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