向量微积分

1.VectorAlgebra

1.1Vectorsandscalars

1.1.1Definitionofavectorandascalar

1.1.2Additionofvectors

1.1.3Componentsofavector

1.2Dotproduct

1.2.1Applicationsofthedotproduct

1.3Crossproduct

1.3.1Applicationsofthecrossproduct

1.4Scalartripleproduct

1.5Vectortripleproduct

1.6Scalarfieldsandvectorfields

2.Line，SurfaceandVolumeIntegrals

2.1Applicationsandmethodsofintegration

2.1.1Examplesoftheuseofintegration

2.1.2Integrationbysubstitution

2.1.3Integrationbyparts

2.2Lineintegrals

2.2.1Introductoryexample：workdoneagainstaforce

2.2.2Evaluationoflineintegrals

2.2.3Conservativevectorfields

2.2.4Otherformsoflineintegrals

2.3Surfaceintegrals

2.3.1Introductoryexample：flowthroughapipe

2.3.2Evaluationofsurfaceintegrals

2.3.30ltherformsofsurfaceintegrals

2.4volumeintegrals

2.4.1Introductoryexample：massofanobjectwithvariabledensity

2.4.2Evaluationofvolumeintegrals

3.Gradient，DivergenceandCurl

3.1PartialdifierentiationandTaylorseries

3.1.1Partialdifierentiation

3.1.2Taylorseriesinmorethanonevariable

3.2Gradientofascalarfield

3.2.1Gradientsconservativefieldsandpotentials

3.2.2Physicalapplicationsofthegradient

3.3Divergenceofavectorfield

3.3.1Physicalinterpretationofdivergence

3.3.2Laplacianofascalarfield

3.4Cllrlofavectorfield

3.4.1Physicalinterpretationofcurl

3.4.2Relationbetweencurlandrotation

3.4.3Curlandconservativevectorfields

4.SuffixNotationanditsApplications

4.1Introductiontosuffixnotation

4.2TheKroneckerdelta

4.3Thealternatingtensor

4.4Relationbetweenijkandij

4.5Grad，divandcurlinsuffixnotation

4.6Combinationsofgrad，divandcurl

4.7Grad，divandcurlappliedtoproductsoffunctions

5.IntegralTheorems

5.1Divergencetheorem

5.1.1C：onservationofmassforafluid

5.1.2Applicationsofthedivergencetheorem

5.1.3Relatedtheoremslinkingsurfaceandvolumeintegrals

5.2StokesStheorem

5.2.1ApplicationsofStokesStheorem

5.2.2Relatedtheoremslinkinglineandsurfaceintegrals

6.CurvilinearCoordinates

6.1Orthogonalcurvilinearcoordinates

6.2Grad，divandcurlinorthogonalcurvilinearcoordinatesystems

6.2.1Gradient

6.2.2Divergence

6.2.3Curl

6.3Cylindricalpolarcoordinates

6.4Sphericalpolarcoordinates

7.CartesianTensors

7.1Coordinatetransformations

7.2Vectorsandscalars

7.3Tensors

7.3.1Thequotientrule

7.3.2Symmetricandanti-symmetrictensors

7.3.3Isotropictensors

7.4Physicalexamplesoftensors

7.4.1Ohmslaw

7.4.2Theinertiatensor

8.ApplicationsofVectorCalculus

8.1Heattransfer

8.2Electromagnetism

8.2.1Electrostatics

8.2.2Electromagneticwavesinavacuum

8.3Continuummechanicsandthestresstensor

8.4Solidmechanics

8.5Fluidmechanics

8.5.1Equationofmotionforafluid

8.5.2Thevorticityequation

8.5.3Bernoullisequation

Solutions

Index

　　Vectorcalculusisthefundamentallanguageofmathematicalphysics.Itprovidesawaytodescribephysicalquantitiesinthree-dimensionalspaceandthewayinwhichthesequantitiesvary.Manytopicsinthephysicalsciencescanbeanalysedmathematicallyusingthetechniquesofvectorcalculus.Thesetopicsincludefluiddynamics,solidmechanicsandelectromagnetism,allofwhichinvolveadescriptionofvectorandscalarquantitiesinthreedimensions.Thisbookassumesnopreviousknowledgeofvectors.However,itisassumedthatthereaderhasaknowledgeofbasiccalculus,includingdifferentiation,integrationandpartialdifferentiation.Someknowledgeoflinearalgebraisalsorequired,particularlytheconceptsofmatricesanddeterminants.