📂Geometry

Total Variation in Differential Geometry

Definition¹

A regular curve $\gamma$ is considered piecewise simple curve with a period of $L$ being a closed curve. Let $\mathbf{Z}(t)$ be a continuous vector field along $\gamma$. Suppose the vector field $\mathbf{V}$ satisfies $\left| \mathbf{V}(p) \right| = 1$ $\forall p \in U$. Let $\alpha$ be a function mapping the angle between $\mathbf{Z}$ and $\mathbf{V}$.

$$\alpha (t) = \angle \left( \mathbf{V}(\gamma (t)), \mathbf{Z}(t) \right)$$

Now, the total angular variation $\delta_{\mathbf{V}} \alpha$ of the vector field $\mathbf{Z}$ along $\gamma$ with respect to $\mathbf{V}$ is defined as follows.

$$\delta_{\mathbf{V}}\alpha := \int_{0}^{L} \dfrac{d \alpha (t)}{d t} dt$$

Explanation

From $t=0$ to $t=L$, it represents how much the direction of $\mathbf{Z}$ changes (in reference to $\mathbf{V}$).

By definition, $\delta_{\mathbf{V}}\alpha$ generally depends on $\mathbf{V}$, but there are cases where it does not depend on $\mathbf{V}$ depending on the nature of the curve $\gamma$.

Theorem

Let the curve $\gamma$ be a null-homotopic enclosing a region $\mathscr{R}$. Then $\delta_{\mathbf{V}}\alpha$ does not depend on the choice of $\mathbf{V}$.