2.2. Vectors and tensors¶

To introduce the concept of vectors and tensors, it is worth considering how a derivative is defined on a manifold.

Given an open subset \(\mathscr{U} \subset \mathcal{M}\), a function \(f: \mathscr{U} \rightarrow \mathbb{R}\) and the curve given by \(Q: T \rightarrow \mathscr{U}\), \(T \subset \mathbb{R}\) , along with the composition \(f \circ Q\), we may examine the derivative of \(f\) a the point, imaged by \(Q(\tau=\tau_0)\), in \(\mathscr{U}\) by considering

\[\begin{split}\left. \left( \frac{\partial f}{\partial \tau} \right)_{Q(\tau)} \right\rvert_{\tau=\tau_0} =& \lim_{\epsilon \rightarrow 0} \Big( \frac{f\left( Q(\tau_0 + \epsilon) \right) - f \left( Q(\tau_0) \right)}{\epsilon} \Big), \\ =& \left. \sum_{i=0}^{n} \frac{\text{d}x^i(\tau)}{\text{d}\tau} \right\rvert_{\tau_0} \left( \frac{\partial f}{\partial x^i} \right)_{Q(t)},\end{split}\]

with the \(x^i\) being the coordinates of the chart \((\mathscr{U}, \phi_\alpha)\). By adopting the summation convention, this may be written compactly as

\[= \left. \left( \frac{\text{d}x^i}{\text{d}\tau} \frac{\partial f}{\partial x^i} \right) \right\rvert_{\tau_0}.\]

2.2.1. Tangent spaces and tangent vectors¶

Definition: Tangent space

The tangent space is an \(n\) dimensions vector space defined at a point \(x \in \mathscr{U}\), where \(\mathscr{U}\) is an open subset of \(n\)-dimensional manifold \(\mathcal{M}\), by the set

(2.1)¶\[\begin{split}T_x \mathscr{U} :=& \ \left\{ x \right\} \times \mathbb{R}^n, \\ =& \ \Big\{ (x, \xi) \ | \ \xi \in \mathbb{R}^n \Big\}.\end{split}\]

The tangent space has linear structure

\[a\left( x, \xi \right) + b\left( x, \eta \right) = \left( x, a \xi + b \eta \right),\]

for \(a,b \in \mathbb{R}\). These tangent spaces are bound to their point of definition in \(\mathscr{U}\), and elements of \(T_x \mathscr{U}\) and \(T_y \mathscr{U}\), \(x \neq y\) live in entirely separate spaces.

By considering various curves \(Q: T \rightarrow \mathscr{U}\), \(T \subset \mathbb{R}\), passing through \(x \in \mathscr{U}\) we may examine tangents to the curve \(Q\) at \(x\): each point along a curve \(Q(\tau)\) defines a velocity vector \(\frac{\text{d}Q}{\text{d}\tau} \in \mathbb{R}^n\), which measures how the point \(Q(\tau)\) changes tangentially with respect to each coordinate of \(\mathscr{U}\) as \(\tau\) changes. It is a velocity in the sense of being a first derivative.

Definition: Tangent vector

A tangent vector is a vector tangent to a point \(Q(\tau) \in \mathscr{U}\) along the curve \(Q\)

(2.2)¶\[\begin{split}\dot{Q}(\tau) &:= \left( Q(\tau), \frac{\text{d}Q}{\text{d}\tau} \right) \in T_{Q(\tau)} \mathscr{U} \\ &= \frac{\text{d}Q^i}{\text{d} \tau} \left. b_i \right\rvert_{Q(\tau)},\end{split}\]

for \(i = 1, ..., n\), where \(\frac{\text{d}Q}{\text{d} \tau} \in \mathbb{R}^n\), and \( \left. b_i \right\rvert_{Q(\tau)} \) forms a basis spanning \(T_x \mathscr{U}\).

We will further explore \(\left. b_i \right\rvert_{Q(\tau)}\) when we discuss frames in the next section, which will connect \(\left. b_i \right\rvert_{Q(\tau)}\) with the \((x, \xi)\) of the tangent space definition eq. (2.1).

