# Christoffel symbols

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In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. [1] The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. [2] [3] However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. [4] Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group O(p, q). As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold. [5] [6] The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.

## Contents

In general, there are an infinite number of metric connections for a given metric tensor; however, there is a unique connection that is free of torsion, the Levi-Civita connection. It is common in physics and general relativity to work almost exclusively with the Levi-Civita connection, by working in coordinate frames (called holonomic coordinates) where the torsion vanishes. For example, in Euclidean spaces, the Christoffel symbols describe how the local coordinate bases change from point to point.

At each point of the underlying n-dimensional manifold, for any local coordinate system around that point, the Christoffel symbols are denoted Γijk for i, j, k = 1, 2, ..., n. Each entry of this n × n × n array is a real number. Under linear coordinate transformations on the manifold, the Christoffel symbols transform like the components of a tensor, but under general coordinate transformations (diffeomorphisms) they do not. Most of the algebraic properties of the Christoffel symbols follow from their relationship to the affine connection; only a few follow from the fact that the structure group is the orthogonal group O(m, n) (or the Lorentz group O(3, 1) for general relativity).

Christoffel symbols are used for performing practical calculations. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In general relativity, the connection plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γijk are zero.

The Christoffel symbols are named for Elwin Bruno Christoffel (1829–1900). [7]

## Note

The definitions given below are valid for both Riemannian manifolds and pseudo-Riemannian manifolds, such as those of general relativity, with careful distinction being made between upper and lower indices (contra-variant and co-variant indices). The formulas hold for either sign convention, unless otherwise noted.

Einstein summation convention is used in this article, with vectors indicated by bold font. The connection coefficients of the Levi-Civita connection (or pseudo-Riemannian connection) expressed in a coordinate basis are called Christoffel symbols.

## Preliminary definitions

Given a coordinate system xi for i = 1, 2, …, n on an n-manifold M, the tangent vectors

${\displaystyle \mathbf {e} _{i}={\frac {\partial }{\partial x^{i}}}=\partial _{i},\quad i=1,\,2,\,\dots ,\,n}$

define what is referred to as the local basis of the tangent space to M at each point of its domain. These can be used to define the metric tensor:

${\displaystyle g_{ij}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}}$

and its inverse:

${\displaystyle g^{ij}=\left(g^{-1}\right)_{ij}}$

which can in turn be used to define the dual basis:

${\displaystyle \mathbf {e} ^{i}=\mathbf {e} _{j}g^{ji},\quad i=1,\,2,\,\dots ,\,n}$

Some texts write ${\displaystyle \mathbf {g} _{i}}$ for ${\displaystyle \mathbf {e} _{i}}$, so that the metric tensor takes the particularly beguiling form ${\displaystyle g_{ij}=\mathbf {g} _{i}\cdot \mathbf {g} _{j}}$. This convention also leaves use of the symbol ${\displaystyle e_{i}}$ unambiguously for the vierbein.

## Definition in Euclidean space

In Euclidean space, the general definition given below for the Christoffel symbols of the second kind can be proven to be equivalent to:

${\displaystyle {\Gamma ^{k}}_{ij}={\frac {\partial \mathbf {e} _{i}}{\partial x^{j}}}\cdot \mathbf {e} ^{k}={\frac {\partial \mathbf {e} _{i}}{\partial x^{j}}}\cdot g^{km}\mathbf {e} _{m}}$

Christoffel symbols of the first kind can then be found via index lowering:

${\displaystyle \Gamma _{kij}={\Gamma ^{m}}_{ij}g_{mk}={\frac {\partial \mathbf {e} _{i}}{\partial x^{j}}}\cdot \mathbf {e} ^{m}g_{mk}={\frac {\partial \mathbf {e} _{i}}{\partial x^{j}}}\cdot \mathbf {e} _{k}.}$

Rearranging, we see that (assuming the partial derivative belongs to the tangent space, which can not occur on a non-Euclidean curved space):

${\displaystyle {\frac {\partial \mathbf {e} _{i}}{\partial x^{j}}}={\Gamma ^{k}}_{ij}\mathbf {e} _{k}=\Gamma _{kij}\mathbf {e} ^{k}}$

In words, the arrays represented by the Christoffel symbols track how the basis changes from point to point. If the derivative doesn't lie on the tangent space, the right expression is the projection of the derivative over the tangent space (see covariant derivative below). Symbols of the second kind decompose the change with respect to the basis, while symbols of the first kind decompose it with respect to the dual basis. In this form, it easy to see the symmetry of the lower or last two indices:

${\displaystyle {\Gamma ^{k}}_{ij}={\Gamma ^{k}}_{ji}}$ and ${\displaystyle \Gamma _{kij}=\Gamma _{kji}}$,

from the definition of ${\displaystyle \mathbf {e} _{i}}$ and the fact that partial derivatives commute (as long as the manifold and coordinate system are well behaved).

