❓How is the gravitational field calculated from the potential?

The gravitational field (acceleration) $g$ is the negative gradient of the potential $\phi: g=-\nabla \phi$. The i-th component is:

$$g^i=-\frac{\partial \phi}{\partial x^i}=-\frac{\partial}{\partial x^i}\left(-\frac{G M}{r}\right)=G M \frac{\partial}{\partial x^i}\left(r^{-1}\right)$$

Using the chain rule $\frac{\partial r}{\partial x^2}=\frac{x^i}{r}$, the derivative is:

$$g^i=G M\left(-\frac{1}{r^2}\right) \frac{\partial r}{\partial x^i}=G M\left(-\frac{1}{r^2}\right) \frac{x^i}{r}=-\frac{G M x^i}{r^3}$$

In vector form, $g=-\frac{G M}{r^2} \hat{r}$, which is the standard Newtonian gravitational field.

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How is the gravitational field calculated from the potential? | FAQsviadean on Notion