Unit 1 - About the course | |||
Introduction to the course | 00:30:00 | ||
Unit 2 - Analytical geometry in the space | |||
The plane R^2 and the 3-space R^3: points and vectors | 00:25:00 | ||
Distance between points | 00:08:00 | ||
Vectors and their products | 00:04:00 | ||
Dot product | 00:14:00 | ||
Cross product | 00:13:00 | ||
Scalar triple product | 00:07:00 | ||
Describing reality with numbers; geometry and physics | 00:06:00 | ||
Straight lines in the plane | 00:08:00 | ||
Planes in the space | 00:13:00 | ||
Straight lines in the space | 00:08:00 | ||
Unit 3 - Conic Units: circle, ellipse, parabola, hyperbola | |||
Conic Units, an introduction | 00:06:00 | ||
Quadratic curves as conic Units | 00:10:00 | ||
Definitions by distance | 00:17:00 | ||
Cheat sheets | 00:04:00 | ||
Circle and ellipse, theory | 00:19:00 | ||
Parabola and hyperbola, theory | 00:12:00 | ||
Completing the square | 00:04:00 | ||
Completing the square, problems 1 and 2 | 00:12:00 | ||
Completing the square, problem 3 | 00:10:00 | ||
Completing the square, problems 4 and 5 | 00:08:00 | ||
Completing the square, problems 6 and 7 | 00:08:00 | ||
Unit 4 - Quadric surfaces: spheres, cylinders, cones, ellipsoids, paraboloids etc | |||
Quadric surfaces, an introduction | 00:16:00 | ||
Degenerate quadrics | 00:17:00 | ||
Ellipsoids | 00:08:00 | ||
Paraboloids | 00:16:00 | ||
Hyperboloids | 00:25:00 | ||
Problems 1 and 2 | 00:09:00 | ||
Problem 3 | 00:07:00 | ||
Problems 4 and 5 | 00:10:00 | ||
Problem 6 | 00:06:00 | ||
Unit 5 - Topology in R^n | |||
Neighborhoods | 00:07:00 | ||
Open, closed, and bounded sets | 00:14:00 | ||
Identify sets, an introduction | 00:04:00 | ||
Example 1 | 00:06:00 | ||
Example 2 | 00:06:00 | ||
Example 3 | 00:05:00 | ||
Example 4 | 00:06:00 | ||
Example 5 | 00:04:00 | ||
Example 6 and 7 | 00:06:00 | ||
Unit 6 - Coordinate systems | |||
Different coordinate systems | 00:02:00 | ||
Polar coordinates in the plane | 00:11:00 | ||
An important example | 00:07:00 | ||
Solving 3 problems | 00:19:00 | ||
Cylindrical coordinates in the space | 00:03:00 | ||
Problem 1 | 00:03:00 | ||
Problem 2 | 00:02:00 | ||
Problem 3 | 00:04:00 | ||
Problem 4 | 00:04:00 | ||
Spherical coordinates in the space | 00:08:00 | ||
Some examples | 00:08:00 | ||
Conversion | 00:08:00 | ||
Problem 1 | 00:08:00 | ||
Problem 2 | 00:12:00 | ||
Problem 3 | 00:11:00 | ||
Problem 4 | 00:07:00 | ||
Unit 7 - Vector-valued functions, introduction | |||
Curves: an introduction | 00:10:00 | ||
Functions: repetition | 00:08:00 | ||
Functions: repetition | 00:08:00 | ||
Vector-valued functions, parametric curves: domain | 00:08:00 | ||
Unit 8 - Some examples of parametrisation | |||
Vector-valued functions, parametric curves | 00:11:00 | ||
An intriguing example | 00:14:00 | ||
Problem 1 | 00:12:00 | ||
Problem 2 | 00:13:00 | ||
Problem 3 | 00:15:00 | ||
Problem 4, helix | 00:09:00 | ||
Unit 9 - Vector-valued calculus; curve: continuous, differentiable, and smooth | |||
Notation | 00:05:00 | ||
Limit and continuity | 00:09:00 | ||
Derivatives | 00:14:00 | ||
Speed, acceleration | 00:08:00 | ||
Position, velocity, acceleration: an example | 00:06:00 | ||
Smooth and piecewise smooth curves | 00:09:00 | ||
Sketching a curve | 00:15:00 | ||
Sketching a curve: an exercise | 00:16:00 | ||
Example 1 | 00:11:00 | ||
Example 2 | 00:16:00 | ||
Example 3 | 00:10:00 | ||
Extra theory: limit and continuity | 00:19:00 | ||
Extra theory: derivative, tangent, and velocity | 00:13:00 | ||
Differentiation rules | 00:27:00 | ||
Differentiation rules, example 1 | 00:19:00 | ||