The same numerical values for Christoffel symbols of the second kind also relate to derivatives of the dual basis, as seen in the expression:

${\displaystyle {\frac {\partial \mathbf {e} ^{i}}{\partial x^{j}}}=-{\Gamma ^{i}}_{jk}\mathbf {e} ^{k}}$,

which we can rearrange as:

${\displaystyle {\Gamma ^{i}}_{jk}=-{\frac {\partial \mathbf {e} ^{i}}{\partial x^{j}}}\cdot \mathbf {e} _{k}}$.

## General definition

### Christoffel symbols of the first kind

The Christoffel symbols of the first kind can be derived either from the Christoffel symbols of the second kind and the metric, [8]

${\displaystyle \Gamma _{cab}=g_{cd}{\Gamma ^{d}}_{ab}\,,}$

or from the metric alone, [8]

${\displaystyle \Gamma _{cab}={\frac {1}{2}}\left({\frac {\partial g_{ca}}{\partial x^{b}}}+{\frac {\partial g_{cb}}{\partial x^{a}}}-{\frac {\partial g_{ab}}{\partial x^{c}}}\right)={\frac {1}{2}}\,\left(g_{ca,b}+g_{cb,a}-g_{ab,c}\right)={\frac {1}{2}}\,\left(\partial _{b}g_{ca}+\partial _{a}g_{cb}-\partial _{c}g_{ab}\right)\,.}$

As an alternative notation one also finds [7] [9] [10]

${\displaystyle \Gamma _{cab}=[ab,c].}$

It is worth noting that [ab, c] = [ba, c]. [11]

### Christoffel symbols of the second kind (symmetric definition)

The Christoffel symbols of the second kind are the connection coefficients—in a coordinate basis—of the Levi-Civita connection. In other words, the Christoffel symbols of the second kind [12] [13] Γkij (sometimes Γk
ij
or {k
ij
}
) [7] [12] are defined as the unique coefficients such that

${\displaystyle \nabla _{i}\mathrm {e} _{j}={\Gamma ^{k}}_{ij}\mathrm {e} _{k}}$,

where i is the Levi-Civita connection on M taken in the coordinate direction ei (i.e., i ≡ ∇ei) and where ei = ∂i is a local coordinate (holonomic) basis. Since this connection has zero torsion, and holonomic vector fields commute (i.e. ${\displaystyle [e_{i},e_{j}]=[\partial _{i},\partial _{j}]=0}$) we have

${\displaystyle \nabla _{i}\mathrm {e} _{j}=\nabla _{j}\mathrm {e} _{i}}$.

Hence in this basis the connection coefficients are symmetric:

Γkij = Γkji. [12]

For this reason, a torsion-free connection is often called symmetric.

The Christoffel symbols can be derived from the vanishing of the covariant derivative of the metric tensor gik:

${\displaystyle 0=\nabla _{l}g_{ik}={\frac {\partial g_{ik}}{\partial x^{l}}}-g_{mk}{\Gamma ^{m}}_{il}-g_{im}{\Gamma ^{m}}_{kl}={\frac {\partial g_{ik}}{\partial x^{l}}}-2g_{m(k}{\Gamma ^{m}}_{i)l}.}$

As a shorthand notation, the nabla symbol and the partial derivative symbols are frequently dropped, and instead a semicolon and a comma are used to set off the index that is being used for the derivative. Thus, the above is sometimes written as

${\displaystyle 0=\,g_{ik;l}=g_{ik,l}-g_{mk}{\Gamma ^{m}}_{il}-g_{im}{\Gamma ^{m}}_{kl}.}$

Using that the symbols are symmetric in the lower two indices, one can solve explicitly for the Christoffel symbols as a function of the metric tensor by permuting the indices and resumming: [11]

${\displaystyle {\Gamma ^{i}}_{kl}={\frac {1}{2}}g^{im}\left({\frac {\partial g_{mk}}{\partial x^{l}}}+{\frac {\partial g_{ml}}{\partial x^{k}}}-{\frac {\partial g_{kl}}{\partial x^{m}}}\right)={\frac {1}{2}}g^{im}\left(g_{mk,l}+g_{ml,k}-g_{kl,m}\right),}$

where (gjk) is the inverse of the matrix (gjk), defined as (using the Kronecker delta, and Einstein notation for summation) gjigik = δjk. Although the Christoffel symbols are written in the same notation as tensors with index notation, they do not transform like tensors under a change of coordinates.

