How to use tensors in Mathematica?

Questions
1

Hello everybody,
I am beginner in Mathematica. I would like to know how to use tensors in Mathematica in general relativity, i.e. when i have defined metric tensor, how to compute tensors that appear in Einstein and Maxwell equations and get exact form of both sides; lowering/uppering four-vectors, covariant derivate etc. Is it built in or is some package needed?

Thank you for any answer.

2

First read a bit

then ask more specific questions …

Regards
Udo.

3

“Tensorial Analysis” was created by Mrs.Renan Cabrera and modified by shinabe munehiro .This tensor calculus package was made for general purpose, easy to use with good looking formula display.

msh_tensorial.m (30.7 KB)
msh_tensorial20240720.nb (1.75 MB)

4

Marty,

The tensor calculus package cannot be downloaded. Can you please repost.

Thanks

5

I have attached “msh_tensorial.m” (the tensor calculus package) to my previous reply, and I am attaching it again.

msh_tensorial.m (30.7 KB)

6

Title: “Tensor Analysis and Calculus” msh_tensorial20241202.nb is a notebook file created for beginners to help them understand introductory books on tensors.
The “msh_tensorial.wl” package is easy to use with good-looking mathematical expressions.

msh_tensorial.wl (36.2 KB)
msh_tensorial20241202 .nb (4.39 MB)

7

Happy New Year. I will upload the latest version. If you have any comments or advice, please feel free to reply.

msh_tensorial.wl (37.6 KB)
msh_tensorial20250102.nb (6.36 MB)

8

“Tensorial Analysis” was originally developed by Renan Cabrera and later modified by Munehiro Shinabe (shinabe.munehiro@hotmail.co.jp). This tensor calculus package is designed for beginners and is well-suited for studying in reference to standard textbooks. Its beautifully formatted equations follow the notation commonly used in academic texts.
Last updated: September 27, 2025

P.S.: I have also attached reference materials for self-study.

New msh_tensorial20250927.nb (17.5 MB)
msh_tensorial.wl (40.4 KB)

9

With the release of version 14.3, examples using the new functions FindShortestCurve and ShortestCurveDistance have been added."

msh_tensorial.wl (43.8 KB)
msh_tensorial20251031.nb (19.1 MB)