This definition admits a local parameterization of the curves \(Q\) by considering

\[x^i \left( x + Q(\tau) \right) = x^i(x) + \frac{\text{d}Q^i}{\text{d} \tau} \tau,\]

over a small interval \(-\varepsilon < \tau < \varepsilon\) .

We define a smooth mapping \(v_\alpha\), called a vector field, as

\[\begin{split}v_\alpha : \mathscr{U} & \rightarrow T_x \mathscr{U}, \\ x & \mapsto (x, \xi(x)) = v(x).\end{split}\]

By considering a curve \(Q: T \rightarrow \mathscr{U}\), \(T \subset \mathbb{R}^p\), such that \(\tau\) now has \(p\) components, we have a \(n \times p\) matrix representation

\[\begin{split}\left( \frac{\partial Q^i}{\partial \tau^j} \right)_{\substack{i = 1, \dots, n \\ j = 1, \dots, p}},\end{split}\]

with rank \(p\) everywhere on \(T\). The tangent space at \(Q(\tau)\) are \(p\) linearly independent vector fields on \(\mathscr{U}\), spanning \(T_{Q(\tau)} \mathscr{U}\) :

\[Q(\tau) \mapsto \left( Q(\tau), \frac{\partial Q}{\partial \tau^j} \right).\]

Such a set of vector fields may be decomposed onto a frame, which we will now take a moment to define.

2.2.2. Tangent mapping¶

Maps between open subsets have an analog in the tangent space.

Definition: Tangent mapping

Let \(\mathscr{U}\) and \(\mathscr{V}\) be open subsets of \(\mathcal{M} \subset \mathbb{R}^n\) and \(\mathcal{K} \subset \mathbb{R}^m\) respectively, and let \(F\) be a map \(F : \mathscr{U} \rightarrow \mathscr{V}\), then the tangent mapping is defined

\[T_x F: T_x \mathscr{U} \rightarrow T_{F(x)} \mathscr{U}.\]

Consider again the curve \(Q : T \rightarrow \mathscr{U}\), with \(T \subset \mathbb{R}\), then we also have the composition \(F \circ Q : T \rightarrow \mathscr{V}\). Let \(x^i\) and \(y^j\) be the coordinates of \(\mathscr{U}\) and \(\mathscr{V}\), then, given the tangent vector in eq. (2.2), we may write

\[\left( Q(\tau), \frac{\text{d}Q}{\text{d}\tau} \right) = \frac{\text{d}Q^i}{\text{d}\tau} \left. \frac{\partial}{\partial x^i} \right\rvert_{Q(\tau)},\]

which is an element of \(T_{Q{\tau}} \mathscr{U}\). The composition \(F \circ Q\) is then decomposed

\[\left( \left( F \circ Q \right)(\tau), \frac{\text{d}}{\text{d}\tau} \left( F \circ Q \right) \right) = \frac{\text{d}}{\text{d}\tau} \left( F \circ Q \right)^j \left. \frac{\partial}{\partial y^j} \right\rvert_{F(Q(\tau))},\]

now an element of \(T_{F(Q(\tau))} \mathscr{V}\) . The tangent map \(T_{Q(\tau)}F\) must then map between these two spaces; for some vector \(\dot{u} \in T_x \mathscr{U}\),

\[\dot{u} = \dot{u}^i \frac{\partial}{\partial x^i},\]

then

\[\left( T_x F \right) \dot{u} = \sum_j^m \left( \sum_i^n \dot{u}^i \frac{\partial}{\partial x^i} F^j \right) \left. \frac{\partial}{\partial y^j} \right\rvert_{F(x)},\]

is the action of the tangent mapping, as by the chain rule we know

\[\frac{\text{d}}{\text{d} \tau} \left( F \circ Q \right)^j = \left. \frac{\partial F^j}{\partial x^i} \right\rvert_{Q(\tau)} \frac{\text{d}Q^i}{\text{d} \tau}.\]

2.2.3. Frames¶

Definition: Frame

An \(n\)-frame is the set of linearly-independent vector fields \(b_i\)

\[\begin{split}b_i : \mathscr{U} &\rightarrow T_{x}\mathscr{U}, \\ x & \mapsto b_i(x),\end{split}\]

where the \(b_i(x)\) form a basis of the co-domain, and \(i = 1, ..., n\).

