# I need a starter kit for Differential Geometry

In Modern Cosmology, the Geodesic Equation is described as:$$\frac {d^2x'^2}{dt}+\left[ \left( \{\frac{\partial x}{\partial x'}\}^{-1} \right)^l_i \frac{\partial^2 x^i}{\partial x'^j \partial x'^k} \right]\frac {dx'^k}{dt}\frac{dx'^j}{dt}=0$$Below this description is the suggestion

You can check that this rather cumbersome expression does indeed give the correct equations of motion in polar coordinates.

That's what I'd like to do in Mathematica, but my skills are not yet sufficient. Would someone please suggest a way to implement this function using a Cartesian to Polar transformation matrix:$$\frac{\partial x^i}{\partial x'^j}=\begin{bmatrix}cos(x'^2) & -x'^1 sin(x'^2)\\sin(x'^2) & x'^1 cos(x'^2)\end{bmatrix}$$Where the Cartesian coordinates are $$x^i$$ and the 2D Polar Coordinates are $$x'^i$$. Specifically x = (x, y) and $$x'=(r,\theta)$$. Ultimately what I hope to do is find the shortest distance between two points in Polar Coordinates. I have some very basic questions like:

• How do you accomplish a partial derivative calculation.
• How do you accomplish a matrix transform.
• What's the best way to represent coordinates that reflect this notation.

I understand these are general questions, but I'm sure the Geodesic has been done so many times in Mathematica, that there should be a consensus on the best way to exploit the language.