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--- dirichlet_energy [2022/09/05 05:43] – spencer
+++ dirichlet_energy [2022/09/05 05:44] (current) – [Euler-Lagrange equations] spencer
@@ Line 18: / Line 18: @@
 Consider a family of maps $u_t \in C^2(M,N)$ such that $u_0 = u$.
 Let $U : \mathbb{R} \times M \to N$ be the map defined by $U(t,x) = u_t(x)$.
-Let $\nabla_{\partial/\partial t}$ be the covariant derivative, and then compute on a vector field $X$ on $M$:
+Let $\nabla_{\mathrm{d}/\mathrm{d}t}$ be the covariant derivative, and then compute on a vector field $X$ on $M$:
 \begin{align*}
-(\nabla_{\partial/\partial t} \mathrm{d} u_t)X &= \nabla_{\partial/\partial t}(\mathrm{d}u_t \cdot X) - \mathrm{d}u_t \cdot \nabla_{\partial/\partial t} X \\
+(\nabla_{\mathrm{d}/\mathrm{d} t} \mathrm{d} u_t)X &= \nabla_{\mathrm{d}/\mathrm{d}t}(\mathrm{d}u_t \cdot X) - \mathrm{d}u_t \cdot \nabla_{\mathrm{d}/\mathrm{d} t} X \\
-&= \nabla_{\partial/\partial t}(\mathrm{d}U \cdot X) - 0\\
+&= \nabla_{\mathrm{d}/\mathrm{d}t}(\mathrm{d}U \cdot X) - 0\\
-&= \nabla_X (\mathrm{d}U \cdot \partial/\partial t) + \mathrm{d}U\left[ \frac{\partial}{\partial t}, X\right] \\
+&= \nabla_X (\mathrm{d}U \cdot \mathrm{d}/\mathrm{d} t) + \mathrm{d}U\left[ \frac{\mathrm{d}}{\mathrm{d}t}, X\right] \\
 &= \nabla_X \frac{\partial U}{\partial t} + 0.
 \end{align*}