Homogenous function

A function f is homogenous of degree n if and only if f(ax) = a^n f(x).

— 1.8 Conserved Quantities

— Structure and Interpretation of Classical Mechanics

.

\displaystyle{\begin{aligned}     D_a f(ax)     &= \frac{d}{da} f(ax) \\     &= \frac{d}{da} a^n f(x) \\     &= f(x) \frac{d}{da} a^n  \\     &= n a^{n-1} f(x) \\     \end{aligned}}

.

\displaystyle{\begin{aligned}     D_a f(ax)     &= \frac{d}{da} f(ax) \\     &= \frac{d(ax)}{da} \frac{d}{d(ax)} f(ax) \\     &= x \frac{d}{d(ax)} f(ax) \\     &= x D_{ax} f(ax) \\     \end{aligned}}

.

\displaystyle{\begin{aligned}     x D_{ax} f(ax) &= n a^{n-1} f(x) \\    x D_{x} f(x) &= n f(x) \\    \end{aligned}}

— Me@2022.09.05 06:28:00 PM

.

.

2022.09.06 Tuesday (c) All rights reserved by ACHK