7. Tensor calculus

7.1 Vectors and covectors

A finite dimensional vector space is denoted by $\cal V, W$. The vector space of linear transformations from $\cal V$ to $\cal W$ is denoted by $\cal L(V,W)$. Consider ${\cal L(V,}I\hspace{-1mm}R):={\cal V}^*$. We name $\cal V^*$ the dual space of $\cal V$. Now we can define vectors in $\cal V$ with basis $\vec{c}$ and covectors in $\cal V^*$ with basis $\hat{\vec{c}}$. Properties of both are:

  1. Vectors: $\vec{x}=x^i\vec{c}_i$ with basis vectors $\vec{c}_i$: \[ \vec{c}_i=\frac{\partial}{\partial x^i} \] Transformation from system $i$ to $i'$ is given by: \[ \vec{c}_{i'}=A_{i'}^i\vec{c}_i=\partial_i\in{\cal V}~~,~~x^{i'}=A_i^{i'}x^i \]
  2. Covectors: $\hat{\vec{x}}=x_i\hat{\vec{c}}^{~i}$ with basis vectors $\hat{\vec{c}}^{~i}$ \[ \hat{\vec{c}}^{~i}=dx^i \] Transformation from system $i$ to $i'$ is given by: \[ \hat{\vec{c}}^{~i'}=A_i^{i'}\hat{\vec{c}}^{~i}\in{\cal V}^*~~,~~\vec{x}_{i'}=A_{i'}^i\vec{x}_i \]

Here the Einstein convention is used: \[ a^ib_i:=\sum_ia^ib_i \] The coordinate transformation is given by: \[ A_{i'}^i=\frac{\partial x^i}{\partial x^{i'}}~~,~~A_i^{i'}=\frac{\partial x^{i'}}{\partial x^i} \] From this follows that $A_k^i\cdot A_l^k=\delta_l^k$ and $A_{i'}^i=(A_i^{i'})^{-1}$.

In differential notation the coordinate transformations are given by: \[ dx^i=\frac{\partial x^i}{\partial x^{i'}}dx^{i'}~~~\mbox{and}~~~\frac{\partial}{\partial x^{i'}}=\frac{\partial x^i}{\partial x^{i'}}\frac{\partial}{\partial x^i} \] The general transformation rule for a tensor $T$ is: \[ T_{s_1...s_m}^{q_1...q_n}=\left|\frac{\partial\vec{x}}{\partial\vec{u}}\right|^\ell \frac{\partial u^{q_1}}{\partial x^{p_1}}\cdots\frac{\partial u^{q_n}}{\partial x^{p_n}}\cdot\frac{\partial x^{r_1}}{\partial u^{s_1}}\cdots\frac{\partial x^{r_m}}{\partial u^{s_m}}T_{r_1...r_m}^{p_1...p_n} \] For an absolute tensor $\ell=0$.

7.2 Tensor algebra

The following holds: \[ a_{ij}(x_i+y_i)\equiv a_{ij}x_i+a_{ij}y_i,~~~\mbox{but:}~~a_{ij}(x_i+y_j)\not\equiv a_{ij}x_i+a_{ij}y_j \] and \[ (a_{ij}+a_{ji})x_ix_j\equiv2a_{ij}x_ix_j,~~~\mbox{but:}~~(a_{ij}+a_{ji})x_iy_j\not\equiv 2a_{ij}x_iy_j \] en $(a_{ij}-a_{ji})x_ix_j\equiv0$.

The sum and difference of two tensors is a tensor of the same rank: $A_q^p\pm B_q^p$. The outer tensor product results in a tensor with a rank equal to the sum of the ranks of both tensors: $A_q^{pr}\cdot B_s^m=C_{qs}^{prm}$. The contraction equals two indices and sums over them. Suppose we take $r=s$ for a tensor $A_{qs}^{mpr}$, this results in: $\sum\limits_r A_{qr}^{mpr}=B_q^{mp}$. The inner product of two tensors is defined by taking the outer product followed by a contraction.