#### Contraction of indices

Contracting the upper index with either of the lower indices (those being symmetric) leads to

${\displaystyle {\Gamma ^{i}}_{ki}={\frac {\partial }{\partial x^{k}}}\ln {\sqrt {|g|}}}$

where ${\displaystyle g=\det g_{ik}}$ is the determinant of the metric tensor. This identity can be used to evaluate divergence of vectors.

### Connection coefficients in a nonholonomic basis

The Christoffel symbols are most typically defined in a coordinate basis, which is the convention followed here. In other words, the name Christoffel symbols is reserved only for coordinate (i.e., holonomic) frames. However, the connection coefficients can also be defined in an arbitrary (i.e., nonholonomic) basis of tangent vectors ui by

${\displaystyle \nabla _{\mathbf {u} _{i}}\mathbf {u} _{j}={\omega ^{k}}_{ij}\mathbf {u} _{k}.}$

Explicitly, in terms of the metric tensor, this is [13]

${\displaystyle {\omega ^{i}}_{kl}={\frac {1}{2}}g^{im}\left(g_{mk,l}+g_{ml,k}-g_{kl,m}+c_{mkl}+c_{mlk}-c_{klm}\right),}$

where cklm = gmpcklp are the commutation coefficients of the basis; that is,

${\displaystyle [\mathbf {u} _{k},\,\mathbf {u} _{l}]={c_{kl}}^{m}\mathbf {u} _{m}}$

where uk are the basis vectors and [ , ] is the Lie bracket. The standard unit vectors in spherical and cylindrical coordinates furnish an example of a basis with non-vanishing commutation coefficients. The difference between the connection in such a frame, and the Levi-Civita connection is known as the contorsion tensor.

### Ricci rotation coefficients (asymmetric definition)

When we choose the basis Xiui orthonormal: gabηab = ⟨Xa, Xb then gmk,lηmk,l = 0. This implies that

${\displaystyle {\omega ^{i}}_{kl}={\frac {1}{2}}\eta ^{im}\left(c_{mkl}+c_{mlk}-c_{klm}\right)}$

and the connection coefficients become antisymmetric in the first two indices:

${\displaystyle \omega _{abc}=-\omega _{bac}\,,}$

where

${\displaystyle \omega _{abc}=\eta _{ad}{\omega ^{d}}_{bc}\,.}$

In this case, the connection coefficients ωabc are called the Ricci rotation coefficients. [14] [15]

Equivalently, one can define Ricci rotation coefficients as follows: [13]

${\displaystyle {\omega ^{k}}_{ij}:=\mathbf {u} ^{k}\cdot \left(\nabla _{j}\mathbf {u} _{i}\right)\,,}$

where ui is an orthonormal nonholonomic basis and uk = ηklul its co-basis.

## Transformation law under change of variable

Under a change of variable from ${\displaystyle \left(x^{1},\,\ldots ,\,x^{n}\right)}$ to ${\displaystyle \left({\bar {x}}^{1},\,\ldots ,\,{\bar {x}}^{n}\right)}$, Christoffel symbols transform as

${\displaystyle {{\bar {\Gamma }}^{i}}_{kl}={\frac {\partial {\bar {x}}^{i}}{\partial x^{m}}}\,{\frac {\partial x^{n}}{\partial {\bar {x}}^{k}}}\,{\frac {\partial x^{p}}{\partial {\bar {x}}^{l}}}\,{\Gamma ^{m}}_{np}+{\frac {\partial ^{2}x^{m}}{\partial {\bar {x}}^{k}\partial {\bar {x}}^{l}}}\,{\frac {\partial {\bar {x}}^{i}}{\partial x^{m}}}}$

where the overline denotes the Christoffel symbols in the ${\displaystyle {\bar {x}}^{i}}$ coordinate system. The Christoffel symbol does not transform as a tensor, but rather as an object in the jet bundle. More precisely, the Christoffel symbols can be considered as functions on the jet bundle of the frame bundle of M, independent of any local coordinate system. Choosing a local coordinate system determines a local section of this bundle, which can then be used to pull back the Christoffel symbols to functions on M, though of course these functions then depend on the choice of local coordinate system.

For each point, there exist coordinate systems in which the Christoffel symbols vanish at the point. [16] These are called (geodesic) normal coordinates, and are often used in Riemannian geometry.

There are some interesting properties which can be derived directly from the transformation law.

• For linear transformation, the inhomogeneous part of the transformation (second term on the right-hand side) vanishes identically and then ${\displaystyle {\Gamma ^{i}}_{jk}}$ behaves like a tensor.
• If we have two fields of connections, say ${\displaystyle {\Gamma ^{i}}_{jk}}$ and ${\displaystyle {{\tilde {\Gamma }}^{i}}_{jk}}$, then their difference ${\displaystyle {\Gamma ^{i}}_{jk}-{{\tilde {\Gamma }}^{i}}_{jk}}$ is a tensor since the inhomogeneous terms cancel each other. The inhomogeneous terms depend only on how the coordinates are changed, but are independent of Christoffel symbol itself.
• If the Christoffel symbol is unsymmetric about its lower indices in one coordinate system i.e., ${\displaystyle {\Gamma ^{i}}_{jk}\neq {\Gamma ^{i}}_{kj}}$, then they remain unsymmetric under any change of coordinates. A corollary to this property is that it is impossible to find a coordinate system in which all elements of Christoffel symbol are zero at a point, unless lower indices are symmetric. This property was pointed out by Albert Einstein [17] and Erwin Schrödinger [18] independently.

## Relationship to parallel transport and derivation of Christoffel symbols in Riemannian space

If a vector ${\displaystyle \xi ^{i}}$ is transported parallel on a curve parametrized by some parameter ${\displaystyle s}$ on a Riemannian manifold, the rate of change of the components of the vector is given by

${\displaystyle {\frac {d\xi ^{i}}{ds}}=-{\Gamma ^{i}}_{mj}{\frac {dx^{m}}{ds}}\xi ^{j}.}$

Now just by using the condition that the scalar product ${\displaystyle g_{ik}\xi ^{i}\eta ^{k}}$ formed by two arbitrary vectors ${\displaystyle \xi ^{i}}$ and ${\displaystyle \eta ^{k}}$ is unchanged is enough to derive the Christoffel symbols. The condition is

${\displaystyle {\frac {d}{ds}}\left(g_{ik}\xi ^{i}\eta ^{k}\right)=0}$

which by product rule expand to

${\displaystyle {\frac {\partial g_{ik}}{\partial x^{l}}}{\frac {dx^{l}}{ds}}\xi ^{i}\eta ^{k}+g_{ik}{\frac {d\xi ^{i}}{ds}}\eta ^{k}+g_{ik}\xi ^{i}{\frac {d\eta ^{k}}{ds}}=0.}$

Applying the parallel transport rule for the two arbitrary vectors and relabelling dummy indices and collecting the coefficients of ${\displaystyle \xi ^{i}\eta ^{k}dx^{l}}$ (arbitrary), we obtain

${\displaystyle {\frac {\partial g_{ik}}{\partial x^{l}}}=g_{rk}{\Gamma ^{r}}_{il}+g_{ir}{\Gamma ^{r}}_{lk}.}$

This is same as the equation obtained by requiring the covariant derivative of the metric tensor to vanish in the General definition section. The derivation from here is simple. By cyclically permuting the indices ${\displaystyle ikl}$ in above equation, we can obtain two more equations and then linearly combining these three equations, we can express ${\displaystyle {\Gamma ^{i}}_{jk}}$ in terms of metric tensor.

## Relationship to index-free notation

Let X and Y be vector fields with components Xi and Yk. Then the kth component of the covariant derivative of Y with respect to X is given by

${\displaystyle \left(\nabla _{X}Y\right)^{k}=X^{i}(\nabla _{i}Y)^{k}=X^{i}\left({\frac {\partial Y^{k}}{\partial x^{i}}}+{\Gamma ^{k}}_{im}Y^{m}\right).}$

Here, the Einstein notation is used, so repeated indices indicate summation over indices and contraction with the metric tensor serves to raise and lower indices:

${\displaystyle g(X,Y)=X^{i}Y_{i}=g_{ik}X^{i}Y^{k}=g^{ik}X_{i}Y_{k}.}$

Keep in mind that gikgik and that gik = δik, the Kronecker delta. The convention is that the metric tensor is the one with the lower indices; the correct way to obtain gik from gik is to solve the linear equations gijgjk = δik.

