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