Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/222

 are called partial differentials. These partial differentials are sometimes denoted by dxu, dyu, dzu, so that (56) is also written

Given $$u = \arctan \tfrac{y}{x}$$, find $$dx$$.

Solution.

Substituting in (55),

The base and altitude of a rectangle are 5 and 4 inches respectively. At a certain instant they are increasing continuously at the rate of 2 inches and 1 inch per second respectively. At what rate is the area of the rectangle increasing at that instant?

Solution. Let $$x =$$ base, $$y =$$ altitude; then $$u = xy =$$ area, $$\tfrac{\partial u}{\partial x} = y, \tfrac{\partial u}{\partial y} = x.$$

Substituting in (51),

But $$x = 5$$ in., $$y = 4$$ in., $$\tfrac{dx}{dt} = 2$$ in. per sec., $$\tfrac{dy}{dt} = 1$$ in. per sec.

NOTE. Considering $$du$$ as an infinitesimal increment of area due to the infinitesimal increments $$dx$$ and $$dy$$, $$du$$ is evidently the sum of two thin strips added on to the two sides. For, in $$du = ydx + xdy$$ (multiplying (A) by $$dt$$),



But the total increment $$\Delta u$$ due to the increments $$dx$$ and $$dy$$ is evidently

Hence the small rectangle in the upper right-hand corner ($$= dxdy$$) is evidently the difference between $$\Delta u$$ and $$du$$. This figure illustrates the fact that the total increment and the total differential of a function of several variables are not in general equal.

The equation

defines either $$x$$ or $$y$$ as an implicit function of the other. It represents any equation containing $$x$$ and $$y$$ when all its terms have been transposed to the first member. Let