Carl Gustav Jacob Jacobi

1. Jacobian Determinant

Used for changing variables in multiple integrals.

$$ J = \det \frac{\partial(y_1, \dots, y_n)}{\partial(x_1, \dots, x_n)} $$

2. Jacobi Identity

A property of the Poisson bracket and Lie algebras.

$$ [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 $$