Multivariable Version of Taylor’s Theorem

Posted on August 9, 2016 by

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.

Advertisements

About mathtuition88

http://mathtuition88.com
This entry was posted in math and tagged , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

Gravatar
WordPress.com Logo

You are commenting using your WordPress.com account. ( Log Out /  Change )

Google photo

You are commenting using your Google account. ( Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. ( Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. ( Log Out /  Change )

Cancel

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.