📂Geometry

Area of a Surface in Differential Geometry

Definition¹

Let’s say $\mathbf{x} : U \to \mathbb{R}^{3}$ is the coordinate chart mapping of a surface. The area of any region $\mathscr{R} \subset \mathbf{x}(U)$ on the surface is defined as follows.

$$\begin{align*} A(\mathscr{R}) &:= \int\int_{\mathbf{x}^{-1}(\mathscr{R})} [\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{n}] du^{1}du^{2} \\ &= \int\int_{\mathbf{x}^{-1}(\mathscr{R})} \sqrt{g} du^{1}du^{2} \end{align*}$$

Here, $(u^{1}, u^{2})$ are the coordinates of $U$, $\mathbf{x}_{i} = \dfrac{\partial \mathbf{x}}{\partial u^{i}}$ is the partial derivative of the $i$-th coordinate, $[\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{n}]$ is the scalar triple product, and $g$ is the determinant of the matrix of coefficients of the first fundamental form.

Explanation

At this time, $\sqrt{g} du^{1}du^{2}$ is called the area element, and is denoted $dA$. For functions defined on the surface like Gaussian curvature $K$, the following notation is also used.

$$\iint_{\mathscr{R}} K dA := \iint_{\mathbf{x}^{-1}(\mathscr{R})} K(u^{1}, u^{2}) \sqrt{g} du^{1} du^{2}$$