# A Mathematical Approach to Tensors

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

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

- “Spin is like a ball that’s spinning, except it’s not a ball and it’s not spinning.”
- “Entropy is disorder.”
- “A vector is an element of a vector space.”
- “A tensor is a thing that transforms like a tensor.”

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

in

- 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.