Mathematics-Online lexicon: Multivariate Chain Rule

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	Mathematics-Online lexicon:
	Multivariate Chain Rule

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Let

be the composition

$\displaystyle h = g \circ f : x \mapsto y =f(x) \mapsto z = g(y)\, ,$

of continuous differentiable functions $f:\mathbb{R}^m \to \mathbb{R}^\ell$ and $g:\mathbb{R}^\ell\to \mathbb{R}^n .$

Then

$\displaystyle h^\prime(x) = g^\prime(y) f^\prime(x)\, ,$

i.e. the jacobian matrix of

ist the product of the jacobian matrices of

and

. The entries of $h^{\prime}$ result from matrix multiplication.

$\displaystyle \frac{\partial h_i}{\partial x_k} = \sum_j \frac{\partial g_i}{\partial y_j} \frac{\partial f_j}{\partial x_k}\, .$

In particular in the special case (i.e. is a parametric curve and a scalar function of variables) the chain rule has the form

$\displaystyle \frac{d\, h}{d \,x} = \left(\operatorname{grad} g\right)^{\operatorname{t}} f^{\prime}(x)$

Examples:

automatically generated 8/ 4/2008