Calculus 3 (multivariable calculus), part 1 of 2.

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Develop a thorough understanding of the fundamental concepts and strategies of multivariate calculus

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                                    Medium of instructions
                                    English

                                    Mode of learning
                                    Self study

                                    Mode of Delivery
                                    Video and Text Based

About the course

Introduction to the course

Analytical geometry in the space

The plane R^2 and the 3-space R^3: points and vectors
Distance between points
Vectors and their products
Dot product
Cross product
Scalar triple product
Describing reality with numbers; geometry and physics
Straight lines in the plane
Planes in the space
Straight lines in the space

Conic sections: circle, ellipse, parabola, hyperbola

Conic sections, an introduction
Quadratic curves as conic sections
Definitions by distance
Cheat sheets
Circle and ellipse, theory
Parabola and hyperbola, theory
Completing the square
Completing the square, problems 1 and 2
Completing the square, problem 3
Completing the square, problems 4 and 5
Completing the square, problems 6 and 7

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

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

Topology in R^n

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

Coordinate systems

Different coordinate systems
Polar coordinates in the plane
An important example
Solving 3 problems
Cylindrical coordinates in the space
Problem 1
Problem 2
Problem 3
Problem 4
Spherical coordinates in the space
Some examples
Conversion
Problem 1
Problem 2
Problem 3
Problem 4

Vector-valued functions, introduction

Curves: an introduction
Functions: repetition
Vector-valued functions, parametric curves
Vector-valued functions, parametric curves: domain

Some examples of parametrization

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

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

Notation
Limit and continuity
Derivatives
Speed, acceleration
Position, velocity, acceleration: an example
Smooth and piecewise smooth curves
Sketching a curve
Sketching a curve: an exercise
Example 1
Example 2
Example 3
Extra theory: limit and continuity
Extra theory: derivative, tangent, and velocity
Differentiation rules
Differentiation rules, example 1
Differentiation rules: example 2
Position, velocity, acceleration, example 3
Position and velocity, one more example
Trajectories of planets

Arc length

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

Arc length parametrization

Parametric curves: parametrization by arc length
Parametrization by arc length, how to do it, example 1
Parametrization by arc length, example 2
Arc length does not depend on parametrization, theory

Real-valued functions of multiple variables

Functions of several variables, introduction
Introduction, continuation 1
Introduction, continuation 2
Domain
Domain, problem solving part 1
Domain, problem solving part 2
Domain, problem solving part 3
Functions of several variables, graphs
Plotting functions of two variables, problems part 1
Plotting functions of two variables, problems part 2
Level curves
Level curves, problem 1
Level curves, problem 2
Level curves, problem 3
Level curves, problem 4
Level curves, problem 5
Level surfaces, definition and problem solving

Limit, continuity

Limit and continuity, part 1
Limit and continuity, part 2
Limit and continuity, part 3
Problem solving 1
Problem solving 2
Problem solving 3
Problem solving 4

Partial derivative, tangent plane, normal line, gradient, Jacobian

Introduction 1: definition and notation
Introduction 2: arithmetical consequences
Introduction 3: geometrical consequences (tangent plane)
Introduction 4: partial derivatives not good enough
Introduction 5: a pretty terrible example
Tangent plane, part 1
Normal vector
Tangent plane part 2: normal equation
Normal line
Tangent planes, problem 1
Tangent planes, problem 2
Tangent planes, problem 3
Tangent planes, problem 4
Tangent planes, problem 5
The gradient
A way of thinking about functions from R^n to R^m
The Jacobian

Higher partial derivatives

Introduction
Definition and notation
Mixed partials, Hessian matrix
The difference between Jacobian matrices and Hessian matrices
Equality of mixed partials; Schwarz' theorem
Schwarz' theorem: Peano's example
Schwarz' theorem: the proof
Partial Differential Equations, introduction
Partial Differential Equations, basic ideas
Partial Differential Equations, problem solving
Laplace equation and harmonic functions 1
Laplace equation and harmonic functions 2
Laplace equation and Cauchy-Riemann equations
Dirichlet problem

