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 P: 6 Hi, ok so practically so I can understand this. What is $grad(F)$ for this specific case? Is it as in the Euclidean case $(2x,2y)$? 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 $g_{ij}=[ax, b; b, cy]$ and a tangent vector $t_{P}=[2,0]$ at some arbitrary point $P=(x_0,y_0)$. The scalar function $F$ is as before. What is $gradF$ for this example? And is $gradF \in T_PM$ ?