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

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Develop a thorough understanding of calculus 3 fundamentals, including multivariate calculus, vector calculus, and integrals.

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                                    Medium of instructions
                                    English

                                    Mode of learning
                                    Self study

                                    Mode of Delivery
                                    Video and Text Based

Introduction to the course

Introduction

Repetition (Riemann integrals, sets in the plane, curves)

Riemann integrals repetition 1
Riemann integrals repetition 2
Riemann integrals repetition 3
Riemann integrals repetition 4
Riemann integrals repetition 5
Curves part 1 general
Curves part 2, arc length
Sets in the plane

Double integrals

Double integrals, notation and applications
APR
Double integrals, definition on APR
Double integrals, definition on compact domains
Multiple integrals generally
Properties of double integrals
Double integrals by inspection 1
Odd functions
Integration by inspection 2
Integration by inspection, Problem 1
Integration by inspection, Problem 2
Integration by inspection, Problem 3
Integration by inspection, Problem 4
Integration by iteration, Fubini on APR
Fubini on APR, Problem 1
Fubini on APR, Problem 2
Fubini on APR, Problem 3
Fubini on APR, rule for products
Fubini on APR, Problem 4
Fubini on APR: an example where order matters
X- and Y-simple sets
Integration by iteration, Fubini on X- and Y-simple sets
Fubini general problem 1
Fubini general problem 2
Fubini general problem 3
Fubini general problem 4
Fubini general problem 5
Fubini general problem 6
Fubini general problem 7
Fubini general problem 8

Change of variables in double integrals

Why change of variables, comparison
Jacobian and the change in area element after substitution
One formula for both substitutions
Inverse substitution
Direct substitution
Change of variables, problem 3
Change of variables, problem 4
Change of variables, problem 5
Change of variables, problem 6
Change of variables, problem 7
Double integrals, wrap-up

Improper integrals

Improper integrals, repetition Calc 2
Improper double integrals
Calc 3 helps Calc 2, problem 1
Improper integrals, problem 2
Improper integrals, problem 3
Improper integrals, problem 4
Improper integrals, problem 5
Improper integrals, problem 6
Mean value theorem
Mean value theorem, example 1
Mean value theorem, example 2

Triple integrals

Triple integrals: notation, definition and properties
Integration by inspection
Fubini
Problem 1
Problem 2
Problem 3
Problem 4
Area and volume in different ways
Volume of a tetrahedron

Change of variables in triple integrals

Change of variables in triple integrals
Change of variables, problem 1
Change of variables, problem 2
Change of variables, problem 3
Change of variables, problem 4
Change of variables, problem 5
Change of variables, wrap-up

Applications of multiple integrals

Applications of multiple integrals, area and volume
Applications of multiple integrals, mass
Applications of multiple integrals, mass centre, centroid
Applications of multiple integrals, surface area
Surface area, problem 1
Surface area, problem 2
Surface area, problem 3
Surface area, problem 4

Vector fields

Different kinds of functions and their visualisation
Vector fields, some examples
Vector fields, definition, notation, plot and domain
Streamlines
Streamlines problem 1
Streamlines problem 2
Streamlines problem 3
Streamlines problem 4
Streamlines problem 5
Streamlines problem 6

Conservative vector fields

Is each vector field a gradient to some function? Computations.
Is each vector field a gradient to some function? Geometry.
Conservative vector fields and equipotential lines
Schwarz' theorem, a repetition
Hessian vs Jacobian
The necessary conditions for conservative vector fields
Example 1: electrostatic field
Example 2: gravitational field
Conservative vector fields and their potentials, problem 1
Conservative vector fields and their potentials, problem 2
Conservative vector fields and their potentials, problem 3
Conservative vector fields and their potentials, problem 4

Line integrals of functions

Line integrals, notation
Line integrals of functions, applications and properties
Line integrals of functions, problem 1
Line integrals of functions, problem 2
Line integrals of functions, problem 3
Line integrals of functions, problem 4

Line integrals of vector fields

Line integrals of vector fields, notation, definition and application
Line integrals of vector fields, properties
Line integrals of vector fields, problem 1 from definition
Line integrals of vector fields, problem 2 from definition
Line integrals of vector fields, problem 3 from definition
Line integrals of vector fields, differential formula
Line integrals of vector fields, differential fomula, problem 4
Fundamental theorem for conservative vector fields
Path independence of line integrals
Path independence, problem 5
Path independence, problem 6
Path independence, problem 7
Path independence, problem 8
Path independence, problem 9
Line integrals of vector fields, wrap-up

Surfaces

Why surfaces and what they are
Different ways of defining surfaces
Boundary of a surface; closed and composite surfaces
Normal vector and orientation of a surface
Normal vectors to some important surfaces
Surface element, both for surfaces defined as graphs and parametric surfaces

Surface integrals

Surface integrals: notation
Surface integrals of functions: definition and applications
Surface integrals of functions: computations and properties
Surface integrals of functions, problem 1
Surface integrals of functions, problem 2
Surface integrals of functions, problem 3
Surface integrals of functions, problem 4

Oriented surfaces and flux integrals

Orientation of a surface which agrees with orientation of its boundary
Flux integrals: notation, definition, computations and applications
Flux integrals: properties
Flux integrals, problem 1
Flux integrals, problem 2
Flux integrals, problem 3

Gradient, divergence and curl

Derivatives: gradient, rotation (curl), divergence
Curl, an interpretation: irrotational vector fields
Rotation (curl) of a 3D vector field, an example
Divergence, an interpretation; solenoidal vector fields
Product rules for gradient, divergence and curl
Product rule for gradient
Product rule for divergence
Product rule for curl
Curl of each vector field is solenoidal; vector potentials
Conservative vector fields are irrotational
Laplacian

Green's theorem in the plane

Green's theorem: our third fundamental theorem
Green's theorem: formulation of the theorem
Green's theorem: proof
Green's theorem: three common issues and how to handle them
Green's theorem: problem 1
Green's theorem: problem 2
Green's theorem: problem 3
Green's theorem: problem 4
Green's theorem: problem 5
Magnetic field and enclosing singularities
Necessary and sufficient condition for (plane) conservative vector fields
Area with help of Green's theorem

Gauss' theorem (Divergence theorem) in 3-space

Gauss' theorem: our fourth fundamental theorem
Gauss' theorem: formulation of the theorem
Gauss' theorem: proof
Gauss' theorem: three common issues and how to handle them
Gauss' theorem: problem 1
Gauss' theorem: problem 2
Gauss' theorem: problem 3
Gauss' theorem: problem 4
An example where Gauss' theorem cannot be applied
Volume of a cone

Stokes' theorem

Stokes' theorem: our fifth fundamental theorem
Stokes' theorem: formulation
Stokes' theorem: proof
Stokes' theorem: how to use it
Stokes' theorem: how it helps; example 1
Stokes' theorem: verification on an example (example 2)
Stokes' theorem: example 3
Stokes' theorem: surface independence, example 4
Stokes' theorem: surface integral of curl over closed surfaces, regular domains
Simply connected sets in space
Necessary and sufficient condition for conservative vector fields
Stokes' theorem, problem 1
Stokes' theorem, problem 2
Stokes' theorem, problem 3
Stokes' theorem, problem 4
Stokes' theorem, problem 5
Stokes' theorem, problem 6
Stokes' theorem for computations of surface integrals, vector potentials

Wrap-up Multivariable calculus / Calculus 3, part 2 of 2

Calculus 3, wrap-up
Final words

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

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