Let $m \geq 2$, let $U \subset \mathbb{R}^m$ be open, let $(N,h)$ be a smooth Riemannian manifold isometrically embedded in some Euclidean space $\mathbb{R}^q$, and let $u \in W^{1,2}(U;N)$ be a stationary harmonic map. Fix $x_0 \in U$ and $R>0$ such that $B(x_0,R) \subset U$. For $x \ne x_0$, define the radial [weak derivative](/page/Weak%20Derivative) of $u$ about $x_0$ by

\begin{align*} \frac{\partial u}{\partial r}(x) := \sum_{\alpha=1}^m \frac{x_\alpha - (x_0)_\alpha}{|x-x_0|}\,\partial_{x_\alpha}u(x) \in \mathbb{R}^q. \end{align*}

Then for every $0<\rho<r<R$,

\begin{align*} r^{2-m}\int_{B(x_0,r)} |\nabla u|^2\,d\mathcal{L}^m(x) - \rho^{2-m}\int_{B(x_0,\rho)} |\nabla u|^2\,d\mathcal{L}^m(x) = 2\int_{B(x_0,r)\setminus B(x_0,\rho)} |x-x_0|^{2-m} \left|\frac{\partial u}{\partial r}(x)\right|^2 \,d\mathcal{L}^m(x). \end{align*}

Consequently, the density function

\begin{align*} \Theta_u(x_0,t) := t^{2-m}\int_{B(x_0,t)} |\nabla u|^2\,d\mathcal{L}^m(x), \qquad 0<t<R, \end{align*}

is nondecreasing.