Multivariable Version of Taylor’s Theorem

Multivariable calculus is an interesting topic that is often neglected in the curriculum. Furthermore it is hard to learn since the existing textbooks are either too basic/computational (e.g. Multivariable Calculus, 7th Edition by Stewart) or too advanced. Many analysis books skip multivariable calculus altogether and just focus on measure and integration.

If anyone has a good book that covers multivariable calculus (preferably rigorously with proofs), do post it in the comments!

The following is a useful multivariable version of Taylor’s Theorem, using the multi-index notation which is regarded as the most efficient way of writing the formula.

Multivariable Version of Taylor’s Theorem

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a $k$ times differentiable function at the point $\mathbf{a}\in\mathbb{R}^n$ . Then there exists $h_\alpha:\mathbb{R}^n\to\mathbb{R}$ such that $\displaystyle f(\mathbf{x})=\sum_{|\alpha|\leq k}\frac{D^\alpha f(\mathbf{a})}{\alpha!}(\mathbf{x}-\mathbf{a})^\alpha+\sum_{|\alpha|=k}h_\alpha(\mathbf{x})(\mathbf{x}-\mathbf{a})^\alpha,$ and $\lim_{\mathbf{x}\to\mathbf{a}}h_\alpha(\mathbf{x})=0$ .

Example ( $n=2$ , $k=1$ )

Write $\mathbf{x}-\mathbf{a}=\mathbf{v}$ .
$\displaystyle f(x,y)=f(\mathbf{a})+\frac{\partial f}{\partial x}(\mathbf{a})v_1+\frac{\partial f}{\partial y}(\mathbf{a})v_2+h_{(1,0)}(x,y)v_1+h_{(0,1)}(x,y)v_2.$