In a classic paper in 1869, Elwin Bruno Christoffel (1829-1900) introduced his famous Christoffel symbols to represent an array of numbers describing a metric connection. They are also known as connection coefficients (and sometimes less respectfully as `Christ-awful symbols’). In differential geometry one usually first encounters them when studying covariant derivatives of tensors in tensor calculus. For example, suppose we try to differentiate the contravariant vector , where denotes a coordinate basis vector (and we are using the Einstein summation convention). We get

In general, the partial derivative in the second term on the right will result in another vector which we can write in terms of its coordinate basis as

This defines the Christoffel symbol . The downstairs indices refer to the rate of change of the basis components with respect to the coordinate variable in the direction of the coordinate basis vector ( being the upstairs index). Substituting the second equation into the first we get

To enable us to factor out the coordinate basis vector we can exchange the symbols and in the second term on the right to get

The expression in the bracket is called the covariant derivative of the contravariant vector , i.e., the rate of change of in each of the directions of the coordinate system . It has the important property that it is itself tensorial (unlike the ordinary partial derivative of the tensor on its own). This covariant derivative is often written using the notation

Having thus established the meaning of the Christoffel symbols, one then goes on to work out that the covariant derivative of a one-form is

and that the covariant derivatives of higher rank tensors are constructed from the building blocks of and by adding a term for each upper index and a term for each lower index . For example, the covariant derivative of the rank-2 tensor is

Christoffel symbols then go on to play vital roles in other areas of differential geometry, perhaps most notably as key components in the definition of the Riemann curvature tensor.

It is possible to have a working knowledge of all of this without truly understanding at a deep level, say geometrically, what Christoffel symbols really mean. In the present note I want to delve a bit more deeply into how one might calculate and interpret Christoffel symbols geometrically. I also want to explore some alternative ways of calculating them in the context of a simple plane polar coordinate system which is related to the usual Cartesian coordinate system via the conversion equations

In an dimensional manifold there are potentially Christoffel symbols to be calculated, though this number is usually reduced by symmetries. In the present plane polar coordinate case, we will need to calculate Christoffel symbols. These are

**Geometric approach**

Consider the situation shown in the diagram below where two vectors and of the basis vector field are drawn emanating from points and respectively:

If we parallel transport the vector from to we end up with the situation shown in the next diagram:

Now, in plane polar coordinates the magnitude of is

Therefore the length of the arc in the diagram is

If is small, we have

where is the vector connecting the endpoints of and , i.e., .

Therefore

Passing to the differential limit as we get

From the diagram we see that points in the opposite direction of . Therefore we have

(note that in plane polar coordinates is of unit length). From this equation we have

But from the definition of Christoffel symbols we have

Therefore we conclude

We have obtained the first two Christoffel symbols on our list from the geometric setup and the nice thing about this approach is that we can see what the underlying changes in the coordinate basis vectors looked like.

To obtain the next two Christoffel symbols on our list, we consider a change in the vector field due to a displacement in the radial direction from to in the following diagram:

We have moved outwards by a small amount and as a result the length of the vectors in the vector field has increased by a small amount shown in the diagram. From the diagram we see that the proportions of the two increases must be same, so we have

or

Passing to the differential limit as we get

Since is directed along the vector we can write the vector equation

so

But

from which we conclude

We have thus found two more Christoffel symbols from the geometrical setup. To get the next two Christoffel symbols on our list we observe that the basis vector field does not change as we move in the radial direction (either in magnitude or direction) so we must have

where the right hand side here denotes a zero vector. But we know that

so we conclude

Finally, to get the last two remaining Christoffel symbols on our list, we consider a change in the vector field due to an angular displacement. In the diagram below two vectors and of the basis vector field are drawn emanating from points and respectively:

If we parallel transport the vector from to we end up with the situation shown in the next diagram:

The arc length is

(since the magnitude of the coordinate basis vector is ). But for small we also have

where is the vector connecting the endpoints of and , i.e., . Therefore

Passing to the differential limit as we have

But has the same direction as . Therefore

where the factor is needed to correct for the magnitude of (we only want the direction of here). Therefore we see that

But

from which we conclude

This completes the geometric calculation of all the Christoffel symbols for plane polar coordinates.

**Algebraic approach**

It is possible to calculate the eight Christoffel symbols quite easily for plane polar coordinates by first expressing the basis components and in terms of the Cartesian components and . Note that these basis components are one-forms, so they transform as

We use the conversion equations

to calculate the coefficients. We get

and therefore

Then we calculate the Christoffel symbols as follows. First,

so

and we conclude

Next,

so

from which we conclude

Next,

so

Therefore we conclude

Finally,

so

from which we conclude

**Metric tensor approach**

The previous approach relied on knowing the functional relationship between the Cartesian coordinates and the plane polar coordinates . There is another more generally useful method of calculating the Christoffel symbols from the components of the metric tensor, using the formula

I will first derive this formula from first principles, then use it to find the Christoffel symbols for the plane polar coordinates case.

The first step is to show that Christoffel symbols are symmetric in their lower indices, i.e.,

as this property will be needed in the derivation of the formula. To prove the symmetry property we start from the defining equation for Christoffel symbols,

Suppose we now decompose the basis vectors in a local Cartesian coordinate system. Then using the transformation rule for one-forms we have

where the are the Cartesian coordinates and the are the coordinate basis vectors (which are constant in both magnitude and direction in the Cartesian system). Differentiating gives

Equating the expressions for we get

But then

so it follows from Young’s Theorem (equality of cross-partials) that

We conclude that Christoffel symbols are symmetric in their lower indices, as claimed.

Note too that the components of the general metric tensor are also symmetric with respect to their indices. This follows from the defining equation of the metric tensor components in terms of the basis vector fields , namely

Since

the metric is symmetric, i.e.,

To derive the formula for the Christoffel symbols in terms of the metric tensor components, we begin again with the defining equation for Christoffel symbols,

Taking the scalar product with another basis vector on both sides we get

Therefore we have

In the second term on the right hand side we can rename and use the fact that the metric is symmetric to reverse the indices. We get

By cyclically renaming the indices , , and we can generate two more similar equations. From the cyclic permutation , , , , we get

and from the cyclic permutation , , , , we get

Now we add the first two equations and subtract the third to get

where we have taken advantage of the symmetry in the lower indices of the Christoffel symbols to cancel some terms. Using the fact that

we multiply both sides by to get the final formula:

This is made easier to remember by noting the following facts. A factor of the inverse metric generates the Christoffel symbol’s upper index. The negative term has the symbol’s lower indices as the indices of the metric. The other two terms in the bracket are cyclic permutations of this last term.

Having derived the formula we can now employ it to calculate the eight Christoffel symbols for plane polar coordinates. We can work out the metric tensor using the distance formula

with the conversion equations

Then

so

Therefore the metric in plane polar coordinates is

The metric tensor is therefore

and the inverse metric is

Now, in the formula for the indices , , and represent the polar coordinates and in various permutations. Inspection of shows that the only partial derivative terms which do not equal zero are

Inspection of shows that this equals zero except when

and

Substituting these values of the metric tensor components into the formula

we get