Table Of ContentRiver Publishers Series in Mathematical, Statistical and
Elementary Vector Calculus Computational Modelling for Engineering
and its Applications with
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Sir Isaac Newton, one of the greatest scientists and mathematicians r
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of all time, introduced the notion of a vector to define the existence of L
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gravitational forces, the motion of the planets around the sun, and the Bl
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motion of the moon around the earth. Vector calculus is a fundamental Pu
scientific tool that allows us to investigate the origins and evolution of rolu
space and time, as well as the origins of gravity, electromagnetism, and gs
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nuclear forces. Vector calculus is an essential language of mathemati- an
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cal physics, and plays a vital role in differential geometry and studies d
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related to partial differential equations widely used in physics, engi- t
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neering, fluid flow, electromagnetic fields, and other disciplines. Vector n A
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calculus represents physical quantities in two or three-d imensional p
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space, as well as the variations in these quantities. l
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The machinery of differential geometry, of which vector calculus a
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is a subset, is used to understand most of the analytic results in a more i
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general form. Many topics in the physical sciences can be mathemati- n
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cally studied using vector calculus techniques. w
This book is designed under the assumption that the readers have it
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no prior knowledge of vector calculus. It begins with an introduction to
vectors and scalars, and also covers scalar and vector products, vector
differentiation and integrals, Gauss’s theorem, Stokes’s theorem, and
Green’s theorem. The MATLAB programming is given in the last chapter.
This book includes many illustrations, solved examples, practice
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Nita H. Shah
River Publishers
River Jitendra Panchal
Elementary Vector Calculus
and its Applications with
MATLAB Programming
RIVER PUBLISHERS SERIES IN MATHEMATICAL,
STATISTICAL AND COMPUTATIONAL MODELLING
FOR ENGINEERING
SeriesEditors:
MANGEYRAM
GraphicEraUniversity,India
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HiroshimaUniversity,Japan
ALIAKBARMONTAZERHAGHIGHI
PrairieViewTexasA&MUniversity,USA
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• StructuralMechanics
• Theoreticalandappliedmechanics
Foralistofotherbooksinthisseries,visitwww.riverpublishers.com
Elementary Vector Calculus
and its Applications with
MATLAB Programming
Nita H. Shah
GujaratUniversity,India
Jitendra Panchal
ParulUniversity,India
Published,soldanddistributedby:
RiverPublishers
Alsbjergvej10
9260Gistrup
Denmark
www.riverpublishers.com
ISBN:978-87-7022-387-4(Hardback)
978-87-7022-386-7(Ebook)
(cid:2)c2022RiverPublishers
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin
aretrievalsystem,ortransmittedinanyformorbyanymeans,mechanical,
photocopying,recordingorotherwise,withoutpriorwrittenpermissionof
thepublishers.
Contents
Preface ix
ListofFigures xi
1 BasicConceptofVectorsandScalars 1
1.1 IntroductionandImportance . . . . . . . . . . . . . . . . . 1
1.2 RepresentationofVectors . . . . . . . . . . . . . . . . . . . 1
