Covariant and Contravariant

Covariant

Power Density is Covariant. Now to explain what that means.

Power density has units of watts per area. Assume we have a power density of 100 watts per square meter. Assume that we change our choice for area from square meter to square foot. We now have about 9.29 watts per square foot.

A decrease to the component corresponds to a decrease to the size of the unit of measurement decreases (a square foot is smaller than a square meter). Since one is going with the other, we say “covariant”.

Contravariant

Velocity is contravariant. Now, to explain what that means.

Assume we have a vector that represents 60 miles per hour. Now, assume we switch from miles to kilometers. The vector doesn’t change, but becomes 100 kilometers per hour.

An increase to the component corresponds to a decrease to the size of the unit of measurement. Since one is going against the other, we say “contravariant”.

Appendix A

Covariant

A'_{ij} = \dfrac {\partial x^l}{\partial x'^i} \dfrac {\partial x^m}{\partial x'^j} A_{lm}

Contravariant

A'^{ij} = \dfrac {\partial x'^i}{\partial x^l} \dfrac {\partial x'^j}{\partial x^m} A^{lm}