Towards and through the vector fields, part 1 of 2: Functions of several real variables and vector-valued functions
What you'll learn
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 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.).

Requirements
Calculus 1 and 2
Some Linear Algebra (a brief summary of some topics is contained in Section 2 of this course)
You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.

Description
Calculus 3 (multivariable calculus), part 1 of 2Towards and through the vector fields, part 1 of 2(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) S1. About the courseS2. Analytical geometry in R^n (n = 2 and n = 3): points, position vectors, lines and planes, distance between points (Ch.10.1)S3. Conic sections (circle, ellipse, parabola, hyperbola)S4. Quadric surfaces (spheres, cylinders, cones, ellipsoids, paraboloids etc) (Ch.10.5)S5. Topology in R^n: distance, open ball, neighbourhood, open and closed set, inner and outer point, boundary point (Ch.10.1)S6. 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)S7. Introduction to vector-valued functionsS8. Some examples of parametrisationS9. Vector-valued calculus; curve: continuous, differentiable and smoothS10. Arc lengthS11. Arc length parametrisationYou 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) S12. Real-valued functions in multiple variables, domain, range, graph surface, level curves, level surfacesYou will learn: describe the domain and range of a function, Illustrate a function f(x,y) with a surface graph or with level curves.S13. Limit, continuityYou 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.S14. Partial derivative, tangent plane, normal line, gradient, JacobianYou 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.S15. Higher partial derivatesYou will learn: compute higher order partial derivatives, use Schwarz' theorem. Solve and verify some simple PDE's.S16. Chain rule: different versionsYou will learn: calculate the chain rule using dependency diagrams and matrix multiplication.S17. Linear approximation, linearisation, differentiability, differentialYou 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.S18. Gradient, directional derivativesYou 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.S19. Implicit functionsYou will learn: calculate the Jacobian determinant, derive partial derivatives with dependent and free variables of implicit functions.S20. Taylor's formula, Taylor's polynomialYou 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)S21. Optimisation on open domains (critical points)S22. Optimisation on compact domainsS23. 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.ExtrasYou will learn: about all the courses we offer. You will also get a glimpse into our plans for future courses, with approximate (very hypothetical!) release dates.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 "001 Outline_Calculus3.pdf" under video 1 ("Introduction to the course"). This content is also presented in video 1.

Overview
Section 1: About the course
Lecture 1 Introduction to the course
Section 2: Analytical geometry in the space
Lecture 2 The plane R^2 and the 3-space R^3: points and vectors
Lecture 3 Distance between points
Lecture 4 Vectors and their products
Lecture 5 Dot product
Lecture 6 Cross product
Lecture 7 Scalar triple product
Lecture 8 Describing reality with numbers; geometry and physics
Lecture 9 Straight lines in the plane
Lecture 10 Planes in the space
Lecture 11 Straight lines in the space
Section 3: Conic sections: circle, ellipse, parabola, hyperbola
Lecture 12 Conic sections, an introduction
Lecture 13 Quadratic curves as conic sections
Lecture 14 Definitions by distance
Lecture 15 Cheat sheets
Lecture 16 Circle and ellipse, theory
Lecture 17 Parabola and hyperbola, theory
Lecture 18 Completing the square
Lecture 19 Completing the square, problems 1 and 2
Lecture 20 Completing the square, problem 3
Lecture 21 Completing the square, problems 4 and 5
Lecture 22 Completing the square, problems 6 and 7
Section 4: Quadric surfaces: spheres, cylinders, cones, ellipsoids, paraboloids etc
Lecture 23 Quadric surfaces, an introduction
Lecture 24 Degenerate quadrics
Lecture 25 Ellipsoids