Differentiation rules: example 2 | 00:19:00 | ||
Position, velocity, acceleration, example 3 | 00:15:00 | ||
Position and velocity, one more example | 00:15:00 | ||
Trajectories of planets | 00:13:00 | ||
Unit 10 - Arc length | |||
Parametric curves: arc length | 00:15:00 | ||
Arc length: problem 1 | 00:11:00 | ||
Arc length: problems 2 and 3 | 00:15:00 | ||
Arc length: problems 4 and 5 | 00:13:00 | ||
Unit 11 - Arc length parametrisation | |||
Parametric curves: parametrisation by arc length | 00:10:00 | ||
Parametrisation by arc length, how to do it, example 1 | 00:12:00 | ||
Parametrisation by arc length, example 2 | 00:22:00 | ||
Arc length does not depend on parametrisation, theory | 00:14:00 | ||
Unit 12 - Real-valued functions of multiple variables | |||
Functions of several variables, introduction | 00:09:00 | ||
Introduction, continuation 1 | 00:14:00 | ||
Introduction, continuation 2 | 00:08:00 | ||
Domain | 00:06:00 | ||
Domain, problem solving part 1 | 00:18:00 | ||
Domain, problem solving part 2 | 00:13:00 | ||
Domain, problem solving part 3 | 00:15:00 | ||
Functions of several variables, graphs | 00:14:00 | ||
Plotting functions of two variables, problems part 1 | 00:16:00 | ||
Plotting functions of two variables, problems part 2 | 00:12:00 | ||
Level curves | 00:14:00 | ||
Level curves, problem 1 | 00:10:00 | ||
Level curves, problem 2 | 00:08:00 | ||
Level curves, problem 3 | 00:09:00 | ||
Level curves, problem 4 | 00:14:00 | ||
Level curves, problem 5 | 00:16:00 | ||
Level surfaces, definition and problem solving | 00:20:00 | ||
Unit 13 - Limit, continuity | |||
Limit and continuity, part 1 | 00:18:00 | ||
Limit and continuity, part 2 | 00:15:00 | ||
Limit and continuity, part 3 | 00:20:00 | ||
Problem solving 1 | 00:25:00 | ||
Problem solving 2 | 00:18:00 | ||
Problem solving 3 | 00:20:00 | ||
Problem solving 4 | 00:15:00 | ||
Unit 14 - Partial derivative, tangent plane, normal line, gradient, Jacobian | |||
Introduction 1: definition and notation | 00:10:00 | ||
Introduction 2: arithmetical consequences | 00:12:00 | ||
Introduction 3: geometrical consequences (tangent plane) | 00:13:00 | ||
Introduction 4: partial derivatives not good enough | 00:06:00 | ||
Introduction 5: a pretty terrible example | 00:15:00 | ||
Tangent plane, part 1 | 00:07:00 | ||
Normal vector | 00:15:00 | ||
Tangent plane part 2: normal equation | 00:09:00 | ||
Normal line | 00:08:00 | ||
Tangent planes, problem 1 | 00:14:00 | ||
Tangent planes, problem 2 | 00:13:00 | ||
Tangent planes, problem 3 | 00:16:00 | ||
Tangent planes, problem 4 | 00:09:00 | ||
Tangent planes, problem 5 | 00:11:00 | ||
The gradient | 00:11:00 | ||
A way of thinking about functions from R^n to R^m | 00:11:00 | ||
The Jacobian | 00:14:00 | ||
Unit 15 - Higher partial derivatives | |||
Introduction | 00:15:00 | ||
Definition and notation | 00:07:00 | ||
Mixed partials, Hessian matrix | 00:13:00 | ||
The difference between Jacobian matrices and Hessian matrices | 00:08:00 | ||
Equality of mixed partials; Schwarz’ theorem | 00:09:00 | ||
Schwarz’ theorem: Peano’s example | 00:06:00 | ||
Schwarz’ theorem: the proof | 00:19:00 | ||
Partial Differential Equations, introduction | 00:04:00 | ||
Partial Differential Equations, basic ideas | 00:11:00 | ||
Partial Differential Equations, problem solving | 00:13:00 | ||
Laplace equation and harmonic functions 1 | 00:08:00 | ||
Laplace equation and harmonic functions 2 | 00:06:00 | ||
Laplace equation and Cauchy-Riemann equations | 00:11:00 | ||
Dirichlet problem | 00:07:00 | ||
