# Intuitive explanation of covariant, contravariant and Lie derivatives

I would be glad if someone could explain in intuitive terms what these different derivatives are, and possibly give some practical, understandable examples of how they would produce different results.

To be clear, I would like to understand the geometrical or physical meaning of these operators more than the mathematical or topological subtleties that lead to them!

Thanks!

• Have you read the relevant wikipedia articles? Jul 28 '10 at 22:32
• Of course, and this question "sort of" stems from that reading :-) Jul 28 '10 at 22:38
• – user23238
Oct 9 '12 at 0:52

The Lie derivative is a derivative of a vector field V along another vector field W. It is defined at a point p as follows: flow the point p along W for some time t and look at the value of V at this point. Then push this forward along the flow of W to a vector at p. Subtract $V_p$ from this, divide by t, and take the limit as $t \to 0$. So this is a measure of how V changes as it gets pushed around by the flow of W.
The covariant derivative is a derivative of a vector field V along a vector W. Unlike the Lie derivative, this does not come for free: we need a connection, which is a way of identifying tangent spaces. The reason we need this extra data is because if we wanted to take the directional derivative of V along the vector W how we do in Euclidean space, we would be taking something like $V_{p+tW} - V_p$, which is the difference of vectors living in different tangent spaces. If we have a metric, then we can impose reasonable conditions that give us a unique connection (the Levi-Civita connection).