7.3 Inner product

Definition: the bilinear transformation $B:{\cal V}\times{\cal V}^*\rightarrow I\hspace{-1mm}R$, $B(\vec{x},\hat{\vec{y}}\,)=\hat{\vec{y}}(\vec{x})$ is denoted by $<\vec{x},\hat{\vec{y}}\,>$. For this pairing operator $<\cdot,\cdot>=\delta$ holds: \[ \hat{\vec{y}}(\vec{x})=<\vec{x},\hat{\vec{y}}>=y_ix^i~~~,~~~<\hat{\vec{c}^{~i}},\vec{c}_j>=\delta_j^i \] Let $G:{\cal V}\rightarrow{\cal V}^*$ be a linear bijection. Define the bilinear forms \begin{eqnarray*} g:{\cal V\times V}\rightarrow I\hspace{-1mm}R&~~~&g(\vec{x},\vec{y})=<\vec{x},G\vec{y}>\\ h:{\cal V^*\times V^*}\rightarrow I\hspace{-1mm}R&~~~&h(\hat{\vec{x}},\hat{\vec{y}}\,)= \end{eqnarray*} Both are not degenerated. The following holds: $h(G\vec{x},G\vec{y})=<\vec{x},G\vec{y}>=g(\vec{x},\vec{y})$. If we identify $\cal V$ and $\cal V^*$ with $G$, than $g$ (or $h$) gives an inner product on $\cal V$.

The inner product $(,)_\Lambda$ on $\Lambda^k(\cal V)$ is defined by: \[ (\Phi,\Psi)_\Lambda=\frac{1}{k!}(\Phi,\Psi)_{T^0_k(\cal V)} \] The inner product of two vectors is than given by: \[ (\vec{x},\vec{y})=x^iy^i<\vec{c}_i,G\vec{c}_j>=g_{ij}x^ix^j \] The matrix $g_{ij}$ of $G$ is given by \[ g_{ij}\hat{\vec{c}}^{~j}=G\vec{c}_i \] The matrix $g^{ij}$ of $G^{-1}$ is given by: \[ g^{kl}\vec{c}_l=G^{-1}\hat{\vec{c}}^{~k} \] For this metric tensor $g_{ij}$ holds: $g_{ij}g^{jk}=\delta_i^k$. This tensor can raise or lower indices: \[ x_j=g_{ij}x^i~~~,~~~x^i=g^{ij}x_j \] and $du^i=\hat{\vec{c}}^{~i}=g^{ij}\vec{c}_j$.

7.4 Tensor product

Definition: let $\cal U$ and $\cal V$ be two finite dimensional vector spaces with dimensions $m$ and $n$. Let $\cal U^*\times V^*$ be the cartesian product of $\cal U$ and $\cal V$. A function $t:{\cal U^*\times V^*}\rightarrow I\hspace{-1mm}R$; $(\hat{\vec{u}};\hat{\vec{v}}\,)\mapsto t(\hat{\vec{u}};\hat{\vec{v}}\,)=t^{\alpha\beta}u_\alpha u_\beta\in I\hspace{-1mm}R$ is called a tensor if $t$ is linear in $\hat{\vec{u}}$ and $\hat{\vec{v}}$. The tensors $t$ form a vector space denoted by $\cal U\otimes V$. The elements $T\in\cal V\otimes V$ are called contravariant 2-tensors: $T=T^{ij}\vec{c}_i\otimes\vec{c}_j=T^{ij}\partial_i\otimes\partial_j$. The elements $T\in\cal V^*\otimes V^*$ are called covariant 2-tensors: $T=T_{ij}\hat{\vec{c}}^{~i}\otimes\hat{\vec{c}}^{~j}=T_{ij}dx^i\otimes dx^j$. The elements $T\in\cal V^*\otimes V$ are called mixed 2 tensors: $T=T_i^{.j}\hat{\vec{c}}^{~i}\otimes\vec{c}_j=T_i^{.j}dx^i\otimes\partial_j$, and analogous for $T\in\cal V\otimes V^*$.

The numbers given by \[ t^{\alpha\beta}=t(\hat{\vec{c}}^{~\alpha},\hat{\vec{c}}^{~\beta}\,) \] with $1\leq\alpha\leq m$ and $1\leq\beta\leq n$ are the components of $t$.

