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I can see in some texts and books that the authors use big letters in order to describe partial derivative of function in $\mathbb{R^n}$ similar to the way we write matrices in linear algebra, for example: $A= f'_x(x_0,y_0)$. I wonder, since it is not being explained, if there's a connection to matrices, and if so, why?


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Jacobian matrix? – HbCwiRoJDp Jan 12 '12 at 21:08
up vote 5 down vote accepted

There is definitely a connection. Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be some sufficiently smooth function, then one can begin to wonder if $f$ has a best linear approximation. This quickly brings up the total derivative which, given a point $p\in\mathbb{R}^n$, can be thought of as the linear transformation $\mathbb{R}^n\to\mathbb{R}^m$ which best approximates $f$ locally at $p$. If we denote this linear transformation by $D_f(p)$, then it's a natural question as to how $D_f(p)$ relates to $D_jf(p)$ where $D_jf(p)$ denotes the $j^\text{th}$ partial derivative. It's a fundamental fact, that $D_f(p)(e_k)=D_kf(p)$ so that if we represent $D_f(p)$ with respect to the standard basis we get a matrix of partial derivatives.

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