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Fellows,

I have been wondering about calculus on manifolds. We know, gradient vector $\mbox{grad}(\phi)$ on flat $\mathbb{R}^n$ space is basically the partial derivative respective to each variable, right!? Let me ask about submanifolds: I have a potential map $\phi$ defined everywhere on manifold $(g, \mathcal{M})$. A submanifold $(h, \mathcal{S})$ has its tangent field $T_p \mathcal{S} \subseteq T_p \mathcal{M}$ on point $p \in \mathcal{S} \subseteq \mathcal{M}$. I have not enough knowledge to understand how to calculate the entries of such vector. I think, it might be easier if manifold $\mathcal{M}$ is rather n real space $\mathbb{R}^n$, which makes metric $g_{ij}$ equal to Dirac entries $\delta_{ij}$.

Let me add an example: we have a classic potential gravitational map defined on attached coordinate frame $\frac{\alpha}{r^2}$, such that constant $\alpha \in \mathbb{R}_+$ corresponds to quantity $GM$, the gravitational constant $G$ and planet mass $M$. It has the formula $\frac{\alpha}{x^2+y^2+z^2}$ in cartesian coordinates. At other hand, we have planet surface $\mathcal{S}$ defined by chart map $\varphi$ given by formula $\rho(\theta, \lambda) \begin{bmatrix} \sin{\lambda} \cos{\theta} \\ \sin{\lambda} \sin{\theta} \\ \cos{\lambda} \end{bmatrix}$. In this case, what would be the gradient $\mbox{grad}_{\mathcal{S}}\phi$ at point $p = \varphi(u, v)$? It is clear: the potential quantity at point $p$ is $\frac{\alpha}{\rho(u, v)^2}$, but is $\mbox{grad}_p \mathcal{S}$ given by vector below?

$$ -2 \, \frac{\alpha}{\rho(u, v)^3} \, \left(\frac{\partial \rho}{\partial u} \, X_u + \frac{\partial \rho}{\partial v} \, X_v \right) $$

User 42
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  • It is true the we cannot calculate anything on abstract manifolds. So we need charts. Try to view $\mathbb R^2$ as an abstract manifold or submanifold of $\mathbb R^n$ and equip it with polar coordinates. Then find the basis for the tangent space. Spoiler – Kurt G. Nov 16 '23 at 13:58
  • Yea, I had such grasp in mind, something like $\phi(p) = (\phi \circ \varphi) (u)$, but did not know how to deal with the composition. It seams, it goes something like $(\phi^{;i} X_i)$, am I right?! – User 42 Nov 16 '23 at 14:08
  • I added a motivation example. – User 42 Nov 16 '23 at 14:34
  • The answer to "something like ... right?" could be anything. You are effectively asking about the gradient in spherical polar coordinates. The formula you can find in that link you can derive by applying the chain rule to the potential function when that is expressed in cartesian resp. in polar coordinates and the two systems are related with the sin cos vector you wrote. I highly recommend that exercise. Compare what you get with Wikipedia. – Kurt G. Nov 16 '23 at 15:51
  • BTW: your notation indicates that you are interested in the surface gradient. Let's get to this later and do the 3 dimensional space gradient first. – Kurt G. Nov 16 '23 at 15:55
  • I like how your nickname is Kurt G., which makes me you trustworthy. – User 42 Nov 16 '23 at 20:45

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