The statement that the connection is torsion-free, namely that

${\displaystyle \nabla _{X}Y-\nabla _{Y}X=[X,\,Y]}$

is equivalent to the statement that—in a coordinate basis—the Christoffel symbol is symmetric in the lower two indices:

${\displaystyle {\Gamma ^{i}}_{jk}={\Gamma ^{i}}_{kj}.}$

The index-less transformation properties of a tensor are given by pullbacks for covariant indices, and pushforwards for contravariant indices. The article on covariant derivatives provides additional discussion of the correspondence between index-free notation and indexed notation.

## Covariant derivatives of tensors

The covariant derivative of a contravariant vector field Vm is

${\displaystyle \nabla _{l}V^{m}={\frac {\partial V^{m}}{\partial x^{l}}}+{\Gamma ^{m}}_{kl}V^{k}.}$

By corollary, divergence of a vector can be obtained as

${\displaystyle \nabla _{i}V^{i}={\frac {1}{\sqrt {-g}}}{\frac {\partial \left({\sqrt {-g}}\,V^{i}\right)}{\partial x^{i}}}.}$

The covariant derivative of a covector field ωm is

${\displaystyle \nabla _{l}\omega _{m}={\frac {\partial \omega _{m}}{\partial x^{l}}}-{\Gamma ^{k}}_{ml}\omega _{k}.}$

The symmetry of the Christoffel symbol now implies

${\displaystyle \nabla _{i}\nabla _{j}\varphi =\nabla _{j}\nabla _{i}\varphi }$

for any scalar field, but in general the covariant derivatives of higher order tensor fields do not commute (see curvature tensor).

The covariant derivative of a type (2, 0) tensor field Aik is

${\displaystyle \nabla _{l}A^{ik}={\frac {\partial A^{ik}}{\partial x^{l}}}+{\Gamma ^{i}}_{ml}A^{mk}+{\Gamma ^{k}}_{ml}A^{im},}$

that is,

${\displaystyle {A^{ik}}_{;l}={A^{ik}}_{,l}+A^{mk}{\Gamma ^{i}}_{ml}+A^{im}{\Gamma ^{k}}_{ml}.}$

If the tensor field is mixed then its covariant derivative is

${\displaystyle {A^{i}}_{k;l}={A^{i}}_{k,l}+{A^{m}}_{k}{\Gamma ^{i}}_{ml}-{A^{i}}_{m}{\Gamma ^{m}}_{kl},}$

and if the tensor field is of type (0, 2) then its covariant derivative is

${\displaystyle A_{ik;l}=A_{ik,l}-A_{mk}{\Gamma ^{m}}_{il}-A_{im}{\Gamma ^{m}}_{kl}.}$

### Contravariant derivatives of tensors

To find the contravariant derivative of a vector field, we must first transform it into a covariant derivative using the metric tensor

${\displaystyle \nabla ^{l}V^{m}=g^{il}\nabla _{i}V^{m}=g^{il}\partial _{i}V^{m}+g^{il}\Gamma _{ki}^{m}V^{k}=\partial ^{l}V^{m}+g^{il}\Gamma _{ki}^{m}V^{k}}$

## Applications

### In general relativity

The Christoffel symbols find frequent use in Einstein's theory of general relativity, where spacetime is represented by a curved 4-dimensional Lorentz manifold with a Levi-Civita connection. The Einstein field equations—which determine the geometry of spacetime in the presence of matter—contain the Ricci tensor, and so calculating the Christoffel symbols is essential. Once the geometry is determined, the paths of particles and light beams are calculated by solving the geodesic equations in which the Christoffel symbols explicitly appear.