If \(x^i\) are identified as the coordinates of \(\mathscr{U}\), then the frame is denoted

\[\left\{ \frac{\partial}{\partial x^i} \right\},\]

which act on points \(x \in \mathscr{U}\) to obtain tangent vectors

\[\frac{\partial}{\partial x^i} (x) := \left( x, \frac{\partial x}{\partial x^i} \right) \in T_{x}\mathscr{U}.\]

The transformation between frames \(b_i\) and \(b^\prime_i\) is

(2.3)¶\[b^\prime_j(x) = \left( \gamma^{-1}(x) \right)^i_{\phantom{i}j} \ b_i(x),\]

with \(\gamma(x) \in \text{GL}_n\). Note that \(\gamma\) is a gauge-group-valued function on \(\mathscr{U}\). The transformation between frames associated with coordinates \(x^i\) and \(y^i\) is

\[\left. \frac{\partial}{\partial y^j} \right\rvert_x = \frac{\partial x^i}{\partial y^j} \left. \frac{\partial}{\partial x^i} \right\rvert_x,\]

identifying \(\left( \gamma^{-1}(x) \right)^i_{\phantom{i}j} = \frac{\partial x^i}{\partial y^j}\).

Any set of coordinates \(x^i\) on \(\mathscr{U}\) determines an \(n\)-frame, but there are generally no coordinates determining a frame coinciding with a given one.

Each of the \(\frac{\partial}{\partial x^i}\) is tangential to the coordinate line of \(x^i\). For the case of Cartesian coordinates \(x^i\), we can express the n-frame

\[\begin{split}\frac{\partial}{\partial x^1} (x) &= (x, (1, 0, ..., 0)), \\ \frac{\partial}{\partial x^2} (x) &= (x, (0, 1, ..., 0)), \\ \vdots & \\ \frac{\partial}{\partial x^n} (x) &= (x, (0, 0, ..., 1)).\end{split}\]

Every vector field \(v\) may be decomposed in terms of a frame \(b_i\)

\[v(x) = v^i(x) b_i(x),\]

where \(v^i\) are smooth functions to \(\mathbb{R}\). For the tangent vector to a curve \(Q(\tau)\) with coordinates \(x^i\), this decomposition, along with the chain-rule, expresses

\[\dot{Q}(\tau) = \frac{\text{d}Q^i}{\text{d}\tau} \frac{\partial}{\partial x^i} (Q (\tau) ).\]

For example, \(n=2\) in Cartesian coordinates:

\[\begin{split}\dot{Q}(\tau) &= \frac{\text{d}Q^1}{\text{d}\tau} \frac{\partial}{\partial x^1} (Q (\tau) ) + \frac{\text{d}Q^2}{\text{d}\tau} \frac{\partial}{\partial x^2} (Q (\tau) ), \\ &= \frac{\text{d}Q^1}{\text{d}\tau} \left( Q(\tau), \left( \frac{ \partial Q(\tau) }{ \partial x^1 }, 0 \right) \right) + \frac{\text{d}Q^2}{\text{d}\tau} \left( Q(\tau), \left( 0, \frac{ \partial Q(\tau) }{ \partial x^2 } \right) \right), \\ &= \left( Q(\tau), \left( \frac{\text{d}Q^1}{\text{d}\tau} \frac{ \partial Q(\tau) }{ \partial x^1 }, \frac{\text{d}Q^2}{\text{d}\tau} \frac{ \partial Q(\tau) }{ \partial x^2 } \right) \right) \in T_{Q(\tau)} \mathscr{U}.\end{split}\]

2.2.4. Connecting tangent spaces with contra- and covariant vectors¶

We have encountered the notion of a contravariant vector in defining tangent vectors, considered as directional derivatives with respect to a frame. The space of contravariant vectors may be thought of as the space of all directions (the tangent space) at a point \(x \in \mathscr{U}\). On the other hand, covariant vectors, sometimes called cotangents, are linear mappings

\[\phi : \ \ T_x \mathscr{U} \rightarrow \mathbb{R}.\]

We will later identify these maps with the far broader concept of differential forms (see Forms), and covariant vectors specifically as one-forms, but for now, we will consider them as the dual space of the tangent space \(T_x \mathscr{U}\) , often denoted as \(T^\ast_x \mathscr{U}\) or \(\Lambda^1 T_x \mathscr{U}\). We will write the basis of the cotangent space \(\beta^i\), with the decomposition

\[\phi = \phi_i \beta^i,\]

where \(\phi_i \in \mathbb{R}\).

