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Total Derivative

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A real function $ f:\mathbb{R}^n\to\mathbb{R}^m$ is called differentiable in $ x$ , if

$\displaystyle f(x+h) = f(x)+f^\prime(x)h + o(\vert h\vert) $

for $ \vert h\vert\to0$.

The total derivative $ f^{\prime}$ is the Jacobi matrix consisting of the partial derivatives of $ f$ :

\begin{displaymath} f^\prime=\operatorname{J}f= \frac{\partial(f_1,\ldots,f_n)... ...tial_1 f_m & \dots & \partial_n f_m \end{array} \right) \,. \end{displaymath}

If $ f$ is a scalar function (i.e. $ m=1$) the total derivative is called the gradient of $ f$,

$\displaystyle f^\prime = \left(\operatorname{grad}\,f\right)^{\operatorname t}\,.$

For the parametrization of a curve ($ n=1$) it is called the tangent vector.

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  automatically generated 5/30/2011