• Covariance and contravariance of vectors
  • Intuition
  • Contravariance
  • Covariance
• Manifolds
  • Tangent vectors
  • Differentials
• $n$-Tensors
  • Metric tensors
• Foonotes

\[\newcommand{\vec}{\mathbf} \newcommand{\d}{\text{d}} \newcommand{\bm}{\pmb} \newcommand{\pd}[1]{\frac{\partial}{\partial #1}} \newcommand{\pdd}[2]{\frac{\partial #1}{\partial #2}}\]

Covariance and contravariance of vectors

Intuition

Covariance and contravariance characterize how quantities (typically vectors) measured in some space transform when the space undergoes a change of basis or coordinate system. A covariant quantity has components that transform in the same way as the basis (to be clarified). A contravariant quantity has components that transform opposite of the basis (to be clarified).

A physical example will illustrate the intuitive sense of the words: imagine a temperature $T$ as a function of position $x$ where position is measured in meters and temperature in degrees Fahrenheit:

\[T = f(x)\]

Consider two quantities:

1. displacement along the $x$-axis between positions (i.e. $\Delta x = x^2 - x^1$)
2. the derivative of $T$ with respect to $x$ at some $x^0$ (i.e. the gradient or the slope)

What happens to each quantity if we rescale the $x$-axis (i.e. change the coordinate system) to be in terms of kilometers instead of meters (i.e. scale the $x$-axis by 1000)?

1. any displacement will shrink by 1000 (i.e. will change opposite of the $x$-axis)
2. the gradient will become steeper by a factor of 1000 (i.e. will change in the same way as the $x$-axis) since, for example, 5 degrees per meter is the same as 5000 degrees per kilometer.

Hence displacement is a contravariant quantity and the gradient is a covariant quantity.

Some conventions:

1. Covariant vectors are written as rows¹ of components with subscripts: $(\alpha_1, \alpha_2, \alpha_3)$

2. Contravariant vector are written as columns of components with superscripts:

   \[\begin{bmatrix} v^1 \\ v^2 \\ v^3 \end{bmatrix}\]
3. Expressions with repeated indices indicate a sum²: $\alpha_i v^i = \sum_i \alpha_i v^i$

In general covariance and contravariance characeterize how components of vector transform under a change of basis; let $V$ be a vector space and let $\bm{e} := \left(\bm{x}_1, \dots, \bm{x}_n\right)$ and $\bm{e}’ := \left(\bm{y}_1, \dots, \bm{y}_n\right)$ be bases for $V$. Then $\bm{e}’$ is related to $\bm{e}$ (and vice-versa) by a change of basis i.e right multiplication by a change of basis matrix $A := \left(a_j^i\right)$

\[\begin{aligned} \bm{e}' &= \bm{e} \cdot A \\ &= \left(a_1^i \bm{x}_i, \dots, a_n^i \bm{x}_i\right) \\ \end{aligned}\]

or terms of components $\bm{y}_j = a_j^i \bm{x}_i$.

Contravariance

Let \(v = v_{\bm{e}}^i \cdot \bm{x}_{i}\) be a vector where $v_{\bm{e}}^i$ are the components of the vector relative to the $\bm{e}$ basis.

If we write the components of $v$ as a column vector

\[\bm{v}_{\bm{e}} = \begin{bmatrix} v_{\bm{e}}^i \\ \vdots \\ v_{\bm{e}}^n \\ \end{bmatrix}\]

we can omit indices and write

\[v = v_{\bm{e}}^i \cdot \bm{x}_i = \bm{e} \cdot \bm{v}_{\bm{e}}\]

or in the $\bm{e}’$ basis

\[v = \bm{e}' \cdot \bm{v}_{\bm{e'}}\]

Equating the two and substituting $\bm{e}’ = \bm{e} \cdot A$

\[\begin{aligned} \bm{e} \cdot \bm{v}_{\bm{e}} &= \bm{e}' \cdot \bm{v}_{\bm{e}'} \\ &= (\bm{e} \cdot A ) \cdot \bm{v}_{\bm{e}'} \end{aligned}\]

Comparing both sides we see that (since $A$ is a change of basis and therefore invertible)

\[\bm{v}_{\bm{e}'} = A^{-1} \cdot \bm{v}_{\bm{e}}\]

where $A^{-1} := (\tilde{a}_j^i)$ is the inverse of the change of basis.

In components

\[v_{\bm{e}'}^i = \tilde{a}_j^i v_{\bm{e}}^j\]

This then is the precise meaning of contravariance; because the components transform with the inverse of the change of basis $A^{-1}$ the components are said to transform contravariantly and the vector $v$ is said to a contravariant vector.

