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Two independent variables.

Theorem 5.21   Let f be a function of two variables (x,y), differentiable on an open domain $\mathcal{D}$. Suppose that x and y are functions of two independent variables u,v, differentiable on an open domain $\mathcal{R} \subset \mathbb{R} ^2$, and such that for every $u,v \in \mathcal{R}, \;
( x(u,v),y(u,v)) \in \mathcal{D}$.

Then w=f ( x(u,v),y(u,v)) is a function of u,v, differentiable on $\mathcal{R}$ and we have:
\begin{align*}\frac {\partial f}{\partial u} &= \frac {\partial f}{\partial x} \...
... v} +\frac {\partial f}{\partial y}+\frac {\partial y}{\partial v}
\end{align*}


  
Figure 3: The chain rule for two intermediate variables.
\begin{figure}
\mbox{\subfigure[]{\epsfig{file=chain3-u.eps,height=4cm} }
\quad...
...}
\qquad
\subfigure[]{\epsfig{file=chain3-v.eps,height=4cm} }
}
\end{figure}

Example 5.22   Let w=f(x,y)=xy, where x=u+v and y=u-v. Then w=(u+v)(u-v)=u2-v2 and we have:

\begin{displaymath}\begin{cases}
\frac {\partial f}{\partial u} = 2u \\ \frac {\partial f}{\partial v}=-2v
\end{cases}\end{displaymath}

By the chain rule, we have:

\begin{displaymath}\begin{cases}
\frac {\partial f}{\partial u} =\frac {\partial...
...\partial v}
=(u-v) \cdot 1 + (u+v) \cdot (-1) = -2v
\end{cases}\end{displaymath}

Theorem 5.23   Let f be a function of three variables (x,y,z), differentiable on an open domain $\mathcal{D}$. Suppose that x, y and z are functions of two independent variables u,v, differentiable on an open domain $\mathcal{R} \subset \mathbb{R} ^2$, and such that for every $(u,v) \in \mathcal{R}, \;
( x(u,v),y(u,v),z(u,v) ) \in \mathcal{D}$.

Then w=f ( x(u,v),y(u,v),z(u,v) ) is a function of u,v, differentiable on $\mathcal{R}$ and we have:
\begin{align*}\frac {\partial f}{\partial u} &= \frac {\partial f}{\partial x} \...
...v}
+ \frac {\partial f}{\partial z} \frac {\partial z}{\partial v}
\end{align*}

Remark 5.24   Actually (x(u,v),y(u,v), z(u,v)) is a parametrization of a surface in the 3-dimenasional space. The chain rule enables us to compute the derivative of a function f defined on a surface.


next up previous contents
Next: Implicit differentiation. Up: The chain rule. Previous: One independent variable.
Noah Dana-Picard
2001-05-30