Taylor Series:
Suppose is a real function on
, is a positive integer, the function
are
continuous on , and exits for all , Let be distinct points in and define
Then there exists such that
Look at function , that is, such that and
(real-valued
functions of arguments)
Gradient: Â We can stack the partial derivatives of in the vector: is called the gradient of
If is we have
that is, the order of differentiation does not matter.
Definition the Hessian Matrix: Â The matrix of all
second partial derivatives of :
Taylor series expansions in : As in the univariate case,
we can expand around a given
point . The
following are the most frequently used results:
First-order Taylor series expansion:
Second-order Taylor series expansion:
Vector-valued functions
Let and
consider the functions , each from . This can be written more
compactly as: , where ,
such function is called a
vector-valued function.
The Jacobian matrix If are differentiable,
we can write the total differential of the vector-valued function  , as:
or, in matrix notation,
The first-order Taylor series expansion for a vector valued function
f takes the form:
