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Feb10-12, 04:48 PM
P: 6
Hi, ok so practically so I can understand this.
What is [itex]grad(F)[/itex] for this specific case? Is it as in the Euclidean case [itex](2x,2y)[/itex]? or it is dependent on some arbitrary tangent vector and the Riemannian metric? I didnt understand your answer completely.

Lets try a practical simple example.
Say I have [itex]g_{ij}=[ax, b; b, cy][/itex] and a tangent vector [itex]t_{P}=[2,0][/itex] at some arbitrary point [itex]P=(x_0,y_0)[/itex]. The scalar function [itex] F [/itex] is as before.

What is [itex]gradF[/itex] for this example?

And is [itex] gradF \in T_PM [/itex] ?