Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/160



Passing to the limit as $$\Delta \theta$$ diminishes towards zero, we get

These relations between $$\rho$$ and the differentials ds, dp, and $$d\theta$$ are correctly represented by a right triangle whose hypotenuse is ds and whose sides are $$d\rho$$ and $$\rho d\theta$$. Then

and dividing by $$d\theta$$ gives (30).

Denoting by $$\psi$$ the angle between $$dp$$ and $$ds$$, we get at once

which is the same as (A), &sect;67.

Find the differential of the arc of the circle $$x^2 + y^2 = r^2$$.


 * Solution. Differentiating, $$\tfrac{dy}{dx} = -\tfrac{x}{y}$$.


 * To find ds in terms of x we substitute in (27), giving


 * To find ds in terms of y we substitute in (28), giving

Find the differential of the arc of the cardioid $$\rho = a (l - \cos \theta)$$ in terms of $$\theta$$.


 * Solution. Differentiating, $$\tfrac{d\rho}{d\theta} = a \sin \theta$$


 * Substituting in (31), gives