## Covariant and contravariant coordinates

Let $\mathbb{E}^n$ be an Euclidean space. To define the vectors of this Euclidean space algebraically one need to choose a coordinate system.
There are two different type of coordinate systems:**homogeneous (Cartesian)** and **inhomogeneous (curvilinear)** coordinate systems.
A Cartesian coordinate system consists of straight line coordinate axes. Curvilinear coordinate systems have curved coordinate axes. Examples are polar coordinates in $2D$ and cylindrical and spherical polar coordinates in $3D$. Many of the concepts in vector calculus such as the gradient, divergence, curl, and the Laplacian can be formulated in a way which is independent of the coordinate system. It alllows physicists to formulate the laws of physics coordinate free.

An important characteristic of a coordinate system is whether the angle between the coordinate axes is **orthogonal** (rectangular), or **non-orthogonal** (oblique). Most often orthogonal coordinate systems are used to model physical problems.

Yet another characteristic is whether the base vectors of a coordinate system are of unit lenght or not. A rectangular system with unit length vectors is called orthonormal.

In two dimensional Euclidean space we can represent a point point $Q$ with a rectangular coordinate system $X$ and an oblique coordinate system $U$. For the oblique system we can define the coordinates in two different ways. One way is to project the point *parallel* to the coordinate axes. The vector components obtained in this way are called a **contravariant** components, and are indicated with superscript indices: $Q=(u^1,u^2)=u^1p^1+u^2p^2$.The vector basis $\{p^1,p^2\}$ is called the contravariant basis set. The other way is to project the point *orthogonal* to the coordinate axes. The vector components obtained by orthogonal projections are called a **covariant** components and are indicated with subscript indices: $Q=(u_1,u_2)=u_1p_1+u_2p_2$. The vector basis $\{p_1,p_2\}$ is called the covariant basis set. A trick to remember the place of the indices for the components is to look at the third letter of the word contravariant (n=upper) or covariant (v=down). In an orthogonal system these two coordinates coincide.

Contravariant and covariant vector representation |