Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/224

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Let P be the point (x, y, z) on the surface given by the equation

(E)

and let PC and AP be sections made by planes through P parallel to the YOZ- and XOZ-planes respectively. Along the curve AP, y is constant; therefore, from (E), z is an implicit function of x alone, and we have, from (57a),

Point P on a surface.
Point P on a surface.
(58)

giving the slope at P of the curve AP, §122.

is used instead of in the first member, since z was originally, from (E), an implicit function of x and y; but (58) is deduced on the hypothesis that y remains constant.

Similarly, the slope at P of the curve PC is

(59)

EXAMPLES

Find the total derivatives, using (51), (52), or (53), in the following six examples:

1. Ans.
2. Ans.
3.
4.
5.
6.
Using (55) or (56), find the total differentials in the next eight examples:
7. Ans.
8.