# Questions tagged [lie-derivative]

The Lie derivative gives a way to define the derivative of a tensor field in the direction of a vector field.

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### Lie derivative of a function w.r.t. vector field and Lie algebra

A note for self-reference: this post continues but differs from another post: Lie derivative of a function (of a point) with respect to a vector field Lie derivative $L_XY$ of vector field $Y$ at a ...
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### Component functions of the exterior derivative

Let $M$ be a smooth manifold, $(E_i)$ a smooth local frame for $M$, and ($\varepsilon^i$) the dual coframe. For each $i$, let $b^{i}_{jk}$ denote the component functions of the exterior derivative of (...
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### interchanging limit and exterior derivative to prove $L_X(df) = d(L_Xf)$

Let $M$ be a smooth manifold, $f: M \to \Bbb{R}$ a smooth function and let $X$ be a smooth vector field on $M$, with flow $(t,p) \mapsto\phi(t,p) \equiv \phi_t(p)$. I'm trying to prove that the Lie ...
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### vertical component of Lie bracket

Let $f:X\to Y$ be a submersion equipped with a connection given by a vertical projection $\mathrm V$. Let $\vec v_1,\vec v_2$ be vector fields on $Y$ with unique horizontal lifts $\vec u_1,\vec u_2$. ...
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