# 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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### Compute the Lie derivative for $X=y \frac{\partial}{\partial x}$ and $Y=x \frac{\partial}{\partial y}$

Question: On $\mathbf{R}^2$, let $X=y \frac{\partial}{\partial x}$ and let $Y=x \frac{\partial}{\partial y}$, with corresponding flows given by $\phi_t(x, y)=(x+t y, y)$ and $\psi_t(x, y)=(x, y+t x)$. ...
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### Formula for the Lie derivative of $fX$ on functions (proving that the Levi-Civita connection is a connection)

I am currently in the process of reading Riemannian Geometry by Gallot, Hulin and Lafontaine in order to learn more about differential and, well, Riemannian geometry. I'm more of a functional analyst ...
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### Lie derivative and partial derivative commute when applied to metric?

I am currently trying to find an expression for the Lagrangian variation of the Christoffel symbols $\Delta \Gamma^\lambda {}_{\mu\nu}$. For the Eulerian variation $\delta \Gamma^\lambda {}_{\mu\nu}$ ...
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### Proof of sum/product formula for matrix exponential

I am reading about IMU error propagation and to pre integrate the IMU one uses the formula. $$Exp(\phi + \delta) \approx Exp(\phi)Exp(J_r(\phi)\delta)$$ I want to understand where it comes from since ...
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### Jacobian of $R^{-1} v$ with respect to $R \in SO(3)$
According to "A micro Lie theory for state estimation in robotics", the Jacobian $J_R^{R\cdot{}\mathbf{v}} = -\mathbf{R}[\mathbf{v}]_\times$. Here non-bolded R is an element of SO(3) or an ...