Exploring the Relationship Between the Chain Rule and Tangent Vectors

[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.