lecture 4a - vector calculus - utrecht university 4a...the curl (rotor) operator • definition:...
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VectorCalculusAprimer
FunctionsofSeveralVariables
• Asinglefunctionofseveralvariables:𝑓: 𝑅$ → 𝑅, 𝑓 𝑥(, 𝑥),⋯ , 𝑥$ = 𝑦.
• Partialderivativevector,orgradient,isavector:𝛻𝑓 =
𝜕𝑦𝜕𝑥(
,⋯ ,𝜕𝑦𝜕𝑥$
Multi-ValuedFunctions• Avector-valued functionofseveralvariables:
𝑓: 𝑅$ → 𝑅/,𝑓 𝑥(, 𝑥), ⋯ , 𝑥$ = 𝑦(, 𝑦),⋯ , 𝑦/ .
• Canbeviewedasachangeofcoordinates,ora mapping.
• Wegetamatrix,denotedastheJacobian:
𝛻𝑓 =
𝜕𝑦(𝜕𝑥(
⋮𝜕