Vectors and Covectors

Joseph Mellor
16 min readDec 26, 2022

Covectors are dual to vectors, and they’re fundamental to understand differential forms and tensors.

This article is stop 14 on The Road to Quantum Mechanics.

Up to this point in the Lagrangian arc, we’ve been looking at the paths objects take through space. These paths are functions of time that output positions in space. As we continue through this series, we’ll want to look at functions of space and time. More specifically, we’ll want to take derivatives and integrals of these functions with respect to our coordinates in space.

While we have taken these spatial derivatives in the PDE arc, we’ve only worked in Cartesian and spherical coordinates. Even then, we had to use a lot of facts without explanation or derivation, such as the form of the Laplacian in spherical coordinates.

Over the next few articles, we’re going to fix those problems by coming up with a way to take these derivatives and integrals in arbitrary coordinates.

Check Your Understanding

We’re doing formalism in this article, so checking your understanding requires that you practice working with the formalism yourself.

Different Representations

A vector field is given by

in standard Cartesian coordinates. Write the vector field in terms of

  • Differentials and Derivatives
  • Bra-Ket Notation
  • Indices
  • Matrices

Convert each of these representations to the corresponding covector representation.

Dual Vectors on a Metric Space

Find the corresponding covector fields for the following vector fields on specific metric spaces.

I’ll be real with you, I don’t know if subscripts actually matter for the metric tensor, but I want to at least try to be consistent. Also, if you see the mistake in the second line, let me know.

If none of the coefficients of the basis vectors change, you’re doing something wrong.

Prerequisites

We’re going to be building on the Math introduced in Bra-Ket Notation and Orthogonality and An

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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/.