Vector Calculus and Partial Differential Equations (English)

   Tuesday, July 25, 2023   

Starting a new Lecture Notes Series on Vector Calculus and Partial Differential Equations

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Vector Calculus and Partial Differential Equations By Lecture Notes together!

Lecture Description of this above Topic:

Lecture 1: Session 1:Notion of double integration with d help of single integration, GeoGebra & rainfall exmple

Lecture 2: Session 2 : Examples on double integration examples over rectangle using Riemann sums.

Lecture 3: Session 3: Examples on Iterated integrals & importance of Fubini's Theorem over rectangular domains.

Lecture 4: Session 4: Examples on Iterated integrals & importance of Fubini's Theorem over arbitrary region.

Lecture 5: Session 5: Examples on Change of order of integration and splitting of domains into disjoint unions.

Lecture 6: Session 6: Why does the example given here is not contradicting Fubini's theorem?

Lecture 7: Session 7: How double integration help us to find area of the region in a plane. Concept & examples.

Lecture 8: Session 8: Why average value of a function is defined in terms of integration?

Lecture 9: Session 9: What are Polar Coordinates? & how Polar Coordinate system is different from Cartesians.

Lecture 10: Session 10: How to solve Double integrals problem using Polar Coordinates. Concept and examples.

Lecture 11: Session 11:Center of Mass, Centroid, Moment of Inertia & Radius of Gyration using Double integration

Lecture 12: Session 12: Important points in Triple integration along with examples.(Part I)

Lecture 13: Session 13: Problems in Triple integration. Sketching surface using GeoGebra ( Part II).

Lecture 14: Session 14: What and why Cylindrical coordinate system. Examples without and with change of order.

Lecture 15: Session 15: Spherical coordinates, Spherical transformations: Concepts and examples.

Lecture 16: Question on Triple Integration in GATE 2021 examination

Lecture 17: Sample question paper on Test 1:Double & Triple Integration & its applications

Lecture 18: Session 16: Jacobian matrix whose entries are first order partial derivatives & Jacobian determinant

Lecture 19: Session 17: Substitution in Multiple Integrals: Its importance, Steps to solve and example(Part I).

Lecture 20: Session 18: Substitution in Multiple integrals: Only Examples ( Part II).

Lecture 21: Session 19: Problem using substitution in triple integration, find volume of mod(xyz) over ellipsoid

Lecture 22: Session 1: What are vector valued functions and Vector Fields. Examples using GeoGebra software.

Lecture 23: Session 2: Parameterization of a curve.

Lecture 24: Session 3: Tangent and unit tangent vector to a parameterized curve. Examples on sketching tangent.

Lecture 25: Session 4: Gradient vector, Gradient operator, Normal fields: Concepts and examples.

Lecture 26: Laplacian operator:Concept, example,properties.Connection with Gradient operator

Lecture 27: Session 6: Divergence operator, examples. Physical meaning of Divergence using Desmos software

Lecture 28: Curl of a vector field. Physical interpretation, example and important properties

Lecture 29: Divergence of cross product of irrotational vectors is always zero.

Lecture 30: Example on how to find angle between the surfaces

Lecture 31: Session 8: Notion of Directional derivatives and its properties followed by examples.

Lecture 32: Question on Direction Derivative in GATE 2021 exam

Lecture 33: Question on Directional Derivative in GATE 2022 exam

Lecture 34: Session 1: Applications of Line integrals.

Lecture 35: Session 2: Parameterization of a curve.

Lecture 36: Session 3 : Line integrals for scalar & vector fields,examples & Path dependence.

Lecture 37: Line integral question (work done) in GATE 2022 examination

Lecture 38: Session:4 - Line integrals over piecewise smooth curves.

Lecture 39: Session 5: Under what conditions line integrals are path independent?

Lecture 40: Session 6: Green's Theorem and it's extended version

Lecture 41: Question on Line Integral using Green's theorem asked in GATE 2021

Lecture 42: Question on Line integral using Green's theorem in GATE 2019 Examination.

Lecture 43: Sample Question Paper on Vector Differential Calculus and Line Integrals.

Lecture 44: Session 1: What do you mean Parametrization of a Surface and Grid curves? Examples using GeoGebra

Lecture 45: Session 2: Type 1 on what are Surface Integrals & how to solve when vector field is given

Lecture 46: Session 3: Type 2 on how to solve Surface integral when Scalar field & Parametrized surface is given

Lecture 47: Session 4: Type 3 on how to Surface integral when a Scalar field and z=f(x,y) is given.

Lecture 48: Session 5: Type 4 on Surface Integrals - when and when not to apply Gauss Divergence Theorem.

Lecture 49: Question on Surface integral using Gauss Divergence Theorem: GATE Exam 2021

Lecture 50: Question on Surface Integral in GATE 2020 Examination

Lecture 51: Session 6: Type 5 on solving Surface Integrals, The Magic of Stoke's Theorem

Lecture 52: Session 1: Partial differential equation and its characterization.

Lecture 53: Session 2: Variable separable method for partial differential equations.

Lecture 54: Solution to family of surfaces u = xy + f(x^2-y^2) in GATE 2021 exam

Lecture 55: Session 3: Modelling of 1- dimensional heat (diffusion) equation.

Lecture 56: Session 4: Solution to 1- dimensional heat (diffusion) equation with zero boundary conditions.

Lecture 57: Session 5: Solution to 1- dimensional heat(diffusion) equation when both ends are insulated.

Lecture 58: Session 6: Solution to 1- dimensional heat equation when one end is insulated.

Lecture 59: Session 7: Steady state solution for 1-dimensional heat equation.

Lecture 60: Session 7: Solution to 1-dimensional heat equation with non zero boundary conditions.

Lecture 61: Session 8: Modelling of 1-dimensional wave equation.

Lecture 62: Session 9: Solution to 1-dimensional wave equation and different types.

Lecture 63: Session 10: Steady two dimensional heat equation: Proof, an example and hints for different types

Lecture 64: Question on Directional Derivative in GATE 2019 Exam

Lecture 65: Heat conduction problem in GATE 2019 examination