Lecture 26 Paraboloids
Lecture 27 Hyperboloids
Lecture 28 Problems 1 and 2
Lecture 29 Problem 3
Lecture 30 Problems 4 and 5
Lecture 31 Problem 6
Section 5: Topology in R^n
Lecture 32 Neighbourhoods
Lecture 33 Open, closed, and bounded sets
Lecture 34 Identify sets, an introduction
Lecture 35 Example 1
Lecture 36 Example 2
Lecture 37 Example 3
Lecture 38 Example 4
Lecture 39 Example 5
Lecture 40 Example 6 and 7
Section 6: Coordinate systems
Lecture 41 Different coordinate systems
Lecture 42 Polar coordinates in the plane
Lecture 43 An important example
Lecture 44 Solving 3 problems
Lecture 45 Cylindrical coordinates in the space
Lecture 46 Problem 1
Lecture 47 Problem 2
Lecture 48 Problem 3
Lecture 49 Problem 4
Lecture 50 Spherical coordinates in the space
Lecture 51 Some examples
Lecture 52 Conversion
Lecture 53 Problem 1
Lecture 54 Problem 2
Lecture 55 Problem 3
Lecture 56 Problem 4
Section 7: Vector-valued functions, introduction
Lecture 57 Curves: an introduction
Lecture 58 Functions: repetition
Lecture 59 Vector-valued functions, parametric curves
Lecture 60 Vector-valued functions, parametric curves: domain
Section 8: Some examples of parametrisation
Lecture 61 Vector-valued functions, parametric curves: parametrisation
Lecture 62 An intriguing example
Lecture 63 Problem 1
Lecture 64 Problem 2
Lecture 65 Problem 3
Lecture 66 Problem 4, helix
Section 9: Vector-valued calculus; curve: continuous, differentiable, and smooth
Lecture 67 Notation
Lecture 68 Limit and continuity
Lecture 69 Derivatives
Lecture 70 Speed, acceleration
Lecture 71 Position, velocity, acceleration: an example
Lecture 72 Smooth and piecewise smooth curves
Lecture 73 Sketching a curve
Lecture 74 Sketching a curve: an exercise
Lecture 75 Example 1
Lecture 76 Example 2
Lecture 77 Example 3
Lecture 78 Extra theory: limit and continuity
Lecture 79 Extra theory: derivative, tangent, and velocity
Lecture 80 Differentiation rules
Lecture 81 Differentiation rules, example 1
Lecture 82 Differentiation rules: example 2
Lecture 83 Position, velocity, acceleration, example 3
Lecture 84 Position and velocity, one more example
Lecture 85 Trajectories of planets
Section 10: Arc length
Lecture 86 Parametric curves: arc length
Lecture 87 Arc length: problem 1
Lecture 88 Arc length: problems 2 and 3
Lecture 89 Arc length: problems 4 and 5
Section 11: Arc length parametrisation
Lecture 90 Parametric curves: parametrisation by arc length
Lecture 91 Parametrisation by arc length, how to do it, example 1
Lecture 92 Parametrisation by arc length, example 2
Lecture 93 Arc length does not depend on parametrisation, theory
Section 12: Real-valued functions of multiple variables
Lecture 94 Functions of several variables, introduction
Lecture 95 Introduction, continuation 1
Lecture 96 Introduction, continuation 2
Lecture 97 Domain
Lecture 98 Domain, problem solving part 1
Lecture 99 Domain, problem solving part 2
Lecture 100 Domain, problem solving part 3
Lecture 101 Functions of several variables, graphs
Lecture 102 Plotting functions of two variables, problems part 1
Lecture 103 Plotting functions of two variables, problems part 2
Lecture 104 Level curves
Lecture 105 Level curves, problem 1
Lecture 106 Level curves, problem 2
Lecture 107 Level curves, problem 3
Lecture 108 Level curves, problem 4
Lecture 109 Level curves, problem 5
Lecture 110 Level surfaces, definition and problem solving
Section 13: Limit, continuity
Lecture 111 Limit and continuity, part 1
Lecture 112 Limit and continuity, part 2
Lecture 113 Limit and continuity, part 3
Lecture 114 Problem solving 1
Lecture 115 Problem solving 2
Lecture 116 Problem solving 3
Lecture 117 Problem solving 4
Section 14: Partial derivative, tangent plane, normal line, gradient, Jacobian
Lecture 118 Introduction 1: definition and notation
Lecture 119 Introduction 2: arithmetical consequences
Lecture 120 Introduction 3: geometrical consequences (tangent plane)
Lecture 121 Introduction 4: partial derivatives not good enough
Lecture 122 Introduction 5: a pretty terrible example
Lecture 123 Tangent plane, part 1
Lecture 124 Normal vector
Lecture 125 Tangent plane part 2: normal equation
Lecture 126 Normal line
Lecture 127 Tangent planes, problem 1
Lecture 128 Tangent planes, problem 2
Lecture 129 Tangent planes, problem 3
Lecture 130 Tangent planes, problem 4
Lecture 131 Tangent planes, problem 5
Lecture 132 The gradient
Lecture 133 A way of thinking about functions from R^n to R^m
Lecture 134 The Jacobian
Section 15: Higher partial derivatives
Lecture 135 Introduction
Lecture 136 Definition and notation
Lecture 137 Mixed partials, Hessian matrix
Lecture 138 The difference between Jacobian matrices and Hessian matrices
Lecture 139 Equality of mixed partials; Schwarz' theorem
Lecture 140 Schwarz' theorem: Peano's example
Lecture 141 Schwarz' theorem: the proof
Lecture 142 Partial Differential Equations, introduction
Lecture 143 Partial Differential Equations, basic ideas
Lecture 144 Partial Differential Equations, problem solving
Lecture 145 Laplace equation and harmonic functions 1
Lecture 146 Laplace equation and harmonic functions 2
Lecture 147 Laplace equation and Cauchy-Riemann equations
Lecture 148 Dirichlet problem
Section 16: Chain rule: different variants
Lecture 149 A general introduction
Lecture 150 Variants 1 and 2
Lecture 151 Variant 3
Lecture 152 Variant 3 (proof)
Lecture 153 Variant 4
Lecture 154 Example with a diagram
Lecture 155 Problem solving
Lecture 156 Problem solving, problem 1
Lecture 157 Problem solving, problem 2
Lecture 158 Problem solving, problem 3
Lecture 159 Problem solving, problem 4
Lecture 160 Problem solving, problem 6
Lecture 161 Problem solving, problem 7
Lecture 162 Problem solving, problem 5
Lecture 163 Problem solving, problem 8
Section 17: Linear approximation, linearisation, differentiability, differential
Lecture 164 Linearisation and differentiability in Calc1
Lecture 165 Differentiability in Calc3: introduction
Lecture 166 Differentiability in two variables, an example
Lecture 167 Differentiability in Calc3 implies continuity
Lecture 168 Partial differentiability does NOT imply differentiability
Lecture 169 An example: continuous, not differentiable
Lecture 170 Differentiability in several variables, a test
Lecture 171 Wrap-up: differentiability, partial differentiability, and continuity in Calc3
Lecture 172 Differentiability in two variables, a geometric interpretation
Lecture 173 Linearization: two examples
Lecture 174 Linearization, problem solving 1
Lecture 175 Linearization, problem solving 2
Lecture 176 Linearization, problem solving 3
Lecture 177 Linearization by Jacobian matrix, problem solving
Lecture 178 Differentials: problem solving 1
Lecture 179 Differentials: problem solving 2
Section 18: Gradient, directional derivatives
Lecture 180 Gradient
Lecture 181 The gradient in each point is orthogonal to the level curve through the point
Lecture 182 The gradient in each point is orthogonal to the level surface through the point
Lecture 183 Tangent plane to the level surface, an example
Lecture 184 Directional derivatives, introduction
Lecture 185 Directional derivatives, the direction
Lecture 186 How to normalize a vector and why it works
Lecture 187 Directional derivatives, the definition
Lecture 188 Partial derivatives as a special case of directional derivatives
Lecture 189 Directional derivatives, an example
Lecture 190 Directional derivatives: important theorem for computations and interpretations
Lecture 191 Directional derivatives: an earlier example revisited
Lecture 192 Geometrical consequences of the theorem about directional derivatives
Lecture 193 Geometical consequences of the theorem about directional derivatives, an example
Lecture 194 Directional derivatives, an example
Lecture 195 Normal line and tangent line to a level curve: how to get their equations
Lecture 196 Normal line and tangent line to a level curve: their equations, an example
Lecture 197 Gradient and directional derivatives, problem 1
Lecture 198 Gradient and directional derivatives, problem 2
Lecture 199 Gradient and directional derivatives, problem 3
Lecture 200 Gradient and directional derivatives, problem 4
Lecture 201 Gradient and directional derivatives, problem 5
Lecture 202 Gradient and directional derivatives, problem 6
Lecture 203 Gradient and directional derivatives, problem 7
Section 19: Implicit functions
Lecture 204 What is the Implicit Function Theorem?
