Curvilinear Coordinates - Orthogonal Curvilinear Coordinates in 3D - Covariant and Contravariant Bases

Covariant and Contravariant Bases

The basis vectors, gradients, and scale factors are all interrelated within a coordinate system by two methods:

1. the basis vectors are unit tangent vectors along the coordinate curves:
   which transform like covariant vectors (denoted by lowered indices), or
2. the basis vectors are unit normal vectors to the coordinate surfaces:
   which transform like contravariant vectors (denoted by raised indices), ∇ is the del operator.

So depending on the method by which they are built, for a general curvilinear coordinate system there are two sets of basis vectors for every point: {b₁, b₂, b₃} is the covariant basis, and {b1, b2, b3} is the contravariant basis.

A vector v can be given in terms either basis, i.e.,

The basis vectors relate to the components by

and

where g is the metric tensor (see below).

A vector is covariant or contravariant if, respectively, its components are covariant (lowered indices, written v_k) or contravariant (raised indices, written vk). From the above vector sums, it can be seen that contravariant vectors are represented with covariant basis vectors, and covariant vectors are represented with contravariant basis vectors.

A key convention in the representation of vectors and tensors in terms of indexed components and basis vectors is invariance in the sense that vector components which transform in a covariant manner (or contravariant manner) are paired with basis vectors that transform in a contravariant manner (or covariant manner).