Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/176



It is often desirable to change both variables simultaneously. An important case is that arising in the transformation from rectangular to polar coördinates. Since

the equation

becomes by substitution an equation between &rho; and &theta;, defining &rho; as a function of &theta;. Hence &rho;, x, y are all functions of &theta;.

Transform the formula for the radius of curvature (42), &sect;103,

into polar coördinates.


 * Solution. Since in (A) and (B), &sect;97, t is any variable on which x and y depend, we may in this case let $$t = \theta$$, giving


 * {| style="width:100%"


 * style="width:3%"|
 * align="center"|$$\frac{dy}{dx} = \frac{ \frac{dy}{d\theta} }{ \frac{dx}{d\theta} }$$, and
 * align="center"|$$\frac{d^2 y}{dx^2} = \frac{ \frac{dx}{d\theta} \frac{d^2 y}{d\theta^2} - \frac{dy}{d\theta} \frac{d^2 x}{d\theta^2} }{ \left( \frac{dx}{d\theta} \right)^3 }$$
 * }
 * align="center"|$$\frac{d^2 y}{dx^2} = \frac{ \frac{dx}{d\theta} \frac{d^2 y}{d\theta^2} - \frac{dy}{d\theta} \frac{d^2 x}{d\theta^2} }{ \left( \frac{dx}{d\theta} \right)^3 }$$
 * }


 * Substituting (B) and (C) in (A), we get


 * {| style="width:100%"


 * align="center"|$$R = \left[ \frac{ \left( \frac{dx}{d\theta} \right)^2 + \left( \frac{dy}{d\theta} \right)^2 }{ \left( \frac{dx}{d\theta} \right)^2 } \right]^{\frac{3}{2}} \div \frac{ \frac{dx}{d\theta} \frac{d^2 y}{d\theta^2} - \frac{dy}{d\theta} \frac{d^2 x}{d\theta^2} }{ \left( \frac{dx}{d\theta} \right)^3 }$$, or
 * align="center"|$$ R = \frac{ \left[ \left( \frac{dx}{d\theta} \right)^2 + \left( \frac{dy}{d\theta} \right)^2 \right]^{\frac{3}{2}} }{ \frac{dx}{d\theta} \frac{d^2 y}{d\theta^2} - \frac{dy}{d\theta} \frac{d^2 x}{d\theta^2} }$$.
 * }
 * align="center"|$$ R = \frac{ \left[ \left( \frac{dx}{d\theta} \right)^2 + \left( \frac{dy}{d\theta} \right)^2 \right]^{\frac{3}{2}} }{ \frac{dx}{d\theta} \frac{d^2 y}{d\theta^2} - \frac{dy}{d\theta} \frac{d^2 x}{d\theta^2} }$$.
 * }
 * }


 * But since $$x = \rho \cos \theta$$ and $$y = \rho \sin \theta$$, we have


 * Substituting these in (D) and reducing,