How Does the Chain Rule Relate to Tangent Vectors in Calculus?

[silverwhale]

Homework Statement

Show that:

\frac{dx^\nu}{d \lambda} \partial_\nu \frac{dx^\mu}{d \lambda} = \frac{d^2 x^\mu}{d \lambda^2}

The Attempt at a Solution

Well, I could simply cancel the dx^nu and get the desired result; that I do understand.
But what about actually looking at this term alone:

\partial_\nu \frac{dx^\mu}{d \lambda},

calculating it and multiplying with dx^nu/dλ, can I get the same result? I get confused by the question: what if the partial derivative acts on the tangent vector; what happens then?


Thanks for your help!

 

[clamtrox]

The problem is that \frac{d}{d\lambda}and \partial_\mu do not commute... So I'm not sure how you could calculate it without knowing what \frac{d}{d\lambda} is.