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Calculus 3 (Multivariable calculus) Part 1 of 2.

Towards and through the vector fields, part 1 of 2.
Instructor:
Hania Uscka-Wehlou
98 students enrolled
English [Auto]
How to solve problems in multivariable calculus (illustrated with more than 200 solved problems) and why these methods work.
Parameterize some curves (straight lines, circles, ellipses, graphs of functions of one variable, intersections of two surfaces).
Describe position, velocity, speed and acceleration; compute arc length of parametric curves; arc length parametrization.
Limits, continuity and differentiability for functions of several variables. Theory, geometric intuitions, and lots of problem solving.
Several variants of the Chain Rule, involving different kinds os functions. You will also learn how to apply these variants of the Chain Rule for problem solving.
Several variants of the Implicit Function Theorem, with various geometrical interpretations; problem solving.
Optimization of functions of several variables, both on open domains and on compact domains (Lagrange multipliers on the boundary, etc.).

Calculus 3 / Multivariable Calculus. Part 1 of 2.

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)

  1. About the course

  2. Analytical geometry in R^n (n = 2 and n = 3): points, position vectors, lines and planes, distance between points (Ch.10.1)

  3. Conic sections (circle, ellipse, parabola, hyperbola) and quadric surfaces (spheres, cylinders, cones, ellipsoids, paraboloids etc) (Ch.10.5)

  4. Topology in R^n: distance, open ball, neighbourhood, open and closed set, inner and outer point, boundary point. (Ch.10.1)

  5. Coordinates: Cartesian, polar, cylindrical, spherical coordinates (Ch.10.6)

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)

  1. Introduction to vector-valued functions

  2. Some examples of parametrisation

  3. Vector-valued calculus; curve: continuous, differentiable and smooth

  4. Arc length

  5. Arc length parametrisation

You will learn: Parametrise some curves (straight lines, circles, ellipses, graphs of functions of one variable); if r(t) = (x(t), y(t), z(t)) is a function describing a particle’s position in R^3 with respect to time t, describe position, velocity, speed and acceleration; compute arc length of parametric curves, arc length parametrisation.

C2 Functions of several variables; differentiability (Chapter 12)

  1. Real-valued functions in multiple variables, domain, range, graph surface, level curves, level surfaces
    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.

  2. 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.

  3. Partial derivative, tangent plane, normal line
    You will learn: calculate first-order partial derivatives, compute scalar products (two formulas) and cross pro- duct, give formulas for normals and tangent planes; understand functions from R^n to R^m, gradients and Jacobians.

  4. Higher partial derivates
    You will learn: compute higher order partial derivatives, use Schwarz’ theorem. Solve and verify some simple PDE’s.

  5. Chain rule: different versions
    You will learn: calculate the chain rule using dependency diagrams and matrix multiplication.

  6. Linear approximation, linearisation, differentiability, differential
    You will learn: determine if a function is differentiable in a point, linearisation of a real-valued function, use linearisation to derive an approximate value of a function, use the test for differentiability (continuous partial derivatives), and properties of differentiable functions.

  7. 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.

  8. Implicit functions
    You will learn: calculate the Jacobian determinant, derive partial derivatives with dependent and free variables of implicit functions.

  9. 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.

C3 Optimisation of functions of several variables (Chapter 13: 13.1–3)

  1. Optimisation on open domains (critical points)

  2. Optimisation on compact domains

  3. 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.

Also make sure that you check with your professor what parts of the course you will need for your midterms. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.

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.

About the course

1
Introduction to the course

Analytical geometry in the space

1
The plane R^2 and the 3-space R^3: points and vectors

This is the description. Where does it show up?

2
Distance between points
3
Vectors and their products
4
Dot product
5
Cross product
6
Scalar triple product
7
Describing reality with numbers; geometry and physics
8
Straight lines in the plane
9
Planes in the space
10
Straight lines in the space

Conic sections: circle, ellipse, parabola, hyperbola

1
Conic sections, an introduction
2
Quadratic curves as conic sections
3
Definitions by distance
4
Cheat sheets
5
Circle and ellipse, theory

Extra material: proof that the definition by distance of an ellipse leads to the right equation.

6
Parabola and hyperbola, theory

Extra material: proof that the definition by distance of a parabola leads to the right equation.

7
Completing the square
8
Completing the square, problems 1 and 2
9
Completing the square, problem 3
10
Completing the square, problems 4 and 5
11
Completing the square, problems 6 and 7

Quadric surfaces: spheres, cylinders, cones, ellipsoids, paraboloids etc

1
Quadric surfaces, an introduction
2
Degenerate quadrics
3
Ellipsoids
4
Paraboloids
5
Hyperboloids
6
Problems 1 and 2
7
Problem 3
8
Problems 4 and 5
9
Problem 6

Topology in R^n

1
Neighborhoods
2
Open, closed, and bounded sets
3
Identify sets, an introduction
4
Example 1
5
Example 2
6
Example 3
7
Example 4
8
Example 5
9
Example 6 and 7

Coordinate systems

1
Different coordinate systems
2
Polar coordinates in the plane
3
An important example
4
Solving 3 problems
5
Cylindrical coordinates in the space
6
Problem 1
7
Problem 2
8
Problem 3
9
Problem 4
10
Spherical coordinates in the space
11
Some examples
12
Conversion
13
Problem 1
14
Problem 2
15
Problem 3
16
Problem 4

Vector-valued functions, introduction

1
Curves: an introduction
2
Functions: repetition
3
Vector-valued functions, parametric curves
4
Vector-valued functions, parametric curves: domain

Some examples of parametrisation

1
Vector-valued functions, parametric curves: parametrisation
2
An intriguing example
3
Problem 1
4
Problem 2
5
Problem 3
6
Problem 4, helix

Vector-valued calculus; curve: continuous, differentiable, and smooth

1
Notation
2
Limit and continuity
3
Derivatives
4
Speed, acceleration
5
Position, velocity, acceleration: an example
6
Smooth and piecewise smooth curves
7
Sketching a curve
8
Sketching a curve: an exercise
9
Example 1
10
Example 2
11
Example 3
12
Extra theory: limit and continuity
13
Extra theory: derivative, tangent, and velocity
14
Differentiation rules
15
Differentiation rules, example 1
16
Differentiation rules: example 2
17
Position, velocity, acceleration, example 3
18
Position and velocity, one more example
19
Trajectories of planets

Arc length

1
Parametric curves: arc length
2
Arc length: problem 1
3
Arc length: problems 2 and 3
4
Arc length: problems 4 and 5

Arc length parametrisation

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