Riemannian Manifold: Integral Formula Explained

[whattttt]

Can someone please explain to me how this formula
integral from 0 to T of
sqrt(g_ij c'(t) c'(t))
I have seen it on wikipedia but don't know how to actually implement the formula.

 

[slider142]

(g_ij) is the metric tensor, c'(t) is, for each t, a tangent vector to the curve c(t).
With respect to a specific choice of coordinates, (g_ij) is a matrix that you premultiply and postmultiply by c'(t) to get a scalar. For example, the standard Euclidean metric in R² with Cartesian coordinates is
\left(\begin{array}{cc}1 & 0\\0 & 1\end{array}\right)
A typical curve in 2-dimensional Euclidean space with Cartesian coordinates has tangent vector
c'(t) = \left(\begin{array}{c}\frac{dx}{dt}\\ \frac{dy}{dt}\end{array}\right)
Premultiplying and postmultiplying gets us the expression:
\left(\begin{array}{cc}\frac{dx}{dt} & \frac{dy}{dt}\end{array}\right) \left(\begin{array}{cc}1 & 0\\0 & 1\end{array}\right) \left(\begin{array}{c}\frac{dx}{dt}\\ \frac{dy}{dt}\end{array}\right)
= \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2
Different metrics will give different notions of length.