Unit 16 - Chain rule: different variants | |||
A general introduction | 00:17:00 | ||
Variants 1 and 2 | 00:10:00 | ||
Variant 3 | 00:18:00 | ||
Variant 3 (proof) | 00:11:00 | ||
Variant 4 | 00:09:00 | ||
Example with a diagram | 00:04:00 | ||
Problem solving | 00:08:00 | ||
Problem solving, problem 1 | 00:04:00 | ||
Problem solving, problem 2 | 00:09:00 | ||
Problem solving, problem 3 | 00:33:00 | ||
Problem solving, problem 4 | 00:15:00 | ||
Problem solving, problem 6 | 00:09:00 | ||
Problem solving, problem 7 | 00:06:00 | ||
Problem solving, problem 5 | 00:28:00 | ||
Problem solving, problem 8 | 00:18:00 | ||
Unit 17 - Linear approximation, linearisation, differentiability, differential | |||
Linearisation and differentiability in Calc1 | 00:11:00 | ||
Differentiability in Calc3: introduction | 00:15:00 | ||
Differentiability in two variables, an example | 00:10:00 | ||
Differentiability in Calc3 implies continuity | 00:10:00 | ||
Partial differentiability does NOT imply differentiability | 00:05:00 | ||
An example: continuous, not differentiable | 00:06:00 | ||
Differentiability in several variables, a test | 00:18:00 | ||
Differentiability, Partial Differentiability, and Continuity in Calc3 | 00:12:00 | ||
Differentiability in two variables, a geometric interpretation | 00:11:00 | ||
Linearization: two examples | 00:16:00 | ||
Linearization, problem solving 1 | 00:11:00 | ||
Linearization, problem solving 2 | 00:11:00 | ||
Linearization, problem solving 3 | 00:12:00 | ||
Linearization by Jacobian matrix, problem solving | 00:16:00 | ||
Differentials: problem solving 1 | 00:11:00 | ||
Differentials: problem solving 2 | 00:10:00 | ||
Unit 18 - Gradient, directional derivatives | |||
Gradient | 00:04:00 | ||
The gradient in each point is orthogonal to the level curve through the point | 00:08:00 | ||
The gradient in each point is orthogonal to the level surface through the point | 00:14:00 | ||
Tangent plane to the level surface, an example | 00:06:00 | ||
Directional derivatives, introduction | 00:06:00 | ||
Directional derivatives, the direction | 00:04:00 | ||
How to normalize a vector and why it works | 00:08:00 | ||
Directional derivatives, the definition | 00:07:00 | ||
Partial derivatives as a special case of directional derivatives | 00:05:00 | ||
Directional derivatives, an example | 00:11:00 | ||
Directional derivatives: important theorem for computations and interpretations | 00:10:00 | ||
Directional derivatives: an earlier example revisited | 00:05:00 | ||
Geometrical consequences of the theorem about directional derivatives | 00:10:00 | ||
Geometical consequences of the theorem about directional derivatives, an example | 00:07:00 | ||
Directional derivatives, an example | 00:11:00 | ||
Normal line and tangent line to a level curve: how to get their equations | 00:06:00 | ||
Normal line and tangent line to a level curve: their equations, an example | 00:14:00 | ||
Gradient and directional derivatives, problem 1 | 00:18:00 | ||
Gradient and directional derivatives, problem 2 | 00:20:00 | ||
Gradient and directional derivatives, problem 3 | 00:09:00 | ||
Gradient and directional derivatives, problem 4 | 00:04:00 | ||
Gradient and directional derivatives, problem 5 | 00:12:00 | ||
Gradient and directional derivatives, problem 6 | 00:10:00 | ||
Gradient and directional derivatives, problem 7 | 00:13:00 | ||
Unit 19 - Implicit functions | |||
What is the Implicit Function Theorem? | 00:13:00 | ||
Jacobian determinant | 00:04:00 | ||
Jacobian determinant for change to polar and to cylindrical coordinates | 00:07:00 | ||