Lecture 205 Jacobian determinant
Lecture 206 Jacobian determinant for change to polar and to cylindrical coordinates
Lecture 207 Jacobian determinant for change to spherical coordinates
Lecture 208 Jacobian determinant and change of area
Lecture 209 The Implicit Function Theorem variant 1
Lecture 210 The Implicit Function Theorem variant 1, an example
Lecture 211 The Implicit Function Theorem variant 2
Lecture 212 The Implicit Function Theorem variant 2, example 1
Lecture 213 The Implicit Function Theorem variant 2, example 2
Lecture 214 The Implicit Function Theorem variant 3
Lecture 215 The Implicit Function Theorem variant 3, an example
Lecture 216 The Implicit Function Theorem variant 4
Lecture 217 The Inverse Function Theorem
Lecture 218 The Implicit Function Theorem, summary
Lecture 219 Notation in some unclear cases
Lecture 220 The Implicit Function Theorem, problem solving 1
Lecture 221 The Implicit Function Theorem, problem solving 2
Lecture 222 The Implicit Function Theorem, problem solving 3
Lecture 223 The Implicit Function Theorem, problem solving 4
Section 20: Taylor's formula, Taylor's polynomial, quadratic forms
Lecture 224 Taylor's formula, introduction
Lecture 225 Quadratic forms and Taylor's polynomial of second degree
Lecture 226 Taylor's polynomial of second degree, theory
Lecture 227 Taylor's polynomial of second degree, example 1
Lecture 228 Taylor's polynomial of second degree, example 2
Lecture 229 Taylor's polynomial of second degree, example 3
Lecture 230 Classification of quadratic forms (positive definite etc)
Lecture 231 Classification of quadratic forms, problem solving 1
Lecture 232 Classification of quadratic forms, problem solving 2
Lecture 233 Classification of quadratic forms, problem solving 3
Section 21: Optimization on open domains (critical points)
Lecture 234 Extreme values of functions of several variables
Lecture 235 Extreme values of functions of two variables, without computations
Lecture 236 Critical points and their classification (max, min, saddle)
Lecture 237 Second derivative test for C^3 functions of several variables
Lecture 238 Second derivative test for C^3 functions of two variables
Lecture 239 Critical points and their classification: some simple examples
Lecture 240 Critical points and their classification: more examples 1
Lecture 241 Critical points and their classification: more examples 2
Lecture 242 Critical points and their classification: more examples 3
Lecture 243 Critical points and their classification: a more difficult example (4)
Section 22: Optimization on compact domains
Lecture 244 Extreme values for continuous functions on compact domains
Lecture 245 Eliminate a variable on the boundary
Lecture 246 Parameterize the boundary
Section 23: Lagrange multipliers (optimization with constraints)
Lecture 247 Lagrange multipliers 1
Lecture 248 Lagrange multipliers 1, an old example revisited
Lecture 249 Lagrange multipliers 1, another example
Lecture 250 Lagrange multipliers 2
Lecture 251 Lagrange multipliers 2, an example
Lecture 252 Lagrange multipliers 3
Lecture 253 Lagrange multipliers 3, an example
Lecture 254 Summary: optimization
Section 24: Final words
Lecture 255 The last one
Section 25: Extras
Lecture 256 Bonus Lecture
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