Partial Derivatives

Functions in differential geometry often take multiple arguments — coordinates on a manifold. The partial derivative $\partial_i f$ differentiates $f$ with respect to its $i$-th argument while holding all others fixed.

In functional notation, $\partial_i$ is itself a higher-order function:

$\partial_i f(x_0, x_1, \ldots) = \lim_{h \to 0} \frac{f(\ldots, x_i + h, \ldots) - f(\ldots, x_i - h, \ldots)}{2h}$

The Lagrangian

The book opens with the Lagrange equations of motion. For a Lagrangian $L(t, q, \dot{q})$:

$\frac{d}{dt}(\partial_2 L) - \partial_1 L = 0$

Here $\partial_1 L$ is the partial derivative of $L$ with respect to its second argument (position $q$), and $\partial_2 L$ is the partial with respect to its third argument (velocity $\dot{q}$). Index 0 is time.

For the harmonic oscillator $L = \frac{1}{2}\dot{q}^2 - \frac{1}{2}kq^2$:

• $\partial_1 L = -kq$
• $\partial_2 L = \dot{q}$

Your Task

Implement partial(i, f) that returns the partial derivative of f with respect to argument i (0-indexed).