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 .
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 .
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 .
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: