Taylor Series for a Function of two variables

Question

I am interested up to the 3rd Order of Taylor expansion.

I was wondering if the following Taylor expansion is correct:

\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} + \frac{1}{2}\Big(u^2 \frac{\partial^2f(x,y)}{\partial x^2}+2uv \frac{\partial^2 f (x,y)}{\partial x \partial y}+v^2 \frac{\partial^2f(x,y)}{\partial y^2}\Big) + \frac{1}{6} \Big(u^3 \frac{\partial^3f(x,y)}{\partial x^3}+v^3 \frac{\partial^3f(x,y)}{\partial y^3}+3u^2v\frac{\partial^2f(x,y)}{\partial x^2}\frac{\partial f (x,y)}{\partial y}+ 3uv^2 \frac{\partial f (x,y)}{\partial x}\frac{\partial^2f(x,y)}{\partial y^2}\Big) \end{equation}

Answer 1

\begin{equation}\label{eq16} \begin{split} f(x + u,y + v) &= \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{\partial ^{n}f (x,y)}{\partial x^{n}}\cdot\frac{\partial ^{k}f (x,y)}{\partial y^{k}}\cdot\frac{(u)^{n}}{n!}\cdot\frac{(v)^{k}}{k!} \\&= \underbrace{f(x,y)}_{n = k = 0} + \underbrace{u \frac{\partial f (x,y)}{\partial x}}_{n = 1\\k = 0}+\underbrace{v \frac{\partial f (x,y)}{\partial y}}_{n = 0\\k = 1} + \Big(\underbrace{\frac{1}{2}u^2 \frac{\partial^2f(x,y)}{\partial x^2}}_{n=2\\k = 0}+ \underbrace{uv \frac{\partial^2 f (x,y)}{\partial x \partial y}}_{n=k=1}+ \underbrace{\frac{1}{2}v^2 \frac{\partial^2f(x,y)}{\partial y^2}}_{n=0\\k=2}\Big) + \Big(\underbrace{\frac{1}{6}u^3 \frac{\partial^3f(x,y)}{\partial x^3}}_{n=3\\k=0}+\underbrace{\frac{1}{6}v^3 \frac{\partial^3f(x,y)}{\partial y^3}}_{n=0\\k=3}+\underbrace{\frac{1}{2}u^2v\frac{\partial^2f(x,y)}{\partial x^2}\frac{\partial f (x,y)}{\partial y}}_{n=2\\k=1}+ \underbrace{\frac{1}{2}uv^2 \frac{\partial f (x,y)}{\partial x}\frac{\partial^2f(x,y)}{\partial y^2}}_{n=1\\k=2}\Big) \\&+ \mathcal{O}(u^4) +\mathcal{O}(v^4) + \mathcal{O}(u^3v) + \mathcal{O}(uv^3)+ \mathcal{O}(u^2v^2) \end{split} \end{equation}

Answer 2

Not exactly: first, it is better to give an exact formula, as $\approx$ has no real mathematical meaning, and the 3rd order terms are: $$\frac16\biggl(u^3\frac{\partial^3f}{\partial x^3}+3u^2v\frac{\partial^3f}{\partial x^2\partial y}+3uv^2\frac{\partial^3f}{\partial x^2\partial y^2}+v^3\frac{\partial^3f}{\partial y^3}\biggr)+o\Bigl(\lVert(x,y\bigr\rVert^3\Bigr)$$

A way to remember this (and to extend it to higher orders and more variables) is to expand the differential operator: $$\biggl(u\frac{\partial\phantom x }{\partial x}+v\frac{\partial\phantom y }{\partial y}\biggr)^{\mkern-5mu3}\cdot f$$