1.3 PositionVectorandVectorComponents . . . . . . . . . . . 2
1.4 ModulusorAbsoluteValueofaVector . . . . . . . . . . . . 3
1.5 ZeroVectorandUnitVector . . . . . . . . . . . . . . . . . 4
1.6 UnitVectorsintheDirectionofAxes . . . . . . . . . . . . 4
1.7 RepresentationofaVectorintermsofUnitVectors . . . . . 5
1.8 AdditionandSubtractionofVectors . . . . . . . . . . . . . 6
1.9 ProductofaVectorwithaScalar . . . . . . . . . . . . . . . 6
1.10 DirectionofaVector . . . . . . . . . . . . . . . . . . . . . 7
1.11 CollinearandCoplanarVectors . . . . . . . . . . . . . . . . 8
1.11.1 CollinearVectors . . . . . . . . . . . . . . . . . . . 8
1.11.2 CoplanarVectors . . . . . . . . . . . . . . . . . . . 8
1.12 GeometricRepresentationofaVectorSum . . . . . . . . . . 8
1.12.1 LawofParallelogramofVectors . . . . . . . . . . . 8
1.12.2 LawofTriangleofVectors . . . . . . . . . . . . . . 9
1.12.3 PropertiesofAdditionofVectors. . . . . . . . . . . 9
1.12.4 PropertiesofScalarProduct . . . . . . . . . . . . . 10
1.12.5 Expression of Any Vector in Terms of the Vectors
AssociatedwithitsInitialPointandTerminalPoint . 10
1.12.6 ExpressionofAnyVectorinTermsofPosition
Vectors . . . . . . . . . . . . . . . . . . . . . . . . 11
1.13 DirectionCosinesofaVector . . . . . . . . . . . . . . . . . 12
1.14 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
v
vi Contents
2 ScalarandVectorProducts 29
2.1 ScalarProduct,orDotProduct,orInnerProduct . . . . . . . 29
2.2 TheMeasureofAngleBetweentwoVectorsand
Projections . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 PropertiesofaDotProduct . . . . . . . . . . . . . . 30
2.3 VectorProductorCrossProductorOuterProductofTwo
Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 GeometricInterpretationofaVectorProduct . . . . . . . . . 38
2.4.1 PropertiesofaVectorProduct . . . . . . . . . . . . 39
2.5 ApplicationofScalarandVectorProducts . . . . . . . . . . 45
2.5.1 WorkDonebyaForce . . . . . . . . . . . . . . . . 46
2.5.2 MomentofaForceAboutaPoint . . . . . . . . . . 46
2.6 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 VectorDifferentialCalculus 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 VectorandScalarFunctionsandFields . . . . . . . . . . . . 55
3.2.1 ScalarFunctionandField . . . . . . . . . . . . . . 56
3.2.2 VectorFunctionandField . . . . . . . . . . . . . . 56
3.2.3 LevelSurfaces . . . . . . . . . . . . . . . . . . . . 56
3.3 CurveandArcLength. . . . . . . . . . . . . . . . . . . . . 57
3.3.1 ParametricRepresentationofCurves . . . . . . . . . 57
3.3.2 CurveswithTangentVector . . . . . . . . . . . . . 58
3.3.2.1 TangentVector . . . . . . . . . . . . . . . 59
3.3.2.2 ImportantConcepts . . . . . . . . . . . . 60
3.3.3 ArcLength . . . . . . . . . . . . . . . . . . . . . . 61
3.3.3.1 UnitTangentVector . . . . . . . . . . . . 61
3.4 CurvatureandTorsion. . . . . . . . . . . . . . . . . . . . . 64
3.4.1 FormulasforCurvatureandTorsion . . . . . . . . . 67
3.5 VectorDifferentiation . . . . . . . . . . . . . . . . . . . . . 70
3.6 GradientofaScalarFieldandDirectionalDerivative . . . . 73
3.6.1 GradientofaScalarField . . . . . . . . . . . . . . 73
3.6.1.1 PropertiesofGradient . . . . . . . . . . . 73
3.6.2 DirectionalDerivative . . . . . . . . . . . . . . . . 74
3.6.2.1 PropertiesofGradient . . . . . . . . . . . 75
3.6.3 EquationsofTangentandNormaltotheLevel
Curves . . . . . . . . . . . . . . . . . . . . . . . . 84
3.6.4 EquationoftheTangentPlanesandNormalLines
totheSurfaces . . . . . . . . . . . . . . . . . . . . 85
Contents vii
3.7 DivergenceandCurlofaVectorField . . . . . . . . . . . . 86
3.7.1 DivergenceofaVectorField . . . . . . . . . . . . . 86
3.7.1.1 PhysicalInterpretationofDivergence . . . 86
3.7.2 CurlofaVectorField . . . . . . . . . . . . . . . . . 89
3.7.2.1 PhysicalInterpretationofCurl. . . . . . . 89
3.7.3 Formulaeforgrad,div,curlInvolvingOperator∇ . 96
