The Jacobian matrix of an autonomous system x'=f(x) of differential equations at a point x0 is the matrix Df(x0) of partial derivatives of the right-hand side evaluated at that point.

The matrix of first-order partial derivatives. For robots, the Jacobian relates the end- effector velocity the joint speeds.

The determinant formed by the 2 partial derivatives of functions of variables, when the derivatives of each function occupy one row of the determinant. For the case of two functions (, ) and (, ), the Jacobian (, ) is sometimes also written The geostrophic advection of any scalar ψ may be written where is the acceleration of gravity, the Coriolis parameter, and the isobaric contour height.

The Jacobian of a nonsingular projective curve is an abelian variety whose points are in bijection with the group Pic of isomorphism classes of invertible sheaves (or divisor classes) of degree 0.

Of or pertaining to James the First, of England, or of his reign or times; especially, pertaining to a style of architecture and decoration popular in the time of James I.; as, Jacobean writers.