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I have a basic question I would like to ask.

I ll start from the Euclidean analogue to try to explain what I want.

Suppose we have a smooth function (real valued scalar field)

[itex]F(x,y)=x^2+y^2[/itex], with [itex]x,y \in ℝ[/itex].

We also have the gradient [itex] \nabla F=\left( \frac{\partial F}{\partial x},\frac{\partial F}{\partial y} \right)=\left( 2x,2y \right)[/itex]

Now lets imagine that F is defined on a Riemannian manifold [itex]M[/itex] with a metric [itex]g_{ij}[/itex].

I would like to calculate the gradient of [itex]F[/itex] for a point [itex] x,y \in M[/itex].

I read that in this case, that the local form of the gradient is

[itex]\nabla F = g^{ij}\frac{∂F}{∂x^k}\frac{∂}{∂x^i}[/itex]

But I do not understand what exactly this formulation means. I have an anyltic expresion for [itex]F, g^{i,j} [/itex] but I am not sure how to calculate the gradient in this case. Can someone perhaps explain the above expresion in layman's terms to me? (I do understand Einstein notation btw).

Also, does the gradient vector live in the tangent space of the point at [itex] x,y[/itex]? because somewhere I read about [itex]\hat{F}=F \circ R [/itex] as the pull back of [itex]F[/itex] through the retraction function [itex]R[/itex] onto the tangent space, and I was confused which "version" lie on the tangent space exactly.

Many thanks

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# Basic diff. geometry question - Gradient of F

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