Zwiebach 210 Hmwk: Commutator not Equal 0

[ehrenfest]

Homework Statement

Zwiebach claims that differentiating 12.12, which is

$$X^I (\tau, \sigma) X^J(\tau, \sigma')-X^J(\tau, \sigma') X^I(\tau, \sigma) = 0$$

w.r.t. tau gives the same commutator with everything dotted. I think that is downright wrong. If you use the product rule to write everything out, it just doesn't work out.

Homework Equations

The Attempt at a Solution

 

[Jimmy Snyder]

Apply equation (12.21) to the second equation in the first line of equations (12.12).

 

[ehrenfest]

You also have to use 12.1 to get the leftmost part, don't you?