Take $\vec{x}\in\cal U$ and $\vec{y}\in\cal V$. Than the function $\vec{x}\otimes\vec{y}$, definied by \[ (\vec{x}\otimes\vec{y})(\hat{\vec{u}},\hat{\vec{v}})=<\vec{x},\hat{\vec{u}}>_U<\vec{y},\hat{\vec{v}}>_V \] is a tensor. The components are derived from: $(\vec{u}\otimes\vec{v})_{ij}=u_iv^j$. The tensor product of 2 tensors is given by: \begin{eqnarray*} {2\choose0}~\mbox{form:}~&&(\vec{v}\otimes\vec{w})(\hat{\vec{p}},\hat{\vec{q}})=v^ip_iw^kq_k=T^{ik}p_iq_k\\ {0\choose2}~\mbox{form:}~&&(\hat{\vec{p}}\otimes\hat{\vec{q}})(\vec{v},\vec{w})=p_iv^iq_kw^k=T_{ik}v^iw^k\\ {1\choose1}~\mbox{form:}~&&(\vec{v}\otimes\hat{\vec{p}})(\hat{\vec{q}},\vec{w})=v^iq_ip_kw^k=T_k^iq_iw^k \end{eqnarray*}

7.5 Symmetric and antisymmetric tensors

A tensor $t\in{\cal V\otimes V}$ is called symmetric resp. antisymmetric if $\forall\hat{\vec{x}},\hat{\vec{y}}\in{\cal V^*}$ holds: $t(\hat{\vec{x}},\hat{\vec{y}}\,)=t(\hat{\vec{y}},\hat{\vec{x}}\,)$ resp. $t(\hat{\vec{x}},\hat{\vec{y}}\,)=-t(\hat{\vec{y}},\hat{\vec{x}}\,)$.

A tensor $t\in{\cal V^*\otimes V^*}$ is called symmetric resp. antisymmetric if $\forall\vec{x},\vec{y}\in{\cal V}$ holds: $t(\vec{x},\vec{y})=t(\vec{y},\vec{x})$ resp. $t(\vec{x},\vec{y})=-t(\vec{y},\vec{x})$. The linear transformations $\cal S$ and $\cal A$ in $\cal V\otimes W$ are defined by: \begin{eqnarray*} {\cal S}t(\hat{\vec{x}},\hat{\vec{y}}\,)&=&\frac{1}{2}(t(\hat{\vec{x}},\hat{\vec{y}})+t(\hat{\vec{y}},\hat{\vec{x}}\,))\\ {\cal A}t(\hat{\vec{x}},\hat{\vec{y}}\,)&=&\frac{1}{2}(t(\hat{\vec{x}},\hat{\vec{y}})-t(\hat{\vec{y}},\hat{\vec{x}}\,)) \end{eqnarray*} Analogous in $\cal V^*\otimes V^*$. If $t$ is symmetric resp. antisymmetric, than ${\cal S}t=t$ resp. ${\cal A}t=t$.

The tensors $\vec{e}_i\vee\vec{e}_j=\vec{e}_i\vec{e}_j=2{\cal S}(\vec{e}_i\otimes\vec{e}_j)$, with $1\leq i\leq j\leq n$ are a basis in $\cal S(V\otimes V)$ with dimension $\frac{1}{2} n(n+1)$.

The tensors $\vec{e}_i\wedge\vec{e}_j=2{\cal A}(\vec{e}_i\otimes\vec{e}_j)$, with $1\leq i\leq j\leq n$ are a basis in $\cal A(V\otimes V)$ with dimension $\frac{1}{2} n(n-1)$.

The complete antisymmetric tensor $\varepsilon$ is given by: $\varepsilon_{ijk}\varepsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$.

The permutation-operators $e_{pqr}$ are defined by: $e_{123}=e_{231}=e_{312}=1$, $e_{213}=e_{132}=e_{321}=-1$, for all other combinations $e_{pqr}=0$. There is a connection with the $\varepsilon$ tensor: $\varepsilon_{pqr}=g^{-1/2}e_{pqr}$ and $\varepsilon^{pqr}=g^{1/2}e^{pqr}$.