3.7.3.1 Formulae for grad, div, curl Involving
Operator∇Once . . . . . . . . . . . . . 96
3.7.3.2 Formulae for grad, div, curl Involving
Operator∇Twice . . . . . . . . . . . . . 100
3.8 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 VectorIntegralCalculus 111
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2 LineIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2.1 Circulation . . . . . . . . . . . . . . . . . . . . . . 112
4.2.2 WorkDonebyaForce . . . . . . . . . . . . . . . . 112
4.3 PathIndependenceofLineIntegrals . . . . . . . . . . . . . 113
4.3.1 Theorem:IndependentofPath . . . . . . . . . . . . 113
4.4 SurfaceIntegrals . . . . . . . . . . . . . . . . . . . . . . . 122
4.4.1 Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.4.2 EvaluationofSurfaceIntegral . . . . . . . . . . . . 123
4.4.2.1 ComponentformofSurfaceIntegral . . . 124
4.5 VolumeIntegrals . . . . . . . . . . . . . . . . . . . . . . . 129
4.5.1 ComponentFormofVolumeIntegral . . . . . . . . 129
4.6 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5 Green’sTheorem,Stokes’Theorem,andGauss’Theorem 135
5.1 Green’sTheorem(inthePlane) . . . . . . . . . . . . . . . . 135
5.1.1 AreaofthePlaneRegion . . . . . . . . . . . . . . . 137
5.2 Stokes’Theorem . . . . . . . . . . . . . . . . . . . . . . . 146
5.3 Gauss’DivergenceTheorem . . . . . . . . . . . . . . . . . 154
5.4 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6 MATLABProgramming 167
6.1 BasicofMATLABProgramming . . . . . . . . . . . . . . . 167
6.1.1 BasicofMATLABProgramming . . . . . . . . . . 167
6.1.1.1 IntroductoryMATLABprogrammes . . . 168
6.1.1.2 RepresentationofaVectorinMATLAB . 183
viii Contents
6.1.1.3 RepresentationofaMatrixinMATLAB . 186
6.2 SomeMiscellaneousExamplesusingMATLAB
Programming . . . . . . . . . . . . . . . . . . . . . . . . . 188
Index 207
AbouttheAuthors 213
Preface
Vector calculus is an essential language of mathematical physics. Vector
calculus plays a vital role in differential geometry, and the study related to
partial differential equations is widely used in physics, engineering, fluid
flow,electromagneticfields,andotherdisciplines.Vectorcalculusrepresents
physicalquantitiesintwoorthree-dimensionalspace,aswellasthevariations
inthesequantities.
The machinery of differential geometry, of which vector calculus is a
subset, is used to understand most of the analytic results in a more general
form. Many topics in the physical sciences can be mathematically studied
usingvectorcalculustechniques.
Descriptionofthebook:
This book is meant for readers who have a basic understanding of vector
calculus. This book is designed to provide accurate information to readers.
The language in the book is kept simple so that all readers can easily
understandeachconcept.
Thisbookbeginswiththeintroductionofvectorsandscalarsinchapter1.
Chapter1containsessentialbasicdefinitionsandconcepts,vectorintermsof
unit vectors, geometric representation of vector sum, and direction cosines.
The scalar and vector products, measurement of angle and projections, geo-
metric interpretation of a vector product, and their applications are given in
chapter 2. In chapter 3, vector and scalar functions and fields, curves, arc
length, formulae for curvature and torsion, and its derivation, curl, diver-
gence,andgradientwithimportantpropertiesandphysicalinterpretation,and
important results are given in vector differential calculus. Chapter 4 vector
integral calculus includes line integrals, circulation, path independence, sur-
faceintegrals,volumeintegrals,anditsapplicationslikefluxandworkdone
by a force are given. In chapter 5, derivation of Green’s theorem, Stokes’s
theorem,andGauss’divergencetheoremaregivenwithvarioussolveexam-
ples. MATLAB programming is given in the last chapter 6 includes basic
informationaboutMATLAB.Initially,basicexamplesaregivenwithproper
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