Jacobian determinant for change to spherical coordinates | 00:09:00 | ||
Jacobian determinant and change of area | 00:10:00 | ||
The Implicit Function Theorem variant 1 | 00:08:00 | ||
The Implicit Function Theorem variant 1, an example | 00:15:00 | ||
The Implicit Function Theorem variant 2 | 00:10:00 | ||
The Implicit Function Theorem variant 2, example 1 | 00:07:00 | ||
The Implicit Function Theorem variant 2, example 2 | 00:14:00 | ||
The Implicit Function Theorem variant 3 | 00:15:00 | ||
The Implicit Function Theorem variant 3, an example | 00:12:00 | ||
The Implicit Function Theorem variant 4 | 00:11:00 | ||
The Inverse Function Theorem | 00:09:00 | ||
The Implicit Function Theorem, summary | 00:04:00 | ||
Notation in some unclear cases | 00:08:00 | ||
The Implicit Function Theorem, problem solving 1 | 00:27:00 | ||
The Implicit Function Theorem, problem solving 2 | 00:13:00 | ||
The Implicit Function Theorem, problem solving 3 | 00:07:00 | ||
The Implicit Function Theorem, problem solving 4 | 00:16:00 | ||
Unit 20 - Taylor’s formula, Taylor’s polynomial, quadratic forms | |||
Taylor’s formula, introduction | 00:10:00 | ||
Quadratic forms and Taylor’s polynomial of second degree | 00:22:00 | ||
Taylor’s polynomial of second degree, theory | 00:11:00 | ||
Taylor’s polynomial of second degree, example 1 | 00:07:00 | ||
Taylor’s polynomial of second degree, example 2 | 00:04:00 | ||
Taylor’s polynomial of second degree, example 3 | 00:11:00 | ||
Classification of quadratic forms (positive definite etc) | 00:12:00 | ||
Classification of quadratic forms, problem solving 1 | 00:08:00 | ||
Classification of quadratic forms, problem solving 2 | 00:14:00 | ||
Classification of quadratic forms, problem solving 3 | 00:10:00 | ||
Unit 21 - Optimization on open domains (critical points) | |||
Extreme values of functions of several variables | 00:12:00 | ||
Extreme values of functions of two variables, without computations | 00:10:00 | ||
Critical points and their classification (max, min, saddle) | 00:09:00 | ||
Second derivative test for C^3 functions of several variables | 00:12:00 | ||
Second derivative test for C^3 functions of two variables | 00:07:00 | ||
Critical points and their classification: some simple examples | 00:06:00 | ||
Critical points and their classification: more examples 1 | 00:05:00 | ||
Critical points and their classification: more examples 2 | 00:08:00 | ||
Critical points and their classification: more examples 3 | 00:10:00 | ||
Critical points and their classification: a more difficult example (4) | 00:47:00 | ||
Unit 22 - Optimization on compact domains | |||
Extreme values for continuous functions on compact domains | 00:06:00 | ||
Eliminate a variable on the boundary | 00:10:00 | ||
Parameterize the boundary | 00:08:00 | ||
Unit 23 - Lagrange multipliers (optimization with constraints) | |||
Lagrange multipliers 1 | 00:13:00 | ||
Lagrange multipliers 1, an old example revisited | 00:08:00 | ||
Lagrange multipliers 1, another example | 00:13:00 | ||
Lagrange multipliers 2 | 00:10:00 | ||
Lagrange multipliers 2, an example | 00:18:00 | ||
Lagrange multipliers 3 | 00:08:00 | ||
Lagrange multipliers 3, an example | 00:09:00 | ||
Summary: optimization | 00:07:00 | ||
Unit 24 - Final words | |||
The last one | 00:05:00 | ||
Assignment | |||
Assignment – Essential Calculus Training Masterclass | 3 weeks, 2 days | ||
Order Your Certificate | |||
Order Your Certificate QLS | 00:00:00 |
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