Gradient, Hessian, and Jacobian

The problem

A set y=f(x) of m equations in n variable x1,...,xn

f:   

The first-order partial derivatives of a vector-valued function, so called Jacobian, or Jacobian Matrix, matrix size is n x m.

http://en.wikipedia.org/wiki/Jacobian

http://mathworld.wolfram.com/Jacobian.html

The problem 

A set y=f(x) of 1 equations in n variable x1,...,xn

f:   

The first-order partial derivatives of a vector-valued function, so called Gradient, or Gradient vector. Matrix size is 1 x n.

http://en.wikipedia.org/wiki/Gradient

The second-order partial derivatives of a vector-valued function, so called Hessain, or Hessain Matrix. Matrix size is n x n.

http://en.wikipedia.org/wiki/Hessian_matrix

http://mathworld.wolfram.com/Hessian.html

Reference

http://www.riskglossary.com/link/gradient_hessian_jacobian.htm