A Mathematical Approach to Tensors

Joseph Mellor
17 min readJun 4, 2023

A matrix is a linear map of a vector to another vector. A tensor is a multilinear map of a tensor to another tensor.

This is part of a matrix. This is stop 20 on the Road to Quantum Mechanics.

There are a lot of bad explanations in science and math.

The last two reflect a common mindset in STEM that you don’t need scaffolding to build a concept just because the final building has none.

While I have some strong words about these explanations, they’re best suited for other articles and videos. In this article, we’re going to unify several mathematical threads that have been running throughout this series.

Check Your Understanding

You’re going to work with tensors both as arbitrary objects and in terms of specific examples.

Tensors in Different Notations

Write out the following tensors

  • A (1, 0)-tensor
  • A (0, 1)-tensor
  • A (1, 1)-tensor
  • A (2, 0)-tensor
  • A (2, 1)-tensor
  • A (2, 2)-tensor


  • Index notation
  • Bra-ket notation
  • Array notation

For each array in the array notation, assume we’re working in 3D space.

Quadrupole Moment

The quadrupole moment is the (0, 2)-tensor in the second-order term in the multipole expansion of some potential. It is defined by

Say you have a disk of radius R centered at the origin with a charge distribution given by

Find the quadrupole moment in Cartesian coordinates. Feel free to use a computer or some other numerical approximation. Then, use the tensor transformation rule to convert write the tensor in Cylindrical coordinates.



Joseph Mellor

BS in Physics, Math, and CS with a minor in High-Performance Computing. You can find all my articles at https://josephmellor.xyz/articles/.