PlanetMath: chain rule (several variables)

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chain rule (several variables)

(Theorem)

The chain rule is a theorem of analysis that governs derivatives of composed functions. The basic theorem is the chain rule for functions of one variables (see here). This entry is devoted to the more general version involving functions of several variables and partial derivatives. Note: the symbol will be used to denote the partial derivative with respect to the $k^{\text{th}}$ variable.

Let $F(x_1,\ldots,x_n)$ and $G_1(x_1,\ldots,x_m),\ldots,G_n(x_1,\ldots,x_m)$ be differentiable functions of several variables, and let

$\displaystyle H(x_1,\ldots,x_m) = F(G_1(x_1,\ldots,x_m),\ldots,G_n(x_1,\ldots,x_m))$

be the function determined by the composition of

with $G_1,\ldots,G_n$ The partial derivatives of

are given by

$\displaystyle (D_k H)(x_1,\ldots,x_m) = \sum_{i=1}^n (D_i F)(G_1(x_1,\ldots,x_m),\ldots) (D_k G_i)(x_1,\ldots,x_m).$

The chain rule can be more compactly (albeit less precisely) expressed in terms of the Jacobi-Legendre partial derivative symbols (historical note). Just as in the Leibniz system, the basic idea is that of one quantity (i.e. variable) depending on one or more other quantities. Thus we would speak about a variable depends differentiably on $y_1,\ldots,y_n$ , which in turn depend differentiably on variables $x_1,\ldots,x_m$ . We would then write the chain rule as

$\displaystyle \frac{\partial z}{\partial x_j} = \sum_{i=1}^n \frac{\partial z}{\partial y_i} \frac{\partial y_i}{\partial x_j},\qquad j=1,\ldots m.$

The most general, and conceptually clear approach to the multi-variable chain is based on the notion of a differentiable mapping, with the Jacobian matrix of partial derivatives playing the role of generalized derivative. Let, $X\subset\mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ be open domains and let

$\displaystyle \mathbf{F}:Y\rightarrow\mathbb{R}^l,\qquad \mathbf{G}:X\rightarrow Y$

be differentiable mappings. In essence, the symbol $\mathbf{F}$ represents

functions of

variables each:

$\displaystyle \mathbf{F}= (F_1,\ldots,F_l),\qquad F_i=F_i(x_1,\ldots,x_n),$

whereas $\mathbf{G}=(G_1,\ldots,G_n)$ represents

functions of

variables each. The derivative of such mappings is no longer a function, but rather a matrix of partial derivatives, customarily called the Jacobian matrix. Thus

$\displaystyle D\mathbf{F}= \begin{pmatrix} D_1 F_1 & \ldots & D_n F_1 \ \vdot... ...m G_1 \ \vdots & \ddots & \vdots \ D_1 G_n & \ldots & D_m G_n \end{pmatrix}$

The chain rule now takes the same form as it did for functions of one variable:

$\displaystyle D(\mathbf{F}\circ \mathbf{G}) = ((D\mathbf{F}) \circ \mathbf{G})\, (D\mathbf{G}),$

albeit with matrix multiplication taking the place of ordinary multiplication.

This form of the chain rule also generalizes quite nicely to the even more general setting where one is interested in describing the derivative of a composition of mappings between manifolds.

"chain rule (several variables)" is owned by rmilson.

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See Also: chain rule, Jacobian, Jacobian matrix

This object's parent.

Attachments:: proof of chain rule (several variables) (Proof) by paolini

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Cross-references: manifolds, even, multiplication, place, matrix multiplication, matrix, mappings, represents, domains, open, Jacobian matrix, chain, clear, terms, composition, differentiable functions, partial derivatives, variables, functions, derivatives, chain rule
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This is version 7 of chain rule (several variables), born on 2002-03-22, modified 2003-10-07.
Object id is 2798, canonical name is ChainRuleSeveralVariables.
Accessed 10671 times total.

Classification:

AMS MSC:

26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)

Pending Errata and Addenda

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