Numerical Analysis 8.1. Multivariable Calculus скачать в хорошем качестве

Numerical Analysis 8.1. Multivariable Calculus

This video, titled "Numerical Analysis 8. 1." by the channel Csoda81, provides a review of multivariable calculus concepts essential for understanding function minimization. Key Concepts Covered: Minimization of Functions: The lecture introduces Chapter 8, which focuses on the problem of finding the minimum of real-valued functions with vector variables [00:03]. The Hessian Matrix: It is defined as an n×n matrix consisting of the second-order partial derivatives of a function [00:19]. Critical Points: A point a is a critical point if the gradient vector (all first-order partial derivatives) is equal to zero [01:00]. Sufficient Conditions for Local Extrema: For a function that is two times continuously differentiable, the following conditions apply at a critical point a [02:30]: Local Minimum: The gradient is zero and the Hessian matrix is positive definite [01:49]. Local Maximum: The gradient is zero and the Hessian matrix is negative definite [02:11]. Special Case: Two-Variable Functions The video details a specific formula for functions with two variables using the value D (the determinant of the Hessian) [03:37]: D=f xx ​ (a,b)⋅f yy ​ (a,b)−[f xy ​ (a,b)] 2 If D 0: The function has a local maximum if f xx ​ 0 [04:17]. The function has a local minimum if f xx ​ 0 [04:40]. If D 0: There is no extremum at that point (often referred to as a saddle point) [04:51]. If D=0: The test is inconclusive, and no general statement can be made [04:58].