You live at some spot in an n dimensional Riemannian manifold and have a convenient coordinate system whose origin is at that spot. Each of the indexes i, j, k and l take on one of n values; there are n⁴ such combinations and as many values for Rⁱ_jkl. For each such set of values Rⁱ_jkl can be evaluated by the following simple procedure. Travel in direction k until coordinate x^k is increased by dx^k. The other coordinates will not have changed. Next travel indirection l for distance dx^l. Now travel backwards in the k direction but a distance –dx^k. Then once more in the direction l for distance –dx^l. You will arrive back near to where you left. That was just practice.

Now travel the same route again but carry along a small vector V that initially points in the direction j with magnitude dx^j. At each step in the journey do not turn V that you carry, even though the coordinates may themselves turn which will cause the coordinate description of your vector to change. This would be a big effect if you were walking near the North pole. There are several ways that the meaning of this can be defined which we do not explore here. When you get back compare the V you carried with a copy that you left behind. The difference between the vector you started with and the one you brought back will be a new vector with components D¹, ...Dⁿ.
Rⁱ_jkl = Dⁱ/(dx^jdx^kdx^l).

You must choose the size of dx^j etc. and this is a practical tradeoff—too small and D will be too small to measure, to big and your curvature result won’t really be local.

Γⁱ_jk tells how fast the coordinates turn.