### In classical (non-relativistic) mechanics

Let ${\displaystyle x^{i}}$ be the generalized coordinates and ${\displaystyle {\dot {x}}^{i}}$ be the generalized velocities, then the kinetic energy for a unit mass is given by ${\displaystyle T={\tfrac {1}{2}}g_{ik}{\dot {x}}^{i}{\dot {x}}^{k}}$, where ${\displaystyle g_{ik}}$ is the metric tensor. If ${\displaystyle V\left(x^{i}\right)}$, the potential function, exists then the contravariant components of the generalized force per unit mass are ${\displaystyle F_{i}=\partial V/\partial x^{i}}$. The metric (here in a purely spatial domain) can be obtained from the line element ${\displaystyle ds^{2}=2Tdt^{2}}$. Substituting the Lagrangian ${\displaystyle L=T-V}$ into the Euler-Lagrange equation, we get [19]

${\displaystyle g_{ik}{\ddot {x}}^{k}+{\frac {1}{2}}\left({\frac {\partial g_{ik}}{\partial x^{l}}}+{\frac {\partial g_{il}}{\partial x^{k}}}-{\frac {\partial g_{lk}}{\partial x^{i}}}\right){\dot {x}}^{l}{\dot {x}}^{k}=F_{i}.}$

Now multiplying by ${\displaystyle g^{ij}}$, we get

${\displaystyle {\ddot {x}}^{j}+{\Gamma ^{j}}_{lk}{\dot {x}}^{l}{\dot {x}}^{k}=F^{j}.}$

When Cartesian coordinates can be adopted (as in inertial frames of reference), we have an Euclidean metrics, the Christoffel symbol vanishes, and the equation reduces to Newton's second law of motion. In curvilinear coordinates [20] (forcedly in non-inertial frames, where the metrics is non-Euclidean and not flat), fictitious forces like the Centrifugal force and Coriolis force originate from the Christoffel symbols, so from the purely spatial curvilinear coordinates.

### In earth surface coordinates

Given a spherical coordinate system, which describes points on the earth surface (approximated as an ideal sphere).

{\displaystyle {\begin{aligned}x(R,\theta ,\varphi )&={\begin{pmatrix}R\cos \theta \cos \varphi &R\cos \theta \sin \varphi &R\sin \theta \end{pmatrix}}\\\end{aligned}}}

For a point x, R is the distance to the earth core (usually approximately the earth radius). θ and φ are the latitude and longitude. Positive θ is the northern hemisphere. To simplify the derivatives, the angles are given in radians (where d sin(x)/dx = cos(x), the degree values introduce an additional factor of 360 / 2 pi).

At any location, the tangent directions are ${\displaystyle e_{R}}$ (up), ${\displaystyle e_{\theta }}$ (north) and ${\displaystyle e_{\varphi }}$ (east) - you can also use indices 1,2,3.

{\displaystyle {\begin{aligned}e_{R}&={\begin{pmatrix}\cos \theta \cos \varphi &\cos \theta \sin \varphi &\sin \theta \end{pmatrix}}\\e_{\theta }&=R\cdot {\begin{pmatrix}-\sin \theta \cos \varphi &-\sin \theta \sin \varphi &\cos \theta \end{pmatrix}}\\e_{\varphi }&=R\cos \theta \cdot {\begin{pmatrix}-\sin \varphi &\cos \varphi &0\end{pmatrix}}\\\end{aligned}}}

The related metric tensor has only diagonal elements (the squared vector lengths). This is an advantage of the coordinate system and not generally true.

{\displaystyle {\begin{aligned}g_{RR}=1\qquad &g_{\theta \theta }=R^{2}\qquad &g_{\varphi \varphi }=R^{2}\cos ^{2}\theta \qquad &g_{ij}=0\quad \mathrm {else} \\g^{RR}=1\qquad &g^{\theta \theta }=1/R^{2}\qquad &g^{\varphi \varphi }=1/(R^{2}\cos ^{2}\theta )\qquad &g^{ij}=0\quad \mathrm {else} \\\end{aligned}}}

Now the necessary quantities can be calculated. Examples:

{\displaystyle {\begin{aligned}e^{R}=1\cdot e_{R}&={\begin{pmatrix}\cos \theta \cos \varphi &\cos \theta \sin \varphi &\sin \theta \end{pmatrix}}\\{\Gamma ^{R}}_{\varphi \varphi }=e^{R}\cdot {\frac {\partial }{\partial \varphi }}e_{\varphi }&=e^{R}\cdot {\begin{pmatrix}-R\cos \theta \cos \varphi &-R\cos \theta \sin \varphi &0\end{pmatrix}}=-R\cos ^{2}\theta \\\end{aligned}}}

The resulting Christoffel symbols of the second kind ${\displaystyle {\Gamma ^{k}}_{ji}=e^{k}\cdot {\frac {\partial e_{j}}{\partial x^{i}}}}$ then are (organized by the "derivative" index i in a matrix):

