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

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Illustrative Example 2. Find the partial derivatives of

Solution. , treating and as constants,
  , treating and as constants,
  , treating and as constants.

Again turning to the function

we have, by (A), §123, defined as the limit of the ratio of the increment of the function ( being constant) to the increment of , as the increment of approaches the limit zero. Similarly, (B), §123, has defined . 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 and when is constant, and

as the ratio of the time rates of change of and when is constant.


124. Partial derivatives interpreted geometrically. Let the equation of the surface shown in the figure be

Plane through point
Plane through point

Pass a plane EFGH through the point P (where and ) 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 and EH as the axis of : In this plane means the same as and we have

slope of section JK at P.