Overview

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COURSE OUTLINE: This course will offer a detailed introduction to integral and vector calculus. We’ll start with the concepts of partition, Riemann sum and Riemann Integrable functions and their properties. We then move to anti-derivatives and will look in to few classical theorems of integral calculus such as fundamental theorem of integral calculus. We’ll then study improper integral, their convergence and learn about few tests which confirm the convergence. Afterwards we’ll look into multiple integrals, Beta and Gamma functions, Differentiation under the integral sign. Finally, we’ll finish the integral calculus part with the calculation of area, rectification, volume and surface integrals. In the next part, we’ll study the vector calculus. We’ll start the first lecture by the collection of vector algebra results. In the following weeks, we’ll learn about scalar and vector fields, level surfaces, limit, continuity, and differentiability, directional derivative, gradient, divergence and curl of vector functions and their geometrical interpretation. We’ll also study the concepts of conservative, irrotational and solenoidal vector fields. We’ll look into the concepts of tangent, normal and binormal and then derive the Serret-Frenet formula. Then we’ll look into the line, volume and surface integrals and finally we’ll learn the three major theorems of vector calculus: Green’s, Gauss’s and Stoke’s theorem

Syllabus

Integral and Vector Calculus.
Lecture 01 : Partition, Riemann intergrability and One example.
Lecture 02 : Partition, Riemann intergrability and One example (Contd.).
Lecture 03 : Condition of integrability.
Lecture 04 : Theorems on Riemann integrations.
Lecture 05 : Examples.
Lecture 06 : Examples (Contd.).
Lecture 07 : Reduction formula.
Lecture 08 : Reduction formula (Contd.).
Lecture 09 : Improper Integral.
Lecture 10 : Improper Integral (Contd.).
Lecture 11 : Improper Integral (Contd.).
Lecture 12 : Improper Integral (Contd.).
Lecture 13 : Introduction to Beta and Gamma Function.
Lecture 14 : Beta and Gamma Function.
Lecture 15 : Differentiation under Integral Sign.
Lecture 16 : Differentiation under Integral Sign (Contd.).
Lecture 17 : Double Integral.
Lecture 18 : Double Integral over a Region E.
Lecture 19 : Examples of Integral over a Region E.
Lecture 20 : Change of variables in a Double Integral.
Lecture 21 : Change of order of Integration.
Lecture 22 : Triple Integral.
Lecture 23 : Triple Integral (Contd.).
Lecture 24 : Area of Plane Region.
Lecture 25 : Area of Plane Region (Contd.).
Lecture 26 : Rectification.
Lecture 27 : Rectification (Contd.).
Lecture 28 : Surface Integral.
Lecture 29 : Surface Integral (Contd.).
Lecture 30 : Surface Integral (Contd.).
Lecture 31 : Volume Integral, Gauss Divergence Theorem.
Lecture 32 : Vector Calculus.
Lecture 33 : Limit, Continuity, Differentiability.
Lecture 34 : Successive Differentiation.
Lecture 35 : Integration of Vector Function.
Lecture 36 : Gradient of a Function.
Lecture 37 : Divergence & Curl.
Lecture 38 : Divergence & Curl Examples.
Lecture 39 : Divergence & Curl important Identities.
Lecture 40 : Level Surface Relevant Theorems.
Lecture 41 : Directional Derivative (Concept & Few Results).
Lecture 42 : Directional Derivative (Concept & Few Results) (Contd.).
Lecture 43 : Directional Derivatives, Level Surfaces.
Lecture 44 : Application to Mechanics.
Lecture 45 : Equation of Tangent, Unit Tangent Vector.
Lecture 46 : Unit Normal, Unit binormal, Equation of Normal Plane.
Lecture 47 : Introduction and Derivation of Serret-Frenet Formula, few results.
Lecture 48 : Example on binormal, normal tangent, Serret-Frenet Formula.
Lecture 49 : Osculating Plane, Rectifying plane, Normal plane.
Lecture 50 : Application to Mechanics, Velocity, speed , acceleration.
Lecture 51 : Angular Momentum, Newton's Law.
Lecture 52 : Example on derivation of equation of motion of particle.
Lecture 53 : Line Integral.
Lecture 54 : Surface integral.
Lecture 55 : Surface integral (Contd.).
Lecture 56 : Green's Theorem & Example.
Lecture 57 : Volume integral, Gauss theorem.
Lecture 58 : Gauss divergence theorem.
Lecture 59 : Stoke's Theorem.
Lecture 60 : Overview of Course.