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Mathematics for Physics

163 Covariant, Contravariant, and Coordinate Systems

The context if this section is the following. In physics and mathematics, points exist independently of the coordinates that is used to represent them.

First, we begin a discussion on covariant and contravariant vectors.

Covariant vectors are linear functionals.

definition (Linear Functional) A linear transformation from a vector space \(V\) to its associated field of scalars \(K\), The mapping \(f: V \rightarrow K\) must satisfy the following linearity condition:

  • \(f(ax + by) = af(x) + bf(y)\)

where

  • \(x, y \in V\) are vectors
  • \(a, b \in K\) are scalars
  • \(V\) is a vector space
  • \(K\) is the field of scalars.

definition (k-Covector) An alternating \(k\)-linear function from the \(k\)-fold product of a vector space to the real numbers, \(T: V^k \rightarrow \mathbb{R}\), where the following conditions apply:

  • The function \(T\) is linear in each of its \(k\) arguments.
  • \(T(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = (\text{sgn } \sigma) T(v_1, \dots, v_k)\) for every permutation \(\sigma \in S_k\).

where

  • \(V\) is a vector space.
  • \(k\) is a positive integer representing the degree of the covector.
  • \(v_1, \dots, v_k \in V\) are vectors.
  • \(A^k(V)\) is the vector space of all \(k\)-covectors on \(V\).
  • \(\text{sgn } \sigma\) is the sign of the permutation.

Note:

  • \(A^k(V)\) is also written \(\Lambda^k(V^*)\).
  • A \(k\)-covector is also called a multicovector of degree \(k\).
  • A \(k\)-covector is also called an alternating \(k\)-tensor.

definition (k-Covector Field) A function assigning a \(k\)-covector to each point of a manifold, \(\omega: M \rightarrow \Lambda^k(T^*M)\), where the following condition applies:

  • For each point \(p\) in \(M\), \(\omega(p)\) is an alternating \(k\)-linear function on the tangent space \(T_pM\).

where

  • \(M\) is a smooth manifold.
  • \(p\) is a point in \(M\).
  • \(T_pM\) is the tangent space of \(M\) at \(p\).
  • \(T^*_pM\) is the cotangent space of \(M\) at \(p\).
  • \(\Lambda^k(T^*_pM)\) is the space of all alternating \(k\)-tensors on the tangent space \(T_pM\).
  • \(\Lambda^k(T^*M)\) is the \(k\)-th exterior power of the cotangent bundle.

Note:

  • \(\Lambda^k(T^*_pM)\) is also written \(A^k(T_pM)\).
  • A \(k\)-covector field is also called a differential \(k\)-form.

example 1 (k-Covector) Let

\[ V = \mathbb{R}^2. \]

Define the function

\[ T : V^2 \rightarrow \mathbb{R} \]

by

\[ T((a,b),(c,d)) = ad-bc. \]

where

  • \[a,b,c,d \in \mathbb{R}\]

Let

\[ v_1=(1,2), \qquad v_2=(3,4). \]

Then

\[ T(v_1,v_2) = (1)(4)-(2)(3) = -2. \]

Interchanging the vectors gives

\[ T(v_2,v_1) = (3)(2)-(4)(1) = 2 = -T(v_1,v_2). \]

Therefore, \(T\) changes sign when its arguments are exchanged, so \(T\) is alternating.

Furthermore, \(T\) is linear in each argument. Hence, \(T\) is an alternating \(2\)-linear function.

Therefore, \(T\) is a \(2\)-covector on \(\mathbb{R}^2\).

163.1 Elementary Example

163.1.1 Simple

A vector uses upper indices. A covector uses lower indices.

\[ V = \{ e_{1},\ e_{2},\ e_{3} \} \]

\[ V^{*} = \{ e^{1},\ e^{2},\ e^{3} \} \]

\[ v = v^{1} e_{1} + v^{2} e_{2},\quad \omega = \omega_{1} e^{1} + \omega_{2} e^{2} \]

where

  • \(v^{i}\) are contravariant components of the vector \(v\).
  • \(\omega_{i}\) are covariant components of the covector \(\omega\).
  • \(V^{*}\) is the dual space of \(V\).

163.1.2 General

Under a change of basis with matrix \(A\), vector components transform with \(A^{-1}\) and covector components transform with \(A\).

\[ v'^{i} = (A^{-1})^{i}_{\ j}\, v^{j} \]

\[ \omega'_{i} = A^{j}_{\ i}\, \omega_{j} \]

\[ A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

where

  • \(A\) is the change-of-basis matrix.
  • \(v'^{i}\) are the new contravariant components.
  • \(\omega'_{i}\) are the new covariant components.