Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/216

 Find the partial derivatives of $$u = \sin (ax + by + cz).$$

Again turning to the function

$$z = f(x,\ y),$$

we have, by (A), §123, defined $$\tfrac{\partial z}{\partial x}$$ as the limit of the ratio of the increment of the function ($$y$$ being constant) to the increment of $$x$$, as the increment of $$x$$ approaches the limit zero. Similarly, (B), §123, has defined $$\tfrac{\partial z}{\partial y}$$. It is evident, however, that if we look upon these partial derivatives from the point of view of § 94, then

may be considered as the ratio of the time rates of change of $$z$$ and $$x$$ when $$y$$ is constant, and

as the ratio of the time rates of change of $$z$$ and $$y$$ when $$x$$ is constant.

Let the equation of the surface shown in the figure be

$z = f(x,\ y).$ Pass a plane EFGH through the point P (where $$x = a$$ and $$y = b$$) on the surface parallel to the XOZ-plane. Since the equation of this plane is

the equation of the section JPK cut out of the surface is

if we consider EF as the axis of $$Z$$ and EH as the axis of $$X$$: In this plane $$\tfrac{\partial z}{\partial x}$$ means the same as $$\tfrac{dz}{dx}$$ and we have