Undergrad About the properties of the Divergence of a vector field

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SUMMARY

The discussion centers on the properties of the divergence of a vector field in the context of Riemannian manifolds. Specifically, the user inquires whether the equation X(exp-f) = -exp(-f)·X(f) holds true, along with the divergence expression div(exp(-f)·X) = exp(-f)〈grad f, X〉 + exp(-f)·div(X). These equations are fundamental in understanding the behavior of vector fields under smooth functions on Riemannian manifolds.

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  • Understanding of Riemannian manifolds
  • Familiarity with vector fields and their properties
  • Knowledge of smooth functions and their derivatives
  • Basic concepts of divergence and gradient in differential geometry
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  • Learn about the relationship between vector fields and smooth functions
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Hello
I have a question if it possible,
Let X a tangantial vector field of a riemannian manifolds M, and f a smooth function define on M.
Is it true that X(exp-f)=-exp(-f).X(f)
And div( exp(-f).X)=exp(-f)〈gradf, X〉+exp(-f)div(X)?
Thank you
 
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These should be fairly simple things to derive yourself.
 

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