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dirichlet_energy [2022/09/05 05:44] – [Euler-Lagrange equations] spencerdirichlet_energy [2022/09/05 05:44] (current) – [Euler-Lagrange equations] spencer
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 Consider a family of maps $u_t \in C^2(M,N)$ such that $u_0 = u$. 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 $U : \mathbb{R} \times M \to N$ 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*} \begin{align*}
 (\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_{\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 \\
dirichlet_energy.1662371060.txt.gz · Last modified: by spencer