Now
that we have some idea on the concept of metric tensor, we now move on to
another very important concept in the curvilinear system (the Christoffel
symbol). Let us consider the curved coordinate system $(u_{1}, u_{2}, u_{3}…. u_{j})$
and their respective direction tangent vector be represented by $\vec{e_{i}}$. We must
remember that the direction vector (basis vector) is not a constant, i.e. it is
position dependent quantities. But, what if we differentiate the basis vector
with respect to the coordinate axes? What can we assume about the answer? Well,
we can still say that the differentiated quantity is still a vector of the same
basis vector, but what about the coefficient of the vector. The coefficient is
what we call as the

**Christoffel Symbol.**
I.e. mathematically we can write as
follows,

This
then is the Christoffel symbol. In the above definition we can calculate the
coefficient by comparing a coefficient to the left hand side of the equation.
However, this method in general is not the method used to calculate the
Christoffel symbol. In general we find the Christoffel symbol from the metric
of the given coordinate system.

We
now derive the relation between the metric tensor and the Christoffel symbol.
We know the relation of the metric tensor and the unit tangent vector $e_{1}, e_{2},
e_{3}….$ is as follows,

We
now differentiate ${g_{pq}}$ with respect to the coordinate axes; we then have
the following,

This is the relation of the
Christoffel to the metric tensor. Notice that we can write the Christoffel
solely in terms of the metric tensor. If we cycle through the indices ${{p,q,k}}$ we then have
different relation for the metric tensor and the Christoffel symbol. In
particular two of the following,

Now
adding $(2),(3)$ and subtracting $(1)$ we have,

Where,
$\partial_{p}$ implies a partial derivative with respect to $p^{th}$ coordinate
axes. And also contracting the metric in the left side of the equation we then
have the Christoffel symbol as generally seen in many text books as,

We
have thus found the relationship between the Christoffel symbol and the metric
tensor.

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