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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 . Suppose that x and y are functions of two independent variables u,v, differentiable on an open domain , and such that for every .

Then w=f ( x(u,v),y(u,v)) is a function of u,v, differentiable on and we have:

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:

By the chain rule, we have:

Theorem 5.23   Let f be a function of three variables (x,y,z), differentiable on an open domain . Suppose that x, y and z are functions of two independent variables u,v, differentiable on an open domain , and such that for every .

Then w=f ( x(u,v),y(u,v),z(u,v) ) is a function of u,v, differentiable on and we have:

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: Implicit differentiation. Up: The chain rule. Previous: One independent variable.
Noah Dana-Picard
2001-05-30