📂Geometry

Differentiable Functions Between Two Surfaces

Definition¹

Let’s say we are given a function $f : M \to N$ between two surfaces $M, N$. Suppose $\mathbf{x} : U \to M$ and $\mathbf{y} : V \to N$ are coordinate chart mappings that include the points $p \in M$, $f(p) \in N$, respectively. If $\mathbf{y}^{-1} \circ f \circ \mathbf{x} : U \to V$ is differentiable, then we say $f$ is differentiable at point $p$.

Explanation

Why don’t we think of the differentiability of $f$, even though the two surfaces $M, N$ are just subsets of $\mathbb{R}^{3}$, like ordinary differentiation? The answer is quite simple as below

$$ f^{\prime}(p) = \lim\limits_{h \to 0} \dfrac{f(p+h) - f(p)}{h} $$

because the numerator’s $f(p+h) - f(p)$ may not belong to $M$. In other words, points on the surface are not generally closed under addition. Even considering a sphere with a radius of $1$, it’s clear that adding any two points can result in leaving the surface of the sphere.

Therefore, think of $\mathbf{y}^{-1} \circ f \circ \mathbf{x}$ as a coordinate piece of $M$ and $N$. Then, this function is a function from $U$, a flat subset of $\mathbb{R}^{2}$, to $V$, so we can talk about ordinary differentiation as we know it.

$$ \mathbf{y}^{-1} \circ f \circ \mathbf{x} : U \subset \mathbb{R}^{2} \to V \subset \mathbb{R}^{2} $$