Chain rule: different variants

A general introduction
Variants 1 and 2
Variant 3
Variant 3 (proof)
Variant 4
Example with a diagram
Problem solving
Problem solving, problem 1
Problem solving, problem 2
Problem solving, problem 3
Problem solving, problem 4
Problem solving, problem 6
Problem solving, problem 7
Problem solving, problem 5
Problem solving, problem 8

Linear approximation, linearisation, differentiability, differential

Linearization and differentiability in Calc1
Differentiability in Calc3: introduction
Differentiability in two variables, an example
Differentiability in Calc3 implies continuity
Partial differentiability does NOT imply differentiability
An example: continuous, not differentiable
Differentiability in several variables, a test
Wrap-up: differentiability, partial differentiability, and continuity in Calc3
Differentiability in two variables, a geometric interpretation
Linearization: two examples
Linearization, problem solving 1
Linearization, problem solving 2
Linearization, problem solving 3
Linearization by Jacobian matrix, problem solving
Differentials: problem solving 1
Differentials: problem solving 2

Gradient, directional derivatives

Gradient
The gradient in each point is orthogonal to the level curve through the point
The gradient in each point is orthogonal to the level surface through the point
Tangent plane to the level surface, an example
Directional derivatives, introduction
Directional derivatives, the direction
How to normalize a vector and why it works
Directional derivatives, the definition
Partial derivatives as a special case of directional derivatives
Directional derivatives, an example
Directional derivatives: important theorem for computations and interpretations
Directional derivatives: an earlier example revisited
Geometrical consequences of the theorem about directional derivatives
Geomatical consequences of the theorem about directional derivatives, an example
Directional derivatives, an example
Normal line and tangent line to a level curve: how to get their equations
Normal line and tangent line to a level curve: their equations, an example
Gradient and directional derivatives, problem 1
Gradient and directional derivatives, problem 2
Gradient and directional derivatives, problem 3
Gradient and directional derivatives, problem 4
Gradient and directional derivatives, problem 5
Gradient and directional derivatives, problem 6
Gradient and directional derivatives, problem 7

Implicit functions

What is the Implicit Function Theorem?
Jacobian determinant
Jacobian determinant for change to polar and to cylindrical coordinates
Jacobian determinant for change to spherical coordinates
Jacobian determinant and change of area
The Implicit Function Theorem variant 1
The Implicit Function Theorem variant 1, an example
The Implicit Function Theorem variant 2
The Implicit Function Theorem variant 2, example 1
The Implicit Function Theorem variant 2, example 2
The Implicit Function Theorem variant 3
The Implicit Function Theorem variant 3, an example
The Implicit Function Theorem variant 4
The Inverse Function Theorem
The Implicit Function Theorem, summary
Notation in some unclear cases
The Implicit Function Theorem, problem solving 1
The Implicit Function Theorem, problem solving 2
The Implicit Function Theorem, problem solving 3
The Implicit Function Theorem, problem solving 4

Taylor's formula, Taylor's polynomial, quadratic forms

Taylor's formula, introduction
Quadratic forms and Taylor's polynomial of second degree
Taylor's polynomial of second degree, theory
Taylor's polynomial of second degree, example 1
Taylor's polynomial of second degree, example 2
Taylor's polynomial of second degree, example 3
Classification of quadratic forms (positive definite etc)
Classification of quadratic forms, problem solving 1
Classification of quadratic forms, problem solving 2
Classification of quadratic forms, problem solving 3

Optimization on open domains (critical points)

Extreme values of functions of several variables
Extreme values of functions of two variables, without computations
Critical points and their classification (max, min, saddle)
Second derivative test for C^3 functions of several variables
Second derivative test for C^3 functions of two variables
Critical points and their classification: some simple examples
Critical points and their classification: more examples 1
Critical points and their classification: more examples 2
Critical points and their classification: more examples 3
Critical points and their classification: a more difficult example (4)

Optimization on compact domains

Extreme values for continuous functions on compact domains
Eliminate a variable on the boundary
Parameterize the boundary

Lagrange multipliers (optimization with constraints)

Lagrange multipliers 1
Lagrange multipliers 1, an old example revisited
Lagrange multipliers 1, another example
Lagrange multipliers 2
Lagrange multipliers 2, an example
Lagrange multipliers 3
Lagrange multipliers 3, an example
Summary: optimization

Final words

The last one

Extras

Bonus Lecture

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