Math 5 minutes

Derivative, Gradient and Jacobian unified

A summary about scalar and vector derivatives.

Functions of scalars

Let a function. For , the derivative of at point is:

This number is the slope of the tangent to the curve described by .

Derivative of scalars

Vector functions of scalars

Let a function. For , the derivative of at point is:

Which is the vector of the partial derivative:

This vector is tangent to the parametric curve described by .

Derivative of vectors

Functions of vectors

Let a function. For , the derivative of at point is:

We call this row vector the gradient of . It is noted:

This vector is tangent to the suface described by and directed along the steepest ascent of .

Gradient

Vector functions of vectors

Let a function. For , the derivative of at point can be writen in two equivalent forms.

Since is the derivative of a scalar function with respect to a vector, we have:

Which corresponds to the following matrix in :

This matrix is called the jacobian of .

Actually, when all the components of are fixed except , can be considered as a function of the scalar . In this view, is the derivative of a vector function with respect to the scalar , and we have: