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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?

$$\begin{align} f(x+u,y+v,z+w) \approx f(x,y,z) &+ u \frac{\partial f (x,y,z)}{\partial x}+v \frac{\partial f (x,y,z)}{\partial y} + w \frac{\partial f (x,y,z)}{\partial z} \\ &+uv \frac{\partial^2 f (x,y,z)}{\partial x \partial y} + vw \frac{\partial^2 f (x,y,z)}{\partial y \partial z}+ uw \frac{\partial^2 f (x,y,z)}{\partial x \partial z} \\ &+ uvw \frac{\partial^3 f (x,y,z)}{\partial x \partial y \partial z} \end{align}$$

Hosein Rahnama
  • 14,882
Gina
  • 353

3 Answers3

10

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.

Hosein Rahnama
  • 14,882
6

It is not correct! We should have the expansion as $f(x+u,y+v,z+w)\approx f(x,y,z) + u \frac{\partial f(x,y,z)}{\partial x} + v\frac{\partial f(x,y,z)}{\partial y} + w\frac{\partial f(x,y,z)}{\partial z} + \frac{1}{2!} \left[u^2 \frac{\partial^2 f(x,y,z)}{\partial x^2} + v^2 \frac{\partial^2 f(x,y,z)}{\partial y^2} + w^2 \frac{\partial^2 f(x,y,z)}{\partial z^2} + 2 uv \frac{\partial^2 f(x,y,z)}{\partial x \partial y} + 2 vw\frac{\partial^2 f(x,y,z)}{\partial y \partial z} + 2 uw\frac{\partial^2 f(x,y,z)}{\partial x \partial z}\right] + \cdots$

adipro
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Kamal Kant Misra
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  • Note, the site supports Latex. Type in $\frac{\delta}{\delta x}$ and you will get $\frac{\delta}{\delta x}$. – peterh May 10 '19 at 05:05
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Let $f$ be an infinitely differentiable function in some open neighborhood around $(x,y,z)=(a,b,c)$.

$f( x,y,z) =f\left( a,b,c \right) +\left( x-a,y-b,z-c \right) \cdot \left( \begin{array}{c} \frac{\partial f}{\partial x}\left( a,b,c \right)\\ \frac{\partial f}{\partial y}\left( a,b,c \right)\\ \frac{\partial f}{\partial z}\left( a,b,c \right)\\ \end{array} \right) +\frac{1}{2}\left( x-a,y-b,z-c \right) \cdot \left( \left[ \begin{matrix} \frac{\partial ^2f}{\partial x^2}& \frac{\partial ^2f}{\partial x\partial y}& \frac{\partial ^2f}{\partial x\partial z}\\ \frac{\partial ^2f}{\partial y\partial x}& \frac{\partial ^2f}{\partial y^2}& \frac{\partial ^2f}{\partial y\partial z}\\ \frac{\partial ^2f}{\partial z\partial x}& \frac{\partial ^2f}{\partial z\partial y}& \frac{\partial ^2f}{\partial z^2}\\ \end{matrix} \right] _{\left( x,y,z \right) =\left( a,b,c \right)}\cdot \left( \begin{array}{c} x-a\\ y-b\\ z-c\\ \end{array} \right) \right)$

Fei PAN
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