{\displaystyle {\begin{aligned}{\begin{pmatrix}{\Gamma ^{R}}_{RR}&{\Gamma ^{R}}_{\theta R}&{\Gamma ^{R}}_{\varphi R}\\{\Gamma ^{\theta }}_{RR}&{\Gamma ^{\theta }}_{\theta R}&{\Gamma ^{\theta }}_{\varphi R}\\{\Gamma ^{\varphi }}_{RR}&{\Gamma ^{\varphi }}_{\theta R}&{\Gamma ^{\varphi }}_{\varphi R}\\\end{pmatrix}}&=\quad {\begin{pmatrix}0&0&0\\0&1/R&0\\0&0&1/R\end{pmatrix}}\\{\begin{pmatrix}{\Gamma ^{R}}_{R\theta }&{\Gamma ^{R}}_{\theta \theta }&{\Gamma ^{R}}_{\varphi \theta }\\{\Gamma ^{\theta }}_{R\theta }&{\Gamma ^{\theta }}_{\theta \theta }&{\Gamma ^{\theta }}_{\varphi \theta }\\{\Gamma ^{\varphi }}_{R\theta }&{\Gamma ^{\varphi }}_{\theta \theta }&{\Gamma ^{\varphi }}_{\varphi \theta }\\\end{pmatrix}}\quad &={\begin{pmatrix}0&-R&0\\1/R&0&0\\0&0&-\tan \theta \end{pmatrix}}\\{\begin{pmatrix}{\Gamma ^{R}}_{R\varphi }&{\Gamma ^{R}}_{\theta \varphi }&{\Gamma ^{R}}_{\varphi \varphi }\\{\Gamma ^{\theta }}_{R\varphi }&{\Gamma ^{\theta }}_{\theta \varphi }&{\Gamma ^{\theta }}_{\varphi \varphi }\\{\Gamma ^{\varphi }}_{R\varphi }&{\Gamma ^{\varphi }}_{\theta \varphi }&{\Gamma ^{\varphi }}_{\varphi \varphi }\\\end{pmatrix}}&=\quad {\begin{pmatrix}0&0&-R\cos ^{2}\theta \\0&0&\cos \theta \sin \theta \\1/R&-\tan \theta &0\end{pmatrix}}\\\end{aligned}}}

These values show how the tangent directions (columns: ${\displaystyle e_{R}}$, ${\displaystyle e_{\theta }}$, ${\displaystyle e_{\varphi }}$) change, seen from an outside perspective (e.g. from space), but given in the tangent directions of the actual location (rows: R, θ, φ).

As an example, take the nonzero derivatives by θ in ${\displaystyle {\Gamma ^{k}}_{j\ \theta }}$, which corresponds to a movement towards north (positive dθ):

• The new north direction ${\displaystyle e_{\theta }}$ changes by -R dθ in the up (R) direction. So the north direction will rotate downwards towards the center of the earth.
• Similarly, the up direction ${\displaystyle e_{R}}$ will be adjusted towards the north. The different lengths of ${\displaystyle e_{R}}$ and ${\displaystyle e_{\theta }}$ lead to a factor of 1/R .
• Moving north, the east tangent vector ${\displaystyle e_{\varphi }}$ changes its length (-tan(θ) on the diagonal), it will shrink (-tan(θ) dθ < 0) on the northern hemisphere, and increase (-tan(θ) dθ > 0) on the southern hemisphere.

These effects are maybe not apparent during the movement, because they are the adjustments that keep the measurements in the coordinates R, θ, φ. Nevertheless, it can affect distances, physics equations, etc. So if e.g. you need the exact change of a magnetic field pointing approximately "south", it can be necessary to also correct your measurement by the change of the north direction using the Christoffel symbols to get the "true" (tensor) value.

The Christoffel symbols of the first kind ${\displaystyle {\Gamma _{l}}_{ji}=g_{lk}{\Gamma ^{k}}_{ji}}$ show the same change using metric-corrected coordinates, e.g. for derivative by φ:

{\displaystyle {\begin{aligned}{\begin{pmatrix}{\Gamma _{R}}_{R\varphi }&{\Gamma _{R}}_{\theta \varphi }&{\Gamma _{R}}_{\varphi \varphi }\\{\Gamma _{\theta }}_{R\varphi }&{\Gamma _{\theta }}_{\theta \varphi }&{\Gamma _{\theta }}_{\varphi \varphi }\\{\Gamma _{\varphi }}_{R\varphi }&{\Gamma _{\varphi }}_{\theta \varphi }&{\Gamma _{\varphi }}_{\varphi \varphi }\\\end{pmatrix}}&=R\cos \theta {\begin{pmatrix}0&0&-\cos \theta \\0&0&R\sin \theta \\\cos \theta &-R\sin \theta &0\end{pmatrix}}\\\end{aligned}}}