From our knowledge of vector spaces, we may already observe

(2.4)¶\[\beta^i(b_j) = \delta^i_{\phantom{i}j},\]

and therefore identify the transformation between dual bases as

(2.5)¶\[\beta^{\prime i} = \gamma^i_{\phantom{i} j} \beta^j,\]

where the matrix \(\gamma\) is the inverse of the \(\gamma^{-1}\) seen in eq. (2.3), which can be easily verified with eq. (2.4).

Notation

There exists often some confusion over the use of the terminology covariant and contravariant, and how it relates to the symbols we discuss. In physics, these words describe how the vector components change when the basis changes.

For example, the vector

\[v = v^i b_i,\]

is a contravariant vector; were the length of the basis \(b_i\) doubled, the vector components would have to be halved, changing in the opposite (contra-) direction. We can then express the covariant vector through

\[v_i = v \cdot b_i,\]

or

\[v = v_i \beta^i,\]

where it is apparent that the covariant vector varies along with (co-) the basis \(b_i\).

2.2.5. Tensors¶

Given a point \(x\) in an open subspace \(\mathscr{U} \subset \mathbb{R}^n\), with tensor space \(T_x \mathscr{U}\) and cotangent space \(T^\ast_x \mathscr{U}\), we may define the cartesian product

\[\Pi^s_{\phantom{s}r} = \underbrace{ \left( T^\ast_x \mathscr{U} \right) \times \dots \times \left( T^\ast_x \mathscr{U} \right) }_{r \ \text{many}} \times \underbrace{ \left( T_x \mathscr{U} \right) \times \dots \times \left( T_x \mathscr{U} \right) }_{s \ \text{many}}.\]

We then define a tensor of type \((r, s)\) as the map

\[T: \Pi^s_{\phantom{s}r} \rightarrow \mathbb{R}.\]

The maps \(T\) are said to be multilinear, i.e. linear in each argument, and span a vector-space of dimension \(n^{r+s}\).

Definition: Tensor space

The space of tensors is the space of tensor products, at a point \(x \in \mathscr{U}\),

\[\Theta^r_{\phantom{r}s} = \underbrace{ \left (T_{x} \mathscr{U} \right) \otimes \dots \otimes \left (T_{x} \mathscr{U} \right) }_{r \ \text{many}} \otimes \underbrace{ \left (T^\ast_{x} \mathscr{U} \right) \otimes \dots \otimes \left (T^\ast_{x} \mathscr{U} \right) }_{s \ \text{many}}.\]

We can identify the basis of a tensor in the tensor space by considering the action of the map \(T\) for \(\phi^i \in T^\ast_x \mathscr{U}\) and \(\dot{x}^i \in T_x \mathscr{U}\)

\[\begin{split}T(\phi^1, ..., \phi^r, \dot{x}_1, \dots, \dot{x}_s) & = T( \phi^1_{\phantom{1}i_1} \beta^{i_1}, \dots, \phi^r_{\phantom{r}i_r} \beta^{i_r}, \dot{x}_1^{\phantom{1}j_1} b_{j_1}, \dots, \dot{x}_s^{\phantom{s}j_s} b_{j_s} ) \\ & = \phi^1_{\phantom{1}i_1}, \dots, \phi^r_{\phantom{r}i_r}, \dot{x}_1^{\phantom{1}j_1} \dots, \dot{x}_s^{\phantom{s}j_s} T( \beta^{i_1}, \dots, \beta^{i_r}, b_{j_1}, \dots, b_{j_s} ),\end{split}\]

letting

\[T( \beta^{i_1}, \dots, \beta^{i_r}, b_{j_1}, \dots, b_{j_s} ) = T^{i_1, \dots, i_r}_{\phantom{i_1, \dots, i_r} j_1, \dots, j_s},\]

