Taylor Series for a Function of $3$ Variables The Taylor expansion of the function $f(x,y)$ is:
\begin{equation}
f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)}{\partial x \partial y}
\end{equation}
When $f=(x,y,z)$ is the following true?
$$\begin{align}
f(x+u,y+v,z+w) \approx f(x,y,z) &+ u \frac{\partial f (x,y,z)}{\partial x}+v \frac{\partial f (x,y,z)}{\partial y} + w \frac{\partial f (x,y,z)}{\partial z}
\\
&+uv \frac{\partial^2 f (x,y,z)}{\partial x \partial y} + vw \frac{\partial^2 f (x,y,z)}{\partial y \partial z}+ uw \frac{\partial^2 f (x,y,z)}{\partial x \partial z} \\
&+ uvw \frac{\partial^3 f (x,y,z)}{\partial x \partial y \partial z}
\end{align}$$
 A: The general formula for the Taylor expansion of a sufficiently smooth real valued function $f:\mathbb{R}^n \to \mathbb{R}$ at $\mathbf{x}_0$ is
$$f({\bf{x}})=f({\bf{x}}_0)+\nabla f({\bf{x}}_0) \cdot ({\bf{x}}-{\bf{x}}_0) + \frac{1}{2} ({\bf{x}}-{\bf{x}}_0) \cdot \nabla \nabla f ({\bf{x}}_0) \cdot ({\bf{x}}-{\bf{x}}_0) + O(\lVert\mathbf{{\bf{x}}-{\bf{x}}_0}\rVert^2)$$
If you call ${\bf{x}}-{\bf{x}}_0:={\bf{h}}$ then the above formula can be rewritten as
$$f({\bf{x}}_0+{\bf{h}})=f({\bf{x}}_0)+\nabla f({\bf{x}}_0) \cdot {\bf{h}} + \frac{1}{2} {\bf{h}} \cdot \nabla \nabla f ({\bf{x}}_0) \cdot {\bf{h}} + O(\lVert\mathbf{h}\rVert^2)$$
In these formulas, $\nabla f$ is the (first) gradient of $f$, $\nabla\nabla f$ is usually called the Hessian (second gradient) of $f$, and $O$ is the famous big O notation. You can extend this formulation for functions like $f:\mathbb{R}^n \to \mathbb{R}^m$. You may also find it useful to take a look at this link.
A: It is not correct! We should have the expansion as
$f(x+u,y+v,z+w)\approx f(x,y,z) + u \frac{\partial f(x,y,z)}{\partial x} + v\frac{\partial f(x,y,z)}{\partial y} + w\frac{\partial f(x,y,z)}{\partial z} + \frac{1}{2!} \left[u^2 \frac{\partial^2 f(x,y,z)}{\partial x^2} + v^2 \frac{\partial^2 f(x,y,z)}{\partial y^2} + w^2 \frac{\partial^2 f(x,y,z)}{\partial z^2} + 2 uv \frac{\partial^2 f(x,y,z)}{\partial x \partial y} + 2 vw\frac{\partial^2 f(x,y,z)}{\partial y \partial z} + 2 uw\frac{\partial^2 f(x,y,z)}{\partial x \partial z}\right] + \cdots$
A: Let $f$ be an infinitely differentiable function in some open neighborhood around $(x,y,z)=(a,b,c)$.
$f( x,y,z) =f\left( a,b,c \right) +\left( x-a,y-b,z-c \right) \cdot \left( \begin{array}{c}
 \frac{\partial f}{\partial x}\left( a,b,c \right)\\
 \frac{\partial f}{\partial y}\left( a,b,c \right)\\
 \frac{\partial f}{\partial z}\left( a,b,c \right)\\
\end{array} \right) +\frac{1}{2}\left( x-a,y-b,z-c \right) \cdot \left( \left[ \begin{matrix}
 \frac{\partial ^2f}{\partial x^2}&  \frac{\partial ^2f}{\partial x\partial y}&  \frac{\partial ^2f}{\partial x\partial z}\\
 \frac{\partial ^2f}{\partial y\partial x}&  \frac{\partial ^2f}{\partial y^2}&  \frac{\partial ^2f}{\partial y\partial z}\\
 \frac{\partial ^2f}{\partial z\partial x}&  \frac{\partial ^2f}{\partial z\partial y}&  \frac{\partial ^2f}{\partial z^2}\\
\end{matrix} \right] _{\left( x,y,z \right) =\left( a,b,c \right)}\cdot \left( \begin{array}{c}
 x-a\\
 y-b\\
 z-c\\
\end{array} \right) \right)$