## Notes

1. See, for instance, ( Spivak 1999 ) and ( Choquet-Bruhat & DeWitt-Morette 1977 )
2. Ronald Adler, Maurice Bazin, Menahem Schiffer, Introduction to General Relativity (1965) McGraw-Hill Book Company ISBN   0-07-000423-4 (See section 2.1)
3. Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation (1973) W. H. Freeman ISBN   0-7167-0334-3 (See chapters 8-11)
4. Misner, Thorne, Wheeler, op. cit. (See chapter 13)
5. Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag ISBN   3-540-42627-2
6. David Bleeker, Gauge Theory and Variational Principles (1991) Addison-Wesely Publishing Company ISBN   0-201-10096-7
7. Christoffel, E.B. (1869), "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades", Journal für die reine und angewandte Mathematik, 70: 46–70
8. Ludvigsen, Malcolm (1999), General Relativity: A Geometrical Approach, p. 88
9. Chatterjee, U.; Chatterjee, N. (2010). Vector and Tensor Analysis. p. 480.
10. Struik, D.J. (1961). Lectures on Classical Differential Geometry (first published in 1988 Dover ed.). p. 114.
11. Bishop, R.L.; Goldberg (1968), Tensor Analysis on Manifolds, p. 241
12. Chatterjee, U.; Chatterjee, N. (2010). Vector & Tensor Analysis. p. 480.
13. G. Ricci-Curbastro (1896). "Dei sistemi di congruenze ortogonali in una varietà qualunque". Mem. Acc. Lincei. 2 (5): 276–322.
14. H. Levy (1925). "Ricci's coefficients of rotation". Bull. Amer. Math. Soc. 31 (3–4): 142–145. doi:.
15. This is assuming that the connection is symmetric (e.g., the Levi-Civita connection). If the connection has torsion, then only the symmetric part of the Christoffel symbol can be made to vanish.
16. Einstein, Albert (2005). "The Meaning of Relativity (1956, 5th Edition)". Princeton University Press (2005).
17. Schrödinger, E. (1950). Space-time structure. Cambridge University Press.
18. Adler, R., Bazin, M., & Schiffer, M. Introduction to General Relativity (New York, 1965).
19. David, Kay, Tensor Calculus (1988) McGraw-Hill Book Company ISBN   0-07-033484-6 (See section 11.4)

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In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In physics, the Navier–Stokes equations are certain partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.

In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in .

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols , , or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf(p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f(p).

In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In special relativity, a four-vector is an object with four components, which transform in a specific way under Lorentz transformation. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts.

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M. The analogous solution for a charged, rotating body is given by the Kerr–Newman metric.

In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.

The rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the linear rotor requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane. The rigid-rotor Schroedinger equation is discussed in Section 11.2 on pages 240-253 of the textbook by Bunker and Jensen.

One of the guiding principles in modern chemical dynamics and spectroscopy is that the motion of the nuclei in a molecule is slow compared to that of its electrons. This is justified by the large disparity between the mass of an electron and the typical mass of a nucleus and leads to the Born-Oppenheimer approximation and the idea that the structure and dynamics of a chemical species are largely determined by nuclear motion on potential energy surfaces. The potential energy surfaces are obtained within the adiabatic or Born–Oppenheimer approximation. This corresponds to a representation of the molecular wave function where the variables corresponding to the molecular geometry and the electronic degrees of freedom are separated. The non separable terms are due to the nuclear kinetic energy terms in the molecular Hamiltonian and are said to couple the potential energy surfaces. In the neighbourhood of an avoided crossing or conical intersection, these terms cannot be neglected. One therefore usually performs one unitary transformation from the adiabatic representation to the so-called diabatic representation in which the nuclear kinetic energy operator is diagonal. In this representation, the coupling is due to the electronic energy and is a scalar quantity that is significantly easier to estimate numerically.

A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism.

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions . There are two kinds: the regular solid harmonics, which vanish at the origin and the irregular solid harmonics, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

Curvilinear coordinates can be formulated in tensor calculus, with important applications in physics and engineering, particularly for describing transportation of physical quantities and deformation of matter in fluid mechanics and continuum mechanics.