TITLE: Taylor Series for a Function of $3$ Variables
QUESTION [7 upvotes]: 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?
$$\begin{align}
f(x+u,y+v,z+w) \approx f(x,y,z) &+ u \frac{\partial f (x,y,z)}{\partial x}+v \frac{\partial f (x,y,z)}{\partial y} + w \frac{\partial f (x,y,z)}{\partial z}
\\
&+uv \frac{\partial^2 f (x,y,z)}{\partial x \partial y} + vw \frac{\partial^2 f (x,y,z)}{\partial y \partial z}+ uw \frac{\partial^2 f (x,y,z)}{\partial x \partial z} \\
&+ uvw \frac{\partial^3 f (x,y,z)}{\partial x \partial y \partial z}
\end{align}$$

REPLY [3 votes]: Let $f$ be an infinitely differentiable function in some open neighborhood around $(x,y,z)=(a,b,c)$.
$f( x,y,z) =f\left( a,b,c \right) +\left( x-a,y-b,z-c \right) \cdot \left( \begin{array}{c}
 \frac{\partial f}{\partial x}\left( a,b,c \right)\\
 \frac{\partial f}{\partial y}\left( a,b,c \right)\\
 \frac{\partial f}{\partial z}\left( a,b,c \right)\\
\end{array} \right) +\frac{1}{2}\left( x-a,y-b,z-c \right) \cdot \left( \left[ \begin{matrix}
 \frac{\partial ^2f}{\partial x^2}&  \frac{\partial ^2f}{\partial x\partial y}&  \frac{\partial ^2f}{\partial x\partial z}\\
 \frac{\partial ^2f}{\partial y\partial x}&  \frac{\partial ^2f}{\partial y^2}&  \frac{\partial ^2f}{\partial y\partial z}\\
 \frac{\partial ^2f}{\partial z\partial x}&  \frac{\partial ^2f}{\partial z\partial y}&  \frac{\partial ^2f}{\partial z^2}\\
\end{matrix} \right] _{\left( x,y,z \right) =\left( a,b,c \right)}\cdot \left( \begin{array}{c}
 x-a\\
 y-b\\
 z-c\\
\end{array} \right) \right)$