdots,a_p)+(h_1,\cdots,h_p)\right)-K(a_1,\cdots,a_p)-\sum_{j=1}^{p}K(a_1,\cdots,h_j,\cdots,a_p)$

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is equal to

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$\displaystyle \sum_{(s_1,\cdots,s_p)\in S}K\left(s_1,\cdots,s_p\right)$

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and so clearly if we can show that

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$\displaystyle \lim_{h\to\bold{0}}\frac{\left\|K(s_1,\cdots,s_p)\right\|}{\|h\|}=0$

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for every $(s_1,\cdots,s_p)\in S$ we can apply the triangle inequality to the definition of the derivative to conclude. To do this we note that since $(s_1,\cdots,s_p)\in S$ there exists (unless $S$ is empty in which case we’re done) $s_{i_1},s_{i_2}$ such that $s_{i_r}=h_{i_r}\;\;r=1,2$. We prove this (for notational convenience) only for when we can choose $i_r=r\;\; r=1,2$ (i.e. there are $h_i$‘s in the first two slots) since the method extends effortlessly to the general case. To do this we note that if $s_t=\bold{0}$ for any $t\in[p]$ then we’re done since then $K(s_1,\cdots,s_p)=\bold{0}$, so assume not. Then,

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\displaystyle \begin{aligned}\lim_{h\to\bold{0}}\frac{\left\|K(h_1,h_2,\cdots,s_p)\right\|}{\|h\|} &\leqslant \lim_{h\to\bold{0}}\frac{|h_1||h_2|\cdots|s_p|}{\|h\|}\left\|K\left(\frac{h_1}{|h_1|},\frac{h_2}{|h_2|},\cdots,\frac{s_p}{|s_p|}\right)\right\|\\ &\leqslant M\lim_{h\to\bold{0}}\frac{|h_1||h_2|\cdots|s_p|}{|h_1|}\\ &=0\end{aligned}

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

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Corollaries

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

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Corollary 1: Let $T\in\text{Hom}\left(\mathbb{R}^n,\mathbb{R}^m\right)$, then $T$ is differentiable everywhere ant $D_T=T$. In particular, the function $f:\mathbb{R}^n\times\cdots\times\mathbb{R}^n\to\mathbb{R}^n:(x_1,\cdots,x_m)\to x_1+\cdots+x_m$ is differentiable everywhere for $n\in\mathbb{N}$

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Corollary 2: The inner product $\langle\cdot,\cdot\rangle:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ is differentiable and $\left(D_{\langle\cdot,\cdot\rangle}(x,y)\right)(z,w)=\langle x,w\rangle+\langle z,y\rangle$ for every $n\in\mathbb{N}$.

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Corollary 3: The function $m:\mathbb{R}^n\to\mathbb{R}:(x_1,\cdots,x_n)\mapsto x_1\cdots x_n$ is differentiable for every $n\in\mathbb{N}$ and has derivative

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$\displaystyle \left(D_m(a_1,\cdots,a_n)\right)(x_1,\cdots,x_n)=x_1\cdots a_n+\cdots+a_1\cdots x_n$

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Corollary: If one identifies $\text{Mat}_n\left(\mathbb{R}\right)$ with $\left(\mathbb{R}^n\right)^n$ then $\det:\text{Mat}_n\left(\mathbb{R}\right)\to\mathbb{R}$ is differentiable and

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$\left(D_{\det}(M)\right)\left(\begin{pmatrix}x_{1,1} & \cdots & x_{1,n}\\ \vdots & \ddots & \vdots\\ x_{n,1} & \cdots & x_{n,n}\end{pmatrix}\right)=\det\begin{pmatrix}m_{1,1} & \cdots & x_{1,n}\\ \vdots & \ddots & \vdots\\ m_{n,1} & \cdots & x_{n,n}\end{pmatrix}+\cdots+\det\begin{pmatrix}x_{1,1} & \cdots & m_{1,n}\\ \vdots & \ddots & \vdots\\ x_{n,1} & \cdots & m_{n,n}\end{pmatrix}$

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where $M=[m_{i,j}]$.

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References:

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

May 23, 2011 -