Covariant

In tensor analysis, a covariant coordinate system is reciprocal to a corresponding contravariant coordinate system.

Roughly speaking, a covariant tensor is a vector field that defines the topology of a space; it is the base which one measures against.

A contravariant vector is thus a measurement or a displacement on this space.

Thus, their relationship can be represented simply as:

<math>\int_{contravariant}^{}covariant=invariant(volume, area, ect.)</math>

If e¹, e², e³ are contravariant basis vectors of R³ (not necessarily orthogonal nor of unit norm) then the covariant basis vectors of their reciprocal system are:

<math> \mathbf{e}_1 = \frac{\mathbf{e}^2 \times \mathbf{e}^3}{\mathbf{e}^1 \cdot \mathbf{e}^2 \times \mathbf{e}^3} ; \qquad \mathbf{e}_2 = \frac{\mathbf{e}^3 \times \mathbf{e}^1}{\mathbf{e}^1 \cdot \mathbf{e}^2 \times \mathbf{e}^3}; \qquad \mathbf{e}_3 = \frac{\mathbf{e}^1 \times \mathbf{e}^2}{\mathbf{e}^1 \cdot \mathbf{e}^2 \times \mathbf{e}^3}

Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors:

<math> q^1 = \mathbf{v \cdot e^1}; \qquad q^2 = \mathbf{v \cdot e^2}; \qquad q^3 = \mathbf{v \cdot e^3} </math>

Likewise, the covariant components of v can be obtained from the dot product of v with covariant basis vectors, viz.

<math> q_1 = \mathbf{v \cdot e_1}; \qquad q_2 = \mathbf{v \cdot e_2}; \qquad q_3 = \mathbf{v \cdot e_3} </math>

Then v can be expressed in two (reciprocal) ways, viz.

<math> \mathbf{v} = q_i \mathbf{e}^i = q_1 \mathbf{e}^1 + q_2 \mathbf{e}^2 + q_3 \mathbf{e}^3 </math>

<math> \mathbf{v} = q^i \mathbf{e}_i = q^1 \mathbf{e}_1 + q^2 \mathbf{e}_2 + q^3 \mathbf{e}_3 </math>.

The indices of covariant coordinates, vectors, and tensors are subscripts. If the contravariant basis vectors are orthonormal then they are equivalent to the covariant basis vectors, so there is no need to distinguish between the covariant and contravariant coordinates, and all indices are subscripts.