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Vector Calculus

Publicat: 19 Apr 2011 00:00

By Chris Tisdell - The University of New South Wales
Course Description:
This is a series of lectures for MATH2111 "Higher Several Variable Calculus" and "Vector Calculus", which is a 2nd-year mathematics subject taught at UNSW, Sydney. This playlist provides a shapshot of some lectures presented in Session 1, 2009. These lectures focus on presenting vector calculus in an applied and engineering context, while maintaining mathematical rigour. Thus, this playlist may be useful to students of mathematics, but also to those of engineering, physics and the applied sciences. There is an emphasis on examples and also on proofs.
Lectures:








Lecture 1 - Applications of Double Integrals



In this lecture I discuss the applications of multiple integrals in an applied mathematics and engineering context. I discuss how to calculate the mass, moments and centre of mass of 2-dimensional thin plates. I also briefly glimpse at applications of triple integrals.












Lecture 2 - Path Integrals - How to Integrate Over Curves



This lecture introduces the idea of a path integral (scalar line integral). Dr Chris Tisdell defines the integral of a function over a curve in space and discusses the need and applications of the idea. Plenty of examples are supplied and special attention is given to the applications of path integrals to engineering and physics, such as calculating the centre of mass of thin springs.












Lecture 3 - Vector Fields



This lecture gently introduces the idea of a vector field. Dr Chris Tisdell discusses the need for a vector field, plus presents many examples.












Lecture 4 - Divergence



This video discusses the 'divergence' of a vector field. Divergence is one of the basic operations of vector calculus and, loosely speaking, may be thought of as a type of derivative in vector calculus.












Lecture 5 - Curl



This lecture gently introduces the idea of a the "curl" of a vector field. The curl is one of the basic operations of vector calculus. Dr Chris Tisdell discusses the definition of the curl and how to compute it. Plently of examples are provided. A physical interpretation of the curl is also presented in terms of circulation density. Basically speaking, curl measures the tendency of a vecotr field to "swirl" around a point.












Lecture 6 - Line Integrals



This lecture discusses how to integrate vector fields over curves, better known as "line integrals".Dr Chris Tisdell defines the concept of a line integral and presents some examples on their calculation. Special attention is given to the applications of line integrals such as: calculating work done by a variable force on a particle moving over curved paths; fluid flow (flux) over closed curves; circulation and flow integrals. Plenty of examples are presented.












Lecture 7 - Applications of Line Integrals



This lecture discusses the applications of line integrals, including calculating work; flux (flow) in the plane over curves; and also circulation around curves in the plane. A number of examples are presented to illustrate the theory.The fundamental theorem of line integrals may be thought of as one of the basic theorems of vector calculus.












Lecture 8 - Fundamental Theorem of Line Integrals



This lecture discusses the "fundamental theorem of line integrals for gradient fields". The topic is motivated and the theorem is stated and proved. A number of examples are presented to illustrate the theory.












Lecture 9 - Green's Theorem



This lecture discusses Green's theorem in the plane. Green's theorem not only gives a relationship between double integrals and line integrals, but it also gives a relationship between "curl" and "circulation".In addition, Gauss' divergence theorem in the plane is also discussed, which gives the relationship between divergence and flux.












Lecture 10 - More on Green's Theorem



This is the 2nd lecture on Green's theorem and it's use. In this lecture we explore some interesting applications of Green's theorem and present several examples. Also some proofs are discussed.












Lecture 11 - Parametrised Surfaces



This lecture gently introduces the idea of parametrizing surfaces in space. The content is a prequel to integration over surfaces that sit in 3D.Many examples are discussed and a method to find tangent vectors and normal vectors to a given surface are presented.












Lecture 12 - Surface Integrals



This lecture gently introduces the idea of a "surface integral" and illustrates how to integral functions over surfaces. The idea is a generalization of double integrals in the plane. The concept of surface integral has a number of important applications such as caculating surface area. In addition, surface integrals find use when calculating the mass of a surface like a cone or bowl.












Lecture 13 - More on Surface Integrals



This lecture continues discussing "surface integrals" and further illustrates how to integral functions over surfaces. The idea is a generalization of double integrals in the plane. The concept of surface integral has a number of important applications in the field of engineering, for example, calculating the mass of a surface like a cone or bowl.












Lecture 14 - Surface Integrals and Vector Fields



This lecture discusses "surface integrals" of vector fields. In particular, we discover how to integrate vector fields over surfaces in 3D space and "flux" integrals. A few examples are presented to illustrate the ideas.Such concepts have important applications in fluid flow and electromagnetics.












Lecture 15 - Partial Differential Equations



This lecture discusses and solves the partial differential equation (PDE) known as 'the heat equation" together with some boundary and initial conditions. The method used involves separation of variables combined with Fourier series. The discussion is in a step-by-step process.





Source: http://academicearth.org/courses/vector-calculus

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