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 be a times differentiable function at the point . Then there exists such that and .
Example (, )