MP4 | Video: h264, 1280×720 | Audio: AAC, 44100 Hz
What you’ll learn
Describe position, velocity, speed and acceleration; compute arc length of parametric curves; arc length parametrization.
Requirements
Calculus 1 and 2
Some linear algebra
Description
Towards and through the vector fields.
(Chapter numbers in Robert A. Adams, Christopher Essex: Calculus, a complete course. 8th or 9th edition.)
C0 Introduction to the course; preliminaries (Chapter 10: very briefly; most of the chapter belongs to prerequisites)
About the course
Analytical geometry in R^n (n = 2 and n = 3): points, position vectors, lines and planes, distance between points (Ch.10.1)
Conic sections (circle, ellipse, parabola, hyperbola) and quadric surfaces (spheres, cylinders, cones, ellipsoids, paraboloids etc) (Ch.10.5)
Topology in R^n: distance, open ball, neighbourhood, open and closed set, inner and outer point, boundary point. (Ch.10.1)
You will learn: to understand which geometrical objects are represented by simpler equations and inequalities in R^2 and R^3, determine whether a set is open or closed, if a point is an inner, outer or boundary point, determine the boundary points, describe points and other geometrical objects in the different coordinate systems.
C1 Vector-valued functions, parametric curves (Chapter 11: 11.1, 11.3)
Introduction to vector-valued functions
Some examples of parametrisation
Vector-valued calculus; curve: continuous, differentiable and smooth
Arc length
Arc length parametrisation
You will learn: describe the domain and range of a function, Illustrate a function f(x,y) with a surface graph or with level curves.
Limit, continuity
You will learn: calculate limit values, determine if a function has limit value or is continuous at one point, use common sum-, product-, . rules for limits.
Chain rule: different versions
You will learn: calculate the chain rule using dependency diagrams and matrix multiplication.
Linear approximation, linearisation, differentiability, differential
Gradient, directional derivatives
You will learn: calculate the gradient, find the direction derivative in a certain direction, properties of gradients, understand the geometric interpretation of the directional derivative, give a formula for the tangent and normal lines to a level curve.
Implicit functions
Taylor’s formula, Taylor’s polynomial
You will learn: derive Taylor’s polynomials and Taylor’s formula. Understand quadratic forms and learn how to determine if they are positive definite, negative definite, or indefinite.
Optimisation on open domains (critical points)
Optimisation on compact domains
Lagrange multipliers (optimisation with constraints)
You will learn: classify critical points: local max and min, saddle points; find max and min values for a given function and region; use Lagrange multipliers with one or more conditions.
A detailed description of the content of the course, with all the 255 videos and their titles, and with the texts of all the 216 problems solved during this course, is presented in the resource file "Outline_Calculus3.pdf" under video 1 ("Introduction to the course"). This content is also presented in video 1.
Who this course is for:
University and college engineering
RapidGator
https://www.keeplinks.org/p27/695c384e39156