Taylor expansion

For a function of one variable, say $g(x)$ we can write down a Taylor expansion about a point $x_{0}$,

where $h=x-x_{0}$. In abbreviated notation, Eq. (1.14) is:

Recall that if we choose $x_{0}=0$, we have the Taylor series about $x=0$, known as the Maclaurin series of $g(x)$. Now, the approximation is valid if the function $g(x)$ and any derivatives used are continuous in the interval $[x, x+h]$ and the values of the function and the derivatives are known at the point $x=x_{0}$.

Under a similar set of conditions, we can use the same idea to expand a function of two variables. For the case of $f=f(x, y)$, the Taylor expansion about the point $\left(x_{0}, y_{0}\right)$ is given by:

where $h=x-x_{0}$ and $k=y-y_{0}$. Equation (1.16) is a second-order expansion. Notice the use of Clairaut's theorem and the equality of mixed partials, i.e. $\frac{\partial^{2} f}{\partial x \partial y}=\frac{\partial^{2} f}{\partial y \partial x}$. This results in the binomial coefficients (shown in bold) in the second-order term. It follows that including a third-order expansion in our approximation of $f(x, y)$, and expressing it in abbreviated notation gives:

Again, these ideas may be extended to functions of three or more variables. Finally, we make use of the notation introduced in Subsec. 1.2.2. namely $\nabla f$ and $\mathcal{H} f$, to express the expansion given by Eq. (1.16):

where $\mathbf{h}=\left(\begin{array}{c}h \\ k\end{array}\right)$ while $\nabla f$ and $\mathcal{H} f$ are given by Eqs. (1.12) and (1.13), respectively.