7.6 Outer product

Let $\alpha\in\Lambda^k(\cal V)$ and $\beta\in\Lambda^l(\cal V)$. Than $\alpha\wedge\beta\in\Lambda^{k+l}(\cal V)$ is defined by: \[ \alpha\wedge\beta=\frac{(k+l)!}{k!l!}{\cal A}(\alpha\otimes\beta) \] If $\alpha$ and $\beta\in\Lambda^1(\cal V)={\cal V}^*$ holds: $\alpha\wedge\beta=\alpha\otimes\beta-\beta\otimes\alpha$

The outer product can be written as: $(\vec{a}\times\vec{b})_i=\varepsilon_{ijk}a^jb^k$, $\vec{a}\times\vec{b}=G^{-1}\cdot*(G\vec{a}\wedge G\vec{b})$.

Take $\vec{a},\vec{b},\vec{c},\vec{d}\in I\hspace{-1mm}R^4$. Than $(dt\wedge dz)(\vec{a},\vec{b})=a_0b_4-b_0a_4$ is the oriented surface of the projection on the $tz$-plane of the parallelogram spanned by $\vec{a}$ and $\vec{b}$.

Further \[ (dt\wedge dy\wedge dz)(\vec{a},\vec{b},\vec{c})=\det\left|\begin{array}{ccc} a_0&b_0&c_0\\ a_2&b_2&c_2\\ a_4&b_4&c_4 \end{array}\right| \] is the oriented 3-dimensional volume of the projection on the $tyz$-plane of the parallelepiped spanned by $\vec{a}$, $\vec{b}$ and $\vec{c}$.

$(dt\wedge dx\wedge dy\wedge dz)(\vec{a},\vec{b},\vec{c},\vec{d})=\det(\vec{a},\vec{b},\vec{c},\vec{d})$ is the 4-dimensional volume of the hyperparellelepiped spanned by $\vec{a}$, $\vec{b}$, $\vec{c}$ and $\vec{d}$.

7.7 The Hodge star operator

$\Lambda^k(\cal V)$ and $\Lambda^{n-k}(\cal V)$ have the same dimension because ${n\choose k}={n\choose{n-k}}$ for $1\leq k\leq n$. Dim$(\Lambda^n({\cal V}))=1$. The choice of a basis means the choice of an oriented measure of volume, a volume $\mu$, in $\cal V$. We can gauge $\mu$ so that for an orthonormal basis $\vec{e}_i$ holds: $\mu(\vec{e}_i)=1$. This basis is than by definition positive oriented if $\mu=\hat{\vec{e}}^{~1}\wedge \hat{\vec{e}}^{~2}\wedge...\wedge \hat{\vec{e}}^{~n}=1$.

Because both spaces have the same dimension one can ask if there exists a bijection between them. If $\cal V$ has no extra structure this is not the case. However, such an operation does exist if there is an inner product defined on $\cal V$ and the corresponding volume $\mu$. This is called the Hodge star operator and denoted by $*$. The following holds: \[ \forall_{w\in\Lambda^k({\cal V})}\exists_{*w\in\Lambda^{k-n}({\cal V})}\forall_{\theta\in\Lambda^k({\cal V})}~~ \theta\wedge*w=(\theta,w)_\lambda\mu \] For an orthonormal basis in $I\hspace{-1mm}R^3$ holds: the volume: $\mu=dx\wedge dy\wedge dz$, $*dx\wedge dy\wedge dz=1$, $*dx=dy\wedge dz$, $*dz=dx\wedge dy$, $*dy=-dx\wedge dz$, $*(dx\wedge dy)=dz$, $*(dy\wedge dz)=dx$, $*(dx\wedge dz)=-dy$.

For a Minkowski basis in $I\hspace{-1mm}R^4$ holds: $\mu=dt\wedge dx\wedge dy\wedge dz$, $G=dt\otimes dt-dx\otimes dx-dy\otimes dy-dz\otimes dz$, and $*dt\wedge dx\wedge dy\wedge dz=1$ and $*1=dt\wedge dx\wedge dy\wedge dz$. Further $*dt=dx\wedge dy\wedge dz$ and $*dx=dt\wedge dy\wedge dz$.

7.8 Differential operations

7.8.1 The directional derivative

The directional derivative in point $\vec{a}$ is given by: \[ {\cal L}_{\vec{a}}f=<\vec{a},df>=a^i\frac{\partial f}{\partial x^i} \]

7.8.2 The Lie-derivative

The Lie-derivative is given by: \[ ({\cal L}_{\vec{v}}\vec{w})^j=w^i\partial_iv^j-v^i\partial_iw^j \]

7.8.3 Christoffel symbols

To each curvelinear coordinate system $u^i$ we add a system of $n^3$ functions $\Gamma^i_{jk}$ of $\vec{u}$, defined by \[ \frac{\partial^2\vec{x}}{\partial u^i\partial u^k}=\Gamma_{jk}^i\frac{\partial\vec{x}}{\partial u^i} \] These are Christoffel symbols of the second kind. Christoffel symbols are no tensors. The Christoffel symbols of the second kind are given by: \[ \left\{\begin{array}{@{}c@{}}i\\ jk \end{array}\right\}:=\Gamma^i_{jk}= \left\langle\frac{\partial^2\vec{x}}{\partial u^k\partial u^j},dx^i\right\rangle \] with $\Gamma^i_{jk}=\Gamma^i_{kj}$. Their transformation to a different coordinate system is given by: \[ \Gamma_{j'k'}^{i'}=A_{i'}^iA_{j'}^jA_{k'}^k\Gamma^i_{jk}+A_i^{i'}(\partial_{j'}A_{k'}^i) \] The first term in this expression is 0 if the primed coordinates are cartesian.

There is a relation between Christoffel symbols and the metric: \[ \Gamma_{jk}^i=\frac{1}{2} g^{ir}(\partial_j g_{kr}+\partial_k g_{rj}-\partial_r g_{jk}) \] and $\Gamma^\alpha_{\beta\alpha}=\partial_\beta(\ln(\sqrt{|g|}))$.

Lowering an index gives the Christoffel symbols of the first kind: $\Gamma^i_{jk}=g^{il}\Gamma_{jkl}$.

7.8.4 The covariant derivative

The covariant derivative $\nabla_j$ of a vector, covector and of rank-2 tensors is given by: \begin{eqnarray*} \nabla_ja^i &=&\partial_ja^i+\Gamma^i_{jk}a^k\\ \nabla_ja_i &=&\partial_ja_i-\Gamma^k_{ij}a_k\\ \nabla_\gamma a^\alpha_\beta &=&\partial_\gamma a^\alpha_\beta -\Gamma^\varepsilon_{\gamma\beta} a^\alpha_\varepsilon+\Gamma^\alpha_{\gamma\varepsilon}a_\beta^\varepsilon\\ \nabla_\gamma a_{\alpha\beta}&=&\partial_\gamma a_{\alpha\beta}-\Gamma^\varepsilon_{\gamma\alpha}a_{\varepsilon\beta}-\Gamma^\varepsilon_{\gamma\beta}a_{\alpha\varepsilon}\\ \nabla_\gamma a^{\alpha\beta}&=&\partial_\gamma a^{\alpha\beta}+\Gamma^\alpha_{\gamma\varepsilon}a^{\varepsilon\beta}+\Gamma^\beta_{\gamma\varepsilon}a^{\alpha\varepsilon} \end{eqnarray*} Ricci's theorem: \[ \nabla_\gamma g_{\alpha\beta}=\nabla_\gamma g^{\alpha\beta}=0 \]

7.9 Differential operators

7.9.1 The Gradient

is given by: \[ {\rm grad}(f)=G^{-1}df=g^{ki}\frac{\partial f}{\partial x^i}\frac{\partial}{\partial x^k} \]

7.9.2 The divergence

is given by: \[ {\rm div}(a^i)=\nabla_ia^i=\frac{1}{\sqrt{g}}\partial_k(\sqrt{g}\,a^k) \]

7.9.3 The curl

is given by: \[ {\rm rot}(a)=G^{-1}\cdot*\cdot d\cdot G\vec{a}=-\varepsilon^{pqr}\nabla_qa_p=\nabla_qa_p-\nabla_pa_q \]

7.9.4 The Laplacian

is given by: \[ \Delta(f)={\rm div~grad}(f)=*d*df=\nabla_ig^{ij}\partial_jf=g^{ij}\nabla_i\nabla_jf= \frac{1}{\sqrt{g}}\frac{\partial}{\partial x^i}\left(\sqrt{g}\,g^{ij}\frac{\partial f}{\partial x^j}\right) \]

7.10 Differential geometry

7.10.1 Space curves

We limit ourselves to $I\hspace{-1mm}R^3$ with a fixed orthonormal basis. A point is represented by the vector $\vec{x}=(x^1,x^2,x^3)$. A space curve is a collection of points represented by $\vec{x}=\vec{x}(t)$. The arc length of a space curve is given by: \[ s(t)=\int\limits_{t_0}^t\sqrt{\left(\frac{dx}{d\tau}\right)^2+\left(\frac{dy}{d\tau}\right)^2+\left(\frac{dz}{d\tau}\right)^2}d\tau \] The derivative of $s$ with respect to $t$ is the length of the vector $d\vec{x}/dt$: \[ \left(\frac{ds}{dt}\right)^2=\left(\frac{d\vec{x}}{dt},\frac{d\vec{x}}{dt}\right) \] The osculation plane in a point $P$ of a space curve is the limiting position of the plane through the tangent of the plane in point $P$ and a point $Q$ when $Q$ approaches $P$ along the space curve. The osculation plane is parallel with $\dot{\vec{x}}(s)$. If $\ddot{\vec{x}}\neq0$ the osculation plane is given by: \[ \vec{y}=\vec{x}+\lambda\dot{\vec{x}}+\mu\ddot{\vec{x}}~~~\mbox{so}~~~ \det(\vec{y}-\vec{x},\dot{\vec{x}},\ddot{\vec{x}}\,)=0 \] In a bending point holds, if $\dddot{\vec{x}}\neq0$: \[ \vec{y}=\vec{x}+\lambda\dot{\vec{x}}+\mu\dddot{\vec{x}} \] The tangent has unit vector $\vec{\ell}=\dot{\vec{x}}$, the main normal unit vector $\vec{n}=\ddot{\vec{x}}$ and the binormal $\vec{b}=\dot{\vec{x}}\times\ddot{\vec{x}}$. So the main normal lies in the osculation plane, the binormal is perpendicular to it.

Let $P$ be a point and $Q$ be a nearby point of a space curve $\vec{x}(s)$. Let $\Delta\varphi$ be the angle between the tangents in $P$ and $Q$ and let $\Delta\psi$ be the angle between the osculation planes (binormals) in $P$ and $Q$. Then the curvature $\rho$ and the torsion $\tau$ in $P$ are defined by: \[ \rho^2=\left(\frac{d\varphi}{ds}\right)^2=\lim_{\Delta s\rightarrow0}\left(\frac{\Delta\varphi}{\Delta s}\right)^2~~~,~~~ \tau^2=\left(\frac{d\psi}{ds}\right)^2 \] and $\rho>0$. For plane curves $\rho$ is the ordinary curvature and $\tau=0$. The following holds: \[ \rho^2=(\vec{\ell},\vec{\ell})=(\ddot{\vec{x}},\ddot{\vec{x}}\,)~~~\mbox{and}~~~ \tau^2=(\dot{\vec{b}},\dot{\vec{b}}) \] Frenet's equations express the derivatives as linear combinations of these vectors: \[ \dot{\vec{\ell}}=\rho\vec{n}~~,~~\dot{\vec{n}}=-\rho\vec{\ell}+\tau\vec{b}~~,~~ \dot{\vec{b}}=-\tau\vec{n} \] From this follows that $\det(\dot{\vec{x}},\ddot{\vec{x}},\dddot{\vec{x}}\,)=\rho^2\tau$.

