Taylor's Theorem

Analogous to the one-dimensional Taylor’s theorem.

Suppose . If all partial derivatives of order are continuous in a closed region and all the partial derivatives of order exist in the open region then:

Here

And . And the below expression is expanded using binomial theorem.

Taylor Series

Linear & quadratic approximations

Linear approximation for a function can be made using the first-order Taylor series expansion:

Quadratic approximation for a function can be made using the second-order Taylor series expansion: