Poisson bracket with the symplectic form

Question

Consider the following extract from a text:

I can't seem to varify the second equality in the definition. Maybe the author meant to write $(\frac{\partial f}{\partial \mathbf{x}})^T J \frac{\partial g}{\partial \mathbf x}$ ?

If someone could clarify this for me I would greatly appreciate it.

Answer 1

Yes, there is a transposition missing. The definition should say $$ \{f,g\}:=\left(\frac{\partial f}{\partial \mathbf{x}}\right)^T J \frac{\partial g}{\partial \mathbf x}. $$