Function Repository Resource:

# ChristoffelSymbol

Return the Christoffel symbol for a given metric

Contributed by: Lars Ulke-Winter
 ResourceFunction["ChristoffelSymbol"][metric,coords] creates the Christoffel symbol of the second kind for metric specified in coordinate variables coords. ResourceFunction["ChristoffelSymbol"][metric,coords,"Kind"→"First"] creates the Christoffel symbol of the first kind.

## Details and Options

Christoffel Symbols are rank-3 objects defined by the relation (with base vectors ek and coordinate variables zj).
Christoffel symbols of the first kind are usually written as , though some text books use the ordering .
Input metric should be a matrix or StructuredArray expression.
ResourceFunction["ChristoffelSymbol"] outputs a triple nested List.

## Examples

### Basic Examples (3)

Christoffel symbol of the second kind for Euclidean cylindrical coordinates:

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=

Christoffel symbol of the first kind for cylindrical coordinates:

 In[3]:=
 Out[4]=

Christoffel symbols of a sphere:

 In[5]:=
 Out[6]=
 In[7]:=
 Out[10]=

### Applications (5)

#### The Christoffel symbol on a torus described with surface coordinates  (5)

Describe a toroidal surface in ambient Cartesian space with surface coordinates:

 In[11]:=

Plot the surface:

 In[12]:=
 Out[12]=

Form a covariant basis on the surface:

 In[13]:=
 Out[13]=

Get the covariant surface metric tensor:

 In[14]:=
 Out[14]=

This leads to the related Christoffel symbol of the second kind of the toroidal surface:

 In[15]:=
 Out[17]=

Lars Ulke-Winter

## Version History

• 1.0.0 – 08 October 2019