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I would like a book that explains how to take the derivative of a function that maps vectors to vectors. Specifically, I would like a book that explains multi-variable differentiation, the multi-variable product rule, and the multi-variable integration by parts.

Here is a simple example of the type of problem I would like to be able to solve with the information in this book: Let $u : \mathbb{R}^n \to \mathbb{R}^n$ such that $u(x) = x^T x$. Find the derivative of u with respect to $x$.

You'd think this material would be in books entitled 'Multivariable Calculus', but I haven't found a book with that title that explains it.

Any help would be appreciated.

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    @celtschk: You're totally right. However, the OP can re-edit if he disagrees with my interpretation of his post.2012-09-17

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The map $\mathbb{R}^{n}\rightarrow \mathbb{R}:x\rightarrow xx^{t}$ gives $(x_{1}...x_{n})\rightarrow \sum x_{i}^{2}$. So the partial derivative with respect to $x_{i}$ is $2x_{i}$, and the total derivative is $2x$. However if you are viewing $x$ as $n\times 1$ matrix and $x^{t}$ as $1\times n$ matrix, then you have a block $n\times n$ matrix whose $j$'th column is $xx_{j}$ and $i$-th row is $xx_{i}$. Since we have dealing with spaces of $n$ and $n^{2}$ dimensions respectively, you would expect the derivative matrix to be in $\mathbb{R}^{n^{3}}$ by specifying each $f_{ij}=x_{i}x_{j}$'s derivative at $x_{k}$ ($0$ when $i,j\not=k$, $x_{i},x_{j}$ otherwise).

The most general way to deal with this is through tensors. But this kind of exercise is standard. I am teaching calculus this semester, and this seems not in Stewart, so taking a look at some analysis texts (like Rudin) might be helpful.

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Hint

You may find derivative mapping $D{u}$ immediately, using definition: $u$ is called differentiable, if $\Delta u=u(x+h)-u(x)$ may be represented as

$u(x+h)-u(x)=Du(x)h+\alpha(x,\,h),$ where $Du(x)$ is linear mapping $Du(x): \; \mathbb{R}^n \to \mathbb{R}^n$ and $\alpha$ satisfies $|| \alpha(x,\,h) ||=o(||h||).$ Extraction linear part from $\Delta u$ gives desired derivative.

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I agree with commenters that any multivariable calculus book ought to cover this sort of problem, but in case your books have been a bit too elementary, perhaps focused on the 2 or 3 variable cases, I'd recommend Hubbard & Hubbard Vector Calculus, Linear Algebra, and Differential Forms for an introduction to what you need for this problem and much more.