A review of the methods of integration of single-variable functions.

A review of the techniques of integration of single-variable real-valued functions.

Techniques of integration (2)

Overview of common functions and their integrals.

Standard integrals

Techniques of integration (1)

Meaning of integration and its associated symbols.

Definition

Welcome to the course and overview of course content.

Welcome

Techniques of integration (3)

Properties of integrals of real-valued single-variable functions.

Properties of integrals

Introduction

Meaning and methods of evaluation of line integrals.

Meaning of line integral, in contrast to a definite integral along the x-axis.

Properties

Different forms of line integrals and how to evaluate  them.

Forms of line integrals

Parametric forms of equations of straight and curved lines.

Parametric curves

Scalar and vector differentials used in line integrals.

Differentials

More worked examples on evaluating line integrals in two and three dimensions.

Worked examples (2)

Worked examples on evaluating line integrals in two and three dimensions.

Worked examples (1)

Worked examples (3)

Meaning and properties of line integrals of conservative vector fields - path independence of their line integrals.

Conservative vector fields

Line Integrals

Meaning, methods of evaluation and applications of double integrals.

How to evaluate double integrals by change of variables.

Change of variables

More worked examples on evaluation of double integrals - involving change of variables.

Worked examples on evaluation of double integrals.

Meaning and symbols for double integrals.

Double Integrals

Applications of double integrals to areas, volumes, total masses, centres of gravity, moments of inertia, etc.

Calculation of area and centroid of a region R in a plane by double integration.

Area and centroid

Calculation of volume beneath a surface and above a region R in a plane by double integration.

Volume

Calculation of total mass and centre of gravity by double integration.

Mass and centre of gravity

Meaning and calculation of mass moments of inertia by double integration.

Mass moments of inertia

Calculating area moments of inertia of a region R in a plane by double integration.

Area moments of inertia

Worked examples on applications of double integrals to areas, volumes, total masses, centres of gravity, moments of inertia, etc.

Applications of Double Integrals

Statement, meaning, proof and applications of Green's theorem in a plane - the two-dimensional form of the fundamental theorem of calculus.

More worked examples on evaluating line or double integrals using Green's theorem.

Statement of Green's theorem in a plane and its use in evaluating line integrals by double integrals, and vice-versa.

Theorem

Worked examples on evaluating line or double integrals using Green's theorem.

Proof

Green's Theorem

Meaning and methods of evaluation of surface integrals.

Meaning and symbols for surface integrals, in contrast to line integrals.

Scalar and vector forms of surface integrals and how to evaluate them.

Forms of surface integrals

Worked examples on evaluation and applications of surface integrals.

More worked examples on evaluation and applications of surface integrals.

How to obtain the differential (elemental) surface area for surface integrals.

Differential surface

Parametric representation of common surfaces.

Parametric surfaces

Surface Integrals

Statement, meaning, proof and applications of Stoke's theorem.

Statement of Stoke's theorem and its application.

Worked examples on applications of Stoke's theorem.

Stoke's Theorem

Meaning and methods of evaluation of triple integrals.

Meaning and symbols for triple integrals, in contrast to double integrals.

Evaluation

Worked examples on evaluation and applications of triple integrals.

More worked examples on evaluation and applications of triple integrals.

Triple Integrals

Statement, meaning, proof and applications of Gauss' theorem.

Statement of Gauss' theorem and its application.

Worked examples on applications of Gauss' theorem.

Gauss' Theorem

Multiple integration; line, surface and volume integrals.

Tutorials for undergraduates in engineering and physical sciences.

Multiple Integration and Its Applications

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