Some curves and their properties are:

Screw line $\tau/\rho=$constant
Circle screw line $\tau=$constant, $\rho=$constant
Plane curves $\tau=0$
Circles $\rho=$constant, $\tau=0$
Lines $\rho=\tau=0$

7.10.2 Surfaces in $I\hspace{-1mm}R^3$

A surface in $I\hspace{-1mm}R^3$ is the collection of end points of the vectors $\vec{x}=\vec{x}(u,v)$, so $x^h=x^h(u^\alpha)$. On the surface are 2 families of curves, one with $u=$constant and one with $v=$constant.

The tangent plane in a point $P$ at the surface has basis: \[ \vec{c}_1=\partial_1\vec{x}~~~\mbox{and}~~~\vec{c}_2=\partial_2\vec{x} \]

7.10.3 The first fundamental tensor

Let $P$ be a point of the surface $\vec{x}=\vec{x}(u^\alpha)$. The following two curves through $P$, denoted by $u^\alpha=u^\alpha(t)$, $u^\alpha=v^\alpha(\tau)$, have as tangent vectors in $P$ \[ \frac{d\vec{x}}{dt}=\frac{du^\alpha}{dt}\partial_\alpha\vec{x}~~~,~~~ \frac{d\vec{x}}{d\tau}=\frac{dv^\beta}{d\tau}\partial_\beta\vec{x} \] The first fundamental tensor of the surface in $P$ is the inner product of these tangent vectors: \[ \left(\frac{d\vec{x}}{dt},\frac{d\vec{x}}{d\tau}\right)= (\vec{c}_\alpha,\vec{c}_\beta)\frac{du^\alpha}{dt}\frac{dv^\beta}{d\tau} \] The covariant components w.r.t.\ the basis $\vec{c}_\alpha=\partial_\alpha\vec{x}$ are: \[ g_{\alpha\beta}=(\vec{c}_\alpha,\vec{c}_\beta) \] For the angle $\phi$ between the parameter curves in $P$: $u=t,v=$constant and $u=$constant, $v=\tau$ holds: \[ \cos(\phi)=\frac{g_{12}}{\sqrt{g_{11}g_{22}}} \] For the arc length $s$ of $P$ along the curve $u^\alpha(t)$ holds: \[ ds^2=g_{\alpha\beta}du^\alpha du^\beta \] This expression is called the line element.

7.10.4 The second fundamental tensor

The 4 derivatives of the tangent vectors $\partial_\alpha\partial_\beta\vec{x}=\partial_\alpha\vec{c}_\beta$ are each linear independent of the vectors $\vec{c}_1$, $\vec{c}_2$ and $\vec{N}$, with $\vec{N}$ perpendicular to $\vec{c}_1$ and $\vec{c}_2$. This is written as: \[ \partial_\alpha\vec{c}_\beta=\Gamma^\gamma_{\alpha\beta}\vec{c}_\gamma+h_{\alpha\beta}\vec{N} \] This leads to: \[ \Gamma^\gamma_{\alpha\beta}=(\vec{c}^{~\gamma},\partial_\alpha\vec{c}_\beta)~~~,~~~ h_{\alpha\beta}=(\vec{N},\partial_\alpha\vec{c}_\beta)=\frac{1}{\sqrt{\det|g|}}\det(\vec{c}_1,\vec{c}_2,\partial_\alpha\vec{c}_\beta) \]

7.10.5 Geodetic curvature

A curve on the surface $\vec{x}(u^\alpha)$ is given by: $u^\alpha=u^\alpha(s)$, than $\vec{x}=\vec{x}(u^\alpha(s))$ with $s$ the arc length of the curve. The length of $\ddot{\vec{x}}$ is the curvature $\rho$ of the curve in $P$. The projection of $\ddot{\vec{x}}$ on the surface is a vector with components \[ p^\gamma=\ddot{u}^\gamma+\Gamma^\gamma_{\alpha\beta}\dot{u}^\alpha\dot{u}^\beta \] of which the length is called the geodetic curvature of the curve in $p$. This remains the same if the surface is curved and the line element remains the same. The projection of $\ddot{\vec{x}}$ on $\vec{N}$ has length \[ p=h_{\alpha\beta}\dot{u}^\alpha\dot{u}^\beta \] and is called the normal curvature of the curve in $P$. The theorem of Meusnier states that different curves on the surface with the same tangent vector in $P$ have the same normal curvature.

