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udemy – Calculus 3 (multivariable calculus), part 2 of 2

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Towards and through the vector fields, part 2 of 2: Integrals and vector calculus

What you'll learn
How to solve problems in multivariable calculus and vector calculus (illustrated with more than 150 solved problems) and why these methods work.
7 types of integrals: double, double improper, and triple integrals; line integrals and surface integrals of functions and of vector fields.
Direct and inverse substitutions for multiple integrals with many examples; Fubini's theorem for various types of domains.
Conservative vector fields and their potentials; fundamental theorem for conservative vector fields.
Green's, Stokes' and Gauss' theorems.
Gradient, curl and divergence.
Surfaces as graphs of functions of two variables and parametric surfaces; normal vectors and orientation of surfaces; boundary of a surface.
Five methods of computing line integrals of vector fields and four methods of computing surface integrals of vector fields (flux integrals).
Requirements
Calculus 1 and 2
Some Linear Algebra (a brief summary of some topics is contained in Section 2 of "Calculus 3, part 1 of 2")
Calculus 3, part 1 or an equivalent (curves, sets in the plane, functions of several variables with limits, continuity and differentiability, partial derivatives, and simple PDEs)
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 2 of 2Towards and through the vector fields, part 2 of 2: Integrals and vector calculus(Chapter numbers in Robert A. Adams, Christopher Essex: Calculus, a complete course. 8th or 9th edition.)C4: Multiple integrals (Chapter 14)S1. Introduction to the courseS2. Repetition (Riemann integrals, sets in the plane, curves)S3. Double integralsYou will learn: compute double integrals on APR (axis-parallel rectangles) by iteration of single integrals; x-simple and y-simple domains; iteration of double integrals (Fubini's theorem).S4. Change of variables in double integralsYou will learn: compute double integrals via variable substitution (mainly to polar coordinates).S5. Improper integralsYou will learn: motivate if an improper integral is convergent or divergent; use the mean-value theorem for double integrals in order to compute the mean value for a two-variable function on a compact connected set.S6. Triple integralsS7. Change of variables in triple integralsYou will learn: compute triple integrals by Fubini's theorem or by variable substitution to spherical or cylindrical coordinates; compute the Jacobian for various kinds of change of variables.S8. Applications of multiple integrals such as mass, surface area, mass centre.You will learn: apply multiple integrals for various aims.C5: Vector fields (Chapter15)S9. Vector fieldsS10. Conservative vector fieldsYou will learn: about vector fields in the plane and in the space; conservative vector fields; use the necessary condition for a vector field to be conservative; compute potential functions for conservative vector fields.S11. Line integrals of functionsS12. Line integral of vector fieldsYou will learn: calculate both kinds of line integrals (the ones of functions, and the ones of vector fields) and use them for computations of mass, arc length, work; three methods for computation of line integrals of vector fields.S13. SurfacesYou will learn: understand surfaces described as graphs to two-variable functions f:R^2-->R and as parametric surfaces, being graphs of r:R^2-->R^3; determine whether a surface is closed and determine surfaces' boundary; determine normal vector to surfaces.S14. Surface integralsYou will learn: calculate surface integrals of scalar functions and use them for computation of mass and area.S15. Oriented surfaces and flux integralsYou will learn: determine orientation of a surface; determine normal vector field; choose orientation of a surface which agrees with orientation of the surface's boundary; calculate flux integrals and use them for computation of the flux of a vector field across a surface.C6: Vector calculus (Chapter16: 16.1--16.5)S16. Gradient, divergence and curl, and some identities involving them; irrotational and solenoidal vector fields (Ch. 16.1--2)S17. Green's theorem in the plane (Ch. 16.3)S18. Gauss' theorem (Divergence Theorem) in 3-space (Ch. 16.4)S19. Stokes' theorem (Ch. 16.5)S20. Wrap-up Multivariable calculus / Calculus 3, part 2 of 2.You will learn: define and compute curl and divergence of (two- and three-dimensional) vector fields and proof some basic formulas involving gradient, divergence and curl; apply Green's, Gauss's and Stokes's theorems, estimate when it is possible (and convenient) to apply these theorems.S21. 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 final exam. 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 200 videos and their titles, and with the texts of all the 152 problems solved during this course, is presented in the resource file "001 Outline_Calculus3_part2.pdf" under video 1 ("Introduction to the course"). This content is also presented in video 1.
Overview
Section 1: Introduction to the course
Lecture 1 Introduction
Section 2: Repetition (Riemann integrals, sets in the plane, curves)
Lecture 2 Riemann integrals repetition 1
Lecture 3 Riemann integrals repetition 2
Lecture 4 Riemann integrals repetition 3
Lecture 5 Riemann integrals repetition 4
Lecture 6 Riemann integrals repetition 5
Lecture 7 Curves part 1 general
Lecture 8 Curves part 2, arc length
Lecture 9 Sets in the plane
Section 3: Double integrals
Lecture 10 Double integrals, notation and applications
Lecture 11 APR
Lecture 12 Double integrals, definition on APR
Lecture 13 Double integrals, definition on compact domains
Lecture 14 Multiple integrals generally
Lecture 15 Properties of double integrals
Lecture 16 Double integrals by inspection 1
Lecture 17 Odd functions
Lecture 18 Integration by inspection 2
Lecture 19 Integration by inspection, Problem 1
Lecture 20 Integration by inspection, Problem 2
Lecture 21 Integration by inspection, Problem 3
Lecture 22 Integration by inspection, Problem 4
Lecture 23 Integration by iteration, Fubini on APR
Lecture 24 Fubini on APR, Problem 1
Lecture 25 Fubini on APR, Problem 2
Lecture 26 Fubini on APR, Problem 3
Lecture 27 Fubini on APR, rule for products
Lecture 28 Fubini on APR, Problem 4
Lecture 29 Fubini on APR: an example where order matters
Lecture 30 X- and Y-simple sets
Lecture 31 Integration by iteration, Fubini on X- and Y-simple sets
Lecture 32 Fubini general problem 1
Lecture 33 Fubini general problem 2
Lecture 34 Fubini general problem 3
Lecture 35 Fubini general problem 4
Lecture 36 Fubini general problem 5
Lecture 37 Fubini general problem 6
Lecture 38 Fubini general problem 7
Lecture 39 Fubini general problem 8
Section 4: Change of variables in double integrals
Lecture 40 Why change of variables, comparison
Lecture 41 Jacobian and the change in area element after substitution
Lecture 42 One formula for both substitutions
Lecture 43 Inverse substitution
Lecture 44 Direct substitution
Lecture 45 Change of variables, problem 3
Lecture 46 Change of variables, problem 4
Lecture 47 Change of variables, problem 5
Lecture 48 Change of variables, problem 6
Lecture 49 Change of variables, problem 7
Lecture 50 Double integrals, wrap-up
Section 5: Improper integrals
Lecture 51 Improper integrals, repetition Calc 2
Lecture 52 Improper double integrals
Lecture 53 Calc 3 helps Calc 2, problem 1
Lecture 54 Improper integrals, problem 2
Lecture 55 Improper integrals, problem 3
Lecture 56 Improper integrals, problem 4
Lecture 57 Improper integrals, problem 5
Lecture 58 Improper integrals, problem 6
Lecture 59 Mean value theorem
Lecture 60 Mean value theorem, example 1
Lecture 61 Mean value theorem, example 2
Section 6: Triple integrals
Lecture 62 Triple integrals: notation, definition and properties
Lecture 63 Integration by inspection
Lecture 64 Fubini
Lecture 65 Problem 1
Lecture 66 Problem 2
Lecture 67 Problem 3
Lecture 68 Problem 4
Lecture 69 Area and volume in different ways
Lecture 70 Volume of a tetrahedron
Section 7: Change of variables in triple integrals
Lecture 71 Change of variables in triple integrals
Lecture 72 Change of variables, problem 1
Lecture 73 Change of variables, problem 2
Lecture 74 Change of variables, problem 3
Lecture 75 Change of variables, problem 4
Lecture 76 Change of variables, problem 5
Lecture 77 Change of variables, wrap-up
Section 8: Applications of multiple integrals
Lecture 78 Applications of multiple integrals, area and volume
Lecture 79 Applications of multiple integrals, mass
Lecture 80 Applications of multiple integrals, mass centre, centroid
Lecture 81 Applications of multiple integrals, surface area
Lecture 82 Surface area, problem 1
Lecture 83 Surface area, problem 2
Lecture 84 Surface area, problem 3
Lecture 85 Surface area, problem 4
Section 9: Vector fields
Lecture 86 Different kinds of functions and their visualisation
Lecture 87 Vector fields, some examples
Lecture 88 Vector fields, definition, notation, plot and domain
Lecture 89 Streamlines
Lecture 90 Streamlines problem 1
Lecture 91 Streamlines problem 2
Lecture 92 Streamlines problem 3
Lecture 93 Streamlines problem 4
Lecture 94 Streamlines problem 5
Lecture 95 Streamlines problem 6
Section 10: Conservative vector fields
Lecture 96 Is each vector field a gradient to some function? Computations.
Lecture 97 Is each vector field a gradient to some function? Geometry.
Lecture 98 Conservative vector fields and equipotential lines
Lecture 99 Schwarz' theorem, a repetition
Lecture 100 Hessian vs Jacobian
Lecture 101 The necessary conditions for conservative vector fields
Lecture 102 Example 1: electrostatic field
Lecture 103 Example 2: gravitational field
Lecture 104 Conservative vector fields and their potentials, problem 1
Lecture 105 Conservative vector fields and their potentials, problem 2
Lecture 106 Conservative vector fields and their potentials, problem 3
Lecture 107 Conservative vector fields and their potentials, problem 4
Section 11: Line integrals of functions
Lecture 108 Line integrals, notation
Lecture 109 Line integrals of functions, applications and properties
Lecture 110 Line integrals of functions, problem 1
Lecture 111 Line integrals of functions, problem 2
Lecture 112 Line integrals of functions, problem 3
Lecture 113 Line integrals of functions, problem 4
Section 12: Line integrals of vector fields
Lecture 114 Line integrals of vector fields, notation, definition and application
Lecture 115 Line integrals of vector fields, properties
Lecture 116 Line integrals of vector fields, problem 1 from definition
Lecture 117 Line integrals of vector fields, problem 2 from definition
Lecture 118 Line integrals of vector fields, problem 3 from definition
Lecture 119 Line integrals of vector fields, differential formula
Lecture 120 Line integrals of vector fields, differential fomula, problem 4
Lecture 121 Fundamental theorem for conservative vector fields
Lecture 122 Path independence of line integrals
Lecture 123 Path independence, problem 5
Lecture 124 Path independence, problem 6
Lecture 125 Path independence, problem 7
Lecture 126 Path independence, problem 8
Lecture 127 Path independence, problem 9
Lecture 128 Line integrals of vector fields, wrap-up
Section 13: Surfaces
Lecture 129 Why surfaces and what they are
Lecture 130 Different ways of defining surfaces
Lecture 131 Boundary of a surface; closed and composite surfaces
Lecture 132 Normal vector and orientation of a surface
Lecture 133 Normal vectors to some important surfaces
Lecture 134 Surface element, both for surfaces defined as graphs and parametric surfaces
Section 14: Surface integrals
Lecture 135 Surface integrals: notation
Lecture 136 Surface integrals of functions: definition and applications
Lecture 137 Surface integrals of functions: computations and properties
Lecture 138 Surface integrals of functions, problem 1
Lecture 139 Surface integrals of functions, problem 2
Lecture 140 Surface integrals of functions, problem 3
Lecture 141 Surface integrals of functions, problem 4
Section 15: Oriented surfaces and flux integrals
Lecture 142 Orientation of a surface which agrees with orientation of its boundary
Lecture 143 Flux integrals: notation, definition, computations and applications
Lecture 144 Flux integrals: properties
Lecture 145 Flux integrals, problem 1
Lecture 146 Flux integrals, problem 2
Lecture 147 Flux integrals, problem 3
Section 16: Gradient, divergence and curl
Lecture 148 Derivatives: gradient, rotation (curl), divergence
Lecture 149 Curl, an interpretation: irrotational vector fields
Lecture 150 Rotation (curl) of a 3D vector field, an example
Lecture 151 Divergence, an interpretation; solenoidal vector fields
Lecture 152 Product rules for gradient, divergence and curl
Lecture 153 Product rule for gradient
Lecture 154 Product rule for divergence
Lecture 155 Product rule for curl
Lecture 156 Curl of each vector field is solenoidal; vector potentials
Lecture 157 Conservative vector fields are irrotational
Lecture 158 Laplacian
Section 17: Green's theorem in the plane
Lecture 159 Green's theorem: our third fundamental theorem
Lecture 160 Green's theorem: formulation of the theorem
Lecture 161 Green's theorem: proof
Lecture 162 Green's theorem: three common issues and how to handle them
Lecture 163 Green's theorem: problem 1
Lecture 164 Green's theorem: problem 2
Lecture 165 Green's theorem: problem 3
Lecture 166 Green's theorem: problem 4
Lecture 167 Green's theorem: problem 5
Lecture 168 Magnetic field and enclosing singularities
Lecture 169 Necessary and sufficient condition for (plane) conservative vector fields
Lecture 170 Area with help of Green's theorem
Section 18: Gauss' theorem (Divergence theorem) in 3-space
Lecture 171 Gauss' theorem: our fourth fundamental theorem
Lecture 172 Gauss' theorem: formulation of the theorem
Lecture 173 Gauss' theorem: proof
Lecture 174 Gauss' theorem: three common issues and how to handle them
Lecture 175 Gauss' theorem: problem 1
Lecture 176 Gauss' theorem: problem 2
Lecture 177 Gauss' theorem: problem 3
Lecture 178 Gauss' theorem: problem 4
Lecture 179 An example where Gauss' theorem cannot be applied
Lecture 180 Volume of a cone
Section 19: Stokes' theorem
Lecture 181 Stokes' theorem: our fifth fundamental theorem
Lecture 182 Stokes' theorem: formulation
Lecture 183 Stokes' theorem: proof
Lecture 184 Stokes' theorem: how to use it
Lecture 185 Stokes' theorem: how it helps; example 1
Lecture 186 Stokes' theorem: verification on an example (example 2)
Lecture 187 Stokes' theorem: example 3
Lecture 188 Stokes' theorem: surface independence, example 4
Lecture 189 Stokes' theorem: surface integral of curl over closed surfaces, regular domains
Lecture 190 Simply connected sets in space
Lecture 191 Necessary and sufficient condition for conservative vector fields
Lecture 192 Stokes' theorem, problem 1
Lecture 193 Stokes' theorem, problem 2
Lecture 194 Stokes' theorem, problem 3
Lecture 195 Stokes' theorem, problem 4
Lecture 196 Stokes' theorem, problem 5
Lecture 197 Stokes' theorem, problem 6
Lecture 198 Stokes' theorem for computations of surface integrals, vector potentials
Section 20: Wrap-up Multivariable calculus / Calculus 3, part 2 of 2
Lecture 199 Calculus 3, wrap-up
Lecture 200 Final words
Section 21: Extras
Lecture 201 Bonus Lecture
University and college engineering
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