and

(2.6)¶\[e_{i_1, \dots, i_r}^{\phantom{i_1, \dots, i_r} j_1, \dots, j_s} \left( \beta^{k_1}, \dots, \beta^{k_r}, b_{l_1}, \dots, b_{l_s} \right) = { \delta^{k_1}_{\phantom{k_1} i_1} \dots \delta^{k_r}_{\phantom{k_r} i_r} \delta^{j_1}_{\phantom{j_1} l_1} \dots \delta^{j_s}_{\phantom{j_s} l_s} },\]

allows the identification of the decomposition of \(T\) as

\[T = T^{i_1, \dots, i_r}_{\phantom{i_1, \dots, i_r} j_1, \dots, j_s} e_{i_1, \dots, i_r}^{\phantom{i_1, \dots, i_r} j_1, \dots, j_s} .\]

The \(e_{i_1, \dots, i_r}^{\phantom{i_1, \dots, i_r} j_1, \dots, j_s}\) are linearly independent, necessitated by eq. (2.6), they provide a basis for tensors of type \((r, s)\). Similarly, it should be apparent that the form of this basis is the direct product

\[e_{i_1, \dots, i_r}^{\phantom{i_1, \dots, i_r} j_1, \dots, j_s} = b_{i_1} \otimes \dots \otimes b_{i_r} \otimes \beta^{j_1} \otimes \dots \otimes \beta^{j_s}.\]

The \(T^{i_1, \dots, i_r}_{\phantom{i_1, \dots, i_r} j_1, \dots, j_s} \in \mathbb{R}\) are the components of the tensor relative to the chosen basis. The change of basis for the basis vectors of a tensor follows from eq. (2.3) and eq. (2.5), which implies the transformation of the components

\[T^{i^\prime_1, \dots, i^\prime_r}_{\phantom{i^\prime_1, \dots, i^\prime_r} j^\prime_1, \dots, j^\prime_s} = { \left( \gamma^{-1} \right)^{i^\prime_1}_{\phantom{i^\prime_1}i_1} \dots \left( \gamma^{-1} \right)^{i^\prime_r}_{\phantom{i^\prime_r}i_r} \gamma^{j_1}_{\phantom{j_1}j^\prime_1} \dots \gamma^{j_s}_{\phantom{j_s}j^\prime_s} } T^{i_1, \dots, i_r}_{\phantom{i_1, \dots, i_r} j_1, \dots, j_s}.\]

2.2.5.1. Tensor symmetry¶

Definition: symmetric or antisymmetric tensor

A tensor of type \((r, s)\) is said to be symmetric or antisymmetric in its contravariant indices if under the exchange of indices

\[T^{i_1, \dots, i_l, \dots, i_m, \dots, i_r}_{\phantom{i_1, \dots, i_l, \dots, i_m, \dots, i_r} j_1, \dots, j_s} = \pm T^{i_1, \dots, i_m, \dots, i_l, \dots, i_r}_{\phantom{i_1, \dots, i_m, \dots, i_l, \dots, i_r} j_1, \dots, j_s},\]

The same property may apply to the covariant indices.

The special class of totally antisymmetric tensors are the \((0, p)\) covariant tensors, which are antisymmetric under every exchange of indices. That is

\[T_{i_1, \dots, i_l, \dots, i_m, \dots, i_p} = - T_{i_1, \dots, i_m, \dots, i_l, \dots, i_p}.\]

Any \((0, s)\) tensor may be made totally antisymmetric by applying an alternating operator \(\mathcal{A}\)

\[\mathcal{A} T_{i_1, \dots, i_p} = \frac{1}{p!} \sum_{\pi \in S_p} \text{sgn}(\pi) \ T_{\pi(i_1), \dots, \pi(i_p)},\]

where \(S_p\) is the set of \(p\) permutations (Wikipedia).

The properties of this operator are

\(\mathcal{A} T = T\) if \(T\) already totally antisymmetric,
\(\mathcal{A} T_{i_1, \dots, i_p} = 0 \ \forall \ p > n\).

These totally antisymmetric tensors are called \(p\)-forms, and will be the subject of study in the next section.

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