A geodetic line of a surface is a curve on the surface for which in each point the main normal of the curve is the same as the normal on the surface. So for a geodetic line is in each point $p^\gamma=0$, so \[ \frac{d^2u^\gamma}{ds^2}+\Gamma^\gamma_{\alpha\beta}\frac{du^\alpha}{ds}\frac{du^\beta}{ds}=0 \] The covariant derivative $\nabla/dt$ in $P$ of a vector field of a surface along a curve is the projection on the tangent plane in $P$ of the normal derivative in $P$.

For two vector fields $\vec{v}(t)$ and $\vec{w}(t)$ along the same curve of the surface follows Leibniz' rule: \[ \frac{d(\vec{v},\vec{w})}{dt}=\left(\vec{v},\frac{\nabla\vec{w}}{dt}\right)+\left(\vec{w},\frac{\nabla\vec{v}}{dt}\right) \] Along a curve holds: \[ \frac{\nabla}{dt}(v^\alpha\vec{c}_\alpha)=\left(\frac{dv^\gamma}{dt}+\Gamma^\gamma_{\alpha\beta}\frac{du^\alpha}{dt}v^\beta\right)\vec{c}_\gamma \]

7.11 Riemannian geometry

The Riemann tensor $R$ is defined by: \[ R^\mu_{\nu\alpha\beta}T^\nu=\nabla_\alpha\nabla_\beta T^\mu-\nabla_\beta\nabla_\alpha T^\mu \] This is a $1\choose 3$ tensor with $n^2(n^2-1)/12$ independent components not identically equal to 0. This tensor is a measure for the curvature of the considered space. If it is 0, the space is a flat manifold. It has the following symmetry properties: \[ R_{\alpha\beta\mu\nu}=R_{\mu\nu\alpha\beta}=-R_{\beta\alpha\mu\nu}=-R_{\alpha\beta\nu\mu} \] The following relation holds: \[ [\nabla_\alpha,\nabla_\beta]T_\nu^\mu=R_{\sigma\alpha\beta}^\mu T_\nu^\sigma+R_{\nu\alpha\beta}^\sigma T_\sigma^\mu \] The Riemann tensor depends on the Christoffel symbols through \[ R^\alpha_{\beta\mu\nu}=\partial_\mu\Gamma^\alpha_{\beta\nu}-\partial_\nu\Gamma^\alpha_{\beta\mu}+\Gamma^\alpha_{\sigma\mu}\Gamma^\sigma_{\beta\nu}-\Gamma^\alpha_{\sigma\nu}\Gamma^\sigma_{\beta\mu} \] In a space and coordinate system where the Christoffel symbols are 0 this becomes: \[ R^\alpha_{\beta\mu\nu}=\frac{1}{2} g^{\alpha\sigma}(\partial_\beta\partial_\mu g_{\sigma\nu}-\partial_\beta\partial_\nu g_{\sigma\mu}+\partial_\sigma\partial_\nu g_{\beta\mu}-\partial_\sigma\partial_\mu g_{\beta\nu}) \] The Bianchi identities are: $\nabla_\lambda R_{\alpha\beta\mu\nu}+\nabla_\nu R_{\alpha\beta\lambda\mu} +\nabla_\mu R_{\alpha\beta\nu\lambda}=0$.

The Ricci tensor is obtained by contracting the Riemann tensor: $R_{\alpha\beta}\equiv R_{\alpha\mu\beta}^\mu$, and is symmetric in its indices: $R_{\alpha\beta}=R_{\beta\alpha}$. The Einstein tensor $G$ is defined by: $G^{\alpha\beta}\equiv R^{\alpha\beta}-\frac{1}{2} g^{\alpha\beta}$. It has the property that $\nabla_\beta G^{\alpha\beta}=0$. The Ricci-scalar is $R=g^{\alpha\beta}R_{\alpha\beta}$.