Taylor Series and Taylor's Theorem for Multivariable Functions, Real Analysis II

In this video we extend the familiar Taylor series concept from single-variable calculus to functions of multiple variables. We begin by revisiting the foundational ideas of Taylor series for functions of one variable (mentioning the convenience of big-O notation for truncated series). This review leads to the formulation of Taylor polynomials in higher dimensions. We first specifically write down the first and second-order Taylor approximations, T1 and T2, for scalar-valued functions of multiple variables. We notice that T1 repeats the first-order approximation that we've been discussing during our study of differentiability, while T2 introduces the quadratic approximation via the Hessian. (Hessian presentation here: https://youtu.be/uWOHsCL7Hik) The core of our discussion is Taylor's theorem itself, with Lagrange's form of the remainder. We dissect the theorem and show how it generalizes the mean value theorem. Finally, we outline a proof strategy, based on our prior understanding of Taylor's theorem for single-variable functions (video here: https://youtu.be/0tbRSpQlMeU). By parameterizing the line segment between p and x, we reduce the problem to a composition, allowing us to apply the one-dimensional theorem. We work out the first and second derivatives of this composition explicitly, and then make a claim regarding the general form of the kth derivative. #multivariablecalculus #taylorstheorem #taylorseries #hessian #gradient #jacobian #realanalysis #mathematicalproofs #advancedcalculus #mathematicalanalysis