Vector Calculus With Applications

Line Integrals, Surface Integrals, Integral Theorems with Applications

What you'll learn

Line integrals with applications, Path independence, important theorems

Double integrals, Triple Integrals, Surface integrals

Green's Theorem, Stokes Theorem, Divergence Theorem with Proofs and Applications

70+ solved examples, 50+ homework assessments, lecture pdfs and eBooks

Derivation of Basic PDEs, including The Continuity Equation, Heat Equation, Wave Equation, Laplace Equation, etc.

Requirements

Precalculus, basic knowledge of computing derivatives and integrals is enough.

A few lectures require the concept of continuity of a function. Otherwise the course is mostly self contained

A little bit of MATLAB knowledge will be needed in a few lectures.

Description

Vector Calculus is one of the most important subject for advanced studies in applied mathematics, physics or engineering. The core of the subject contains the "Integral Theorems" as they have a wide range of applications to these disciplines. Keeping in view the course outlines of most of the (local) institutes and universities, the following topics are included in this course:Line Integrals (of both scalar and vector fields) Work done as a Line integralPath Independent Line integralsConservative FieldsSimply Connected and Multiply Connected Domains (with Animations for illustration)Theorems (with proofs) on Path independence of Line Integrals and Conservative Fields (five theorems)Applications of Line integrals (Calculating the area of fence whose base is a curve C in space, calculating the Mass, the Center of Mass, and Moment of inertia of a wire)The Double Integrals with applications (calculating mass, c.o.m, moment of inertia of a lamina)Parametric Representation of SurfacesSurface area (with alternative formulas)Tangent Plane and Normal to a Surface Surface Integrals (of vector and scalar fields)Integral Theorems (Green's Theorem, Stokes's Theorem, Gauss's (Divergence) Theorem)Applications of Integral Theorems Derivation of Heat equation, Wave Equation, Laplace Equation, Continuity Equation etc. More than 70 worked examples are included to illustrate these topics. Most of the topics are illustrated with Slides and Animations (where needed), and a few examples are worked on digital white board  (One Note) using Tablet.PDFs of lectures are also attached with the video lectures, Student Practice Material is also included with necessary hints. Most of the lectures are in the form of slides so that the learners time is marginally saved as compared to the white board lectures.  More examples will be included (some examples can be included on demand of the learners).Hopes the students may learn and enjoy from this course.There may be mistakes and I will be extremely thankful to those who point out mistakes in any lecture (Check out the description of each video for corrections). Suggestions are always welcomed because we always need improvements.

Overview

Section 1: Line Integrals, Path independence, Theorems, Applications of Line Integrals

Lecture 1 Line integrals of Scalar and Vector fields with Geometrical interpretation

Lecture 2 How to evaluate Line integrals of Scalar Fields, Example#1, Example#2

Lecture 3 How to evaluate Line integrals of Vector Fields, Example#1, Example#2

Lecture 4 Line integral along piecewise smooth curves, Example#1

Lecture 5 Example#2, Example#3

Lecture 6 Work done as line Integral, Example#1

Lecture 7 Example#2, Example#3

Lecture 8 Example#4 (Using MATLAB for plotting), Example#5

Lecture 9 Effect of Reversing the direction of C on the Line Integral

Lecture 10 Path independent Line integrals, Conservative Fields, Nonconservative Fields

Lecture 11 Theorem #1

Lecture 12 Theorem #2, with consequences

Lecture 13 How to find Scalar Potential for a Conservative Field, Example #1, Example #2

Lecture 14 Simply Connected Domains (part 1)

Lecture 15 Simply Connected Domains (part 2)

Lecture 16 Theorem #3, Theorem #4

Lecture 17 How to use Theorem #2 and Theorem #3

Lecture 18 How to use Theorem #4, Example #3 and Example #4

Lecture 19 Importance of Simply Connected domains for Path independence of Line integrals

Lecture 20 Theorem #5 (Law of Conservation of Energy)

Lecture 21 Applications of Line integrals (Part-1) : Area of a Fence, Example #2, Example#3

Lecture 22 Applications of Line integrals (Part-2) : Mass, Example #3

Lecture 23 Applications of Line integrals (Part-3) : Center of Mass, Example #4

Lecture 24 Applications of Line integrals (Part-4) : Moment of Inertia, Example #5

Section 2: Double Integrals and Surface Integrals with Applications

Lecture 25 Double integrals, Geometrical Interpretetion, How to evaluate, example

Lecture 26 Example #4, Example #5, Example #6

Lecture 27 Change of Variables in Double integrals, Example #7

Lecture 28 Applications of Double integrals (part-1), Area, Example #1, 2, 3

Lecture 29 Applications (part2), Calculating the Mass and C.O.M, Example #1,2,3,4

Lecture 30 Applications (part-3), Calculating the Moment of Inertia, Example #1, 2

Lecture 31 Parametric Representation of Surfaces, Examples (Paraboloid, Sphere, Cylinder)

Lecture 32 Surface Area with Alternative Formulas

Lecture 33 Example#1, Example#2, Example#3

Lecture 34 Example#4, Example#5

Lecture 35 Definition of Surface Integral, Flux through a Surface

Lecture 36 Tangent Plane and Normal to a Surface, Orientable and Non Orientable Surfaces

Lecture 37 Normal vector and Equation of Tangent plane to a given Surface : Example #1,2

Lecture 38 Surface Integrals of Scalar Fields: Example#1, Example #2

Lecture 39 Example#3, Example #4

Lecture 40 Surface Integrals of Vector Fields: Example#1, Example #2

Lecture 41 Example#3, Example#4

Section 3: Chapter#3 : Integral Theorems (Green's -, Stoke's -, Divergence Theorem)

Lecture 42 Theorem#1 : The Green's Theorem in Plane

Lecture 43 How to Apply the Green's Theorem, Example#1, Example#2

Lecture 44 Stokes's Theorem

Lecture 45 How to Verify the Stokes Theorem, Example#1, Example#2

This course is for Undergraduate students of Mathematics, Physics, Computer, Engineering etc.,Graduate Students and Instructors may also profit from this course