Vectors

Vector: size and direction

Scalar: size but no direction

Matrix

Vectors can be represented with matrix.

A Jacobian matrix is a matrix of first partial derivatives.

$$J=\left[\begin{array}{c}\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\end{array}\right]$$

A Hessian matrix is a matrix of second partial derivatives

$$H_{i,j}=\frac{\partial {}^{2}f}{\partial x_i\partial x_j}$$