Covariance

A linear functional $\alpha$ on the vector space $V$ is a linear map from $V$ to $\mathbb{R}$. Such a functional is represented in terms of its components with respect to the dual basis $\bm{f} := \left(\bm{u}^1, \dots, \bm{u}^n\right)$ (note the upper indices) which are defined by their action on the basis $\bm{e}$

\[\bm{u}^i (\bm{x}_j) = \delta_j^i\]

where $\delta_j^i$ the Kronecker delta is defined

\[\delta_j^i := \begin{cases} 1 \text{ if } i = j \\ 0 \text{ otherwise} \end{cases}\]

The components of $\alpha$ with respect to the dual basis $\bm{f}$ can be seen to be $\alpha_i^{\bm{f}} := \alpha(\bm{x}_i)$. Naturally, under a change of basis for $V$, i.e. $\bm{e} \rightarrow \bm{e}’$, the components of $\alpha$ also transform due to the change of basis $\bm{f} \rightarrow \bm{f}’$:

\[\begin{aligned} \alpha_i^{\bm{f}'} &:= \alpha(\bm{y}_i) \\ &\;= \alpha( a_i^j \bm{x}_j) \\ &\;= a_i^j \alpha( \bm{x}_j) \\ &\;= a_i^j \alpha_j^{\bm{f}} \end{aligned}\]

Define $\bm{\alpha}^{\bm{f}} := \left(\alpha_1^{\bm{f}}, \dots, \alpha_n^{\bm{f}} \right)$ to be the row vector of components of $\alpha$ and we can write the above index free as $\bm{\alpha}^{\bm{f}’} = \bm{\alpha}^{\bm{f}} \cdot A$. This then is the precise meaning of covariance; because the components of linear functionals transform with $A$ (rather than $A^{-1}$) they are said to transform covariantly and $\alpha$ is said to be a covariant vector or just covector.

Manifolds

Let $M$ be a $n$-dimensional manifold with an atlas of coordinate charts, i.e. overlapping coordinate systems. For a particular coordinate system ${x^1, \dots, x^n}$ the tangent space $TM_p$ at a point $p \in M$ has a basis comprised by partial derivatives with respect to those coordinate functions:

\[\bm{e} = \left\{ \pd{x^1}, \dots, \pd{x^n} \right\}\]

In this context a basis is called a frame.

Tangent vectors

If

\[\bm{e}' = \left\{ \pd{y^1}, \dots, \pd{y^n} \right\}\]

is another frame then the frame are related by the Jacobian

\[\mathbf{J} := \begin{bmatrix}{\dfrac {\partial y_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial y_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial y_{n}}{\partial x_{1}}}&\cdots &{\dfrac {\partial y_{n}}{\partial x_{n}}}\end{bmatrix}\]

through its inverse

\[\bm{e}' = \bm{e} \cdot \mathbf{J}^{-1}\]

This is evident by considering the chain rule

\[\pdd{f}{y^i} = \sum_i \pdd{f}{x^j} \pdd{x^j}{y^i}\]

or

\[\pd{y^i} = \pdd{x^j}{y^i}\pd{x^j}\]

It bears repeating: the basis for the tangent space transform with $\mathbf{J}^{-1}$. A tangent vector $v$ is a linear combination of coordinate partials

\[v = v_{\bm{e}}^i \pd{x^i} = \bm{e} \cdot \bm{v}_{\bm{e}}\]

Under a change in coordinate system $\bm{e} \rightarrow \bm{e}’$

\[\bm{e} \cdot \bm{v}_{\bm{e}} = \bm{e}' \cdot \bm{v}_{\bm{e}'} = \left( \bm{e} \cdot \mathbf{J}^{-1} \right) \cdot \bm{v}_{\bm{e}'}\]

and hence

\[\bm{v}_{\bm{e}'} = \mathbf{J} \cdot \bm{v}_{\bm{e}}\]

i.e. the components of $\bm{v}_{\bm{e}}$ transform contravariantly.

Differentials

For a particular coordinate system ${x^1, \dots, x^n}$ the cotangent space $TM_p^$ at a point $p \in M$ has a dual basis comprised by differentials (also called *1-forms) \(\;\bm{f} = \left\{\d x^1, \dots ,\d x^n \right\}\) defined by the relation

\[\d x^i \left(\pd{x_j}\right) = \delta_j^i\]

The dual basis elements transform contravariantly:

\[\d y^i = \pdd{y^i}{x^j} \d x^j\]

In opposition to the dual basis, the components of a 1-form $\alpha$ transform covariantly under a change of basis $\bm{f} \rightarrow \bm{f}’$ (induced by a change of basis $\bm{e} \rightarrow \bm{e}’$):

\[\begin{aligned} \alpha_i^{\bm{f}'} &:= \alpha \left(\pd{y_i}\right) \\ &\;= \alpha \left(\pdd{x^j}{y^i}\pd{x^j}\right) \\ &\;= \pdd{x^j}{y^i} \alpha \left(\pd{x^j}\right) \\ &\;= \pdd{x^j}{y^i} \alpha_j^{\bm{f}} \\ \end{aligned}\]

Hence $\bm{\alpha}^{\bm{f}’} = \bm{\alpha}^{\bm{f}} \cdot \mathbf{J}^{-1}$ and therefore 1-forms transform covariantly.

$n$-Tensors

Vectors can be combined³ to form higher rank tensors; the combination of two contravariant vectors $v,w$, sometimes denoted in a coordinate free fashion as $T := v \otimes w$ or in coordinates as $T^{mn} := v^m w^n$, is called a rank 2 tensor or 2-tensor. Contravariant and covariant vectors can also be combined; the combination of a covariant $v$ and a contravariant $w$, denoted $T_m^n := v_m w^n$ is called a rank 2 mixed tensor. Note that the dimension of the tensor space is the product of the dimensions of the factor spaces, i.e. $\dim(V \otimes W) = \dim(V) \times \dim(W)$ but the arity of the tensor is the number of factor spaces. The preceding constructions naturally extend to higher arities. Incidentally a scalar is a rank 0.

Higher rank tensors transforms according to how their factors transform

\[\begin{aligned} T^{mn}_{\bm{e}'} &:= v_{\bm{e}'}^m w_{\bm{e}'}^n \\ &\;= \sum_{i} \pdd{y^m}{x^i} v_{\bm{e}}^i \sum_{j} \pdd{y^n}{x^j} w_{\bm{e}}^j \\ &\;= \sum_{ij} \pdd{y^m}{x^i} \pdd{y^n}{x^j} v_{\bm{e}}^i w_{\bm{e}}^j \\ &\;= \sum_{ij} \pdd{y^m}{x^i} \pdd{y^n}{x^j} T^{ij}_{\bm{e}} \\ \end{aligned}\]

and so we see that $T^{ij}_{\bm{e}}$ transforms with two applications of $\mathbf{J}$ and so it is twice contravariant (hence the two contravariant indices). Analogously a covariant rank 2 tensor transforms with two applications of $\mathbf{J}^{-1}$

\[T_{mn}^{\bm{e}'} = \pdd{x^i}{y^m} \pdd{x^j}{y^n} T_{ij}^{\bm{e}}\]

Metric tensors

A metric tensor (or just metric) is a bilinear form⁴ $g: V \times V \rightarrow K$. Since 1-forms map vectors to numbers it should be no surprise that a metric is the tensor product of 1-forms:

\[g := g_{mn} \d x^m \d x^n \\\]

For intuition consider $n$-dimensional Pythagoras’ theorem

\[\begin{aligned} \d s^2 &:= \left(\d x^1\right)^2 + \left(\d x^2\right)^2 + \cdots + \left(\d x^n\right)^2 \\ &\;= \d x^m \d x^m \\ &\;= \d x^m \d x^n \delta_{mn} \\ \end{aligned}\]

where $\delta_{mn}$ is called the flat metric (non-zero terms only on the diagonal); it’s exactly the non-flat metrics that are interesting in GR (non-zero terms off the diagonal).

How does the metric transform under a change of coordinates? Like a rank 2 covariant tensor:

\[g_{mn}^{\bm{e}'} = \pdd{x^i}{y^m} \pdd{x^j}{y^n} g_{ij}^{\bm{e}}\]

A metric enables identification identification of covariant and contravariant vectors: a contravariant vector $v$ uniquely determines a covariant vector $\alpha$ via the metric

\[\alpha(\cdot) := g (v, \cdot)\]

Conversely each a covariant $\alpha$ uniquely determines a contravariant $v$.

As a result of this identification we can speak of the covariant components or contravariant components of a vector and one can pass between them using the musical isomorphism: Given a vector $v = v^i \bm{x_i}$ , we define its flat by

\[v^{\flat } := (g_{ij}v^{i})\mathbf {x}^{j}=v_{j}\mathbf{x} ^{j}\]

where $g = g_{ij} x^i \otimes x^j$ is 2-covariant metric that is symmetric and nondegenerate. This is referred to as “lowering an index”. In the same way, given a covector field $\alpha = \alpha_i \bm{u}^i$ , we define its sharp by

\[\alpha^{\sharp }:=(g^{ij}\alpha_{i})\bm {u} _{j}=\alpha^{j}\bm{u}_{j}\]

where $g^{ij}$ are the components of the inverse metric tensor (given by the entries of the inverse matrix to $g_{ij}$). Taking the sharp of a covector field is referred to as “raising an index”.

Foonotes