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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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    $u (x) = x^T x$ is a scalar-valued function, not a vector-valued function.2012-09-17
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    Depending on whether $x$ is a column or row vector, $x^tx$ will be a $1\times1$ or $n\times n$ matrix (respectively), so your codomain for the map $u$ is incorrect. Interestingly, I *would* think any book entitled multivariable calculus (or even any comprehensive Calculus book, which should include multivariable calculus) would have it, and I wonder about what kind of book(s) you've looked at.2012-09-17
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    What exactly do you mean with $x^t x$? Multiplying the vector with its $t$-th component? (BTW, you should use LaTeX formatting; in this case, it would just mean to surround your formulas by dollar signs.)2012-09-17
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    @RodCarvalho: Your edit changed the lower-case into an uppercase $T$. Now it reads as a transpose. Which may or may not be what user24205 meant. The fact that this way $u(x)$ doesn't map to $\mathbb R^n$ speaks against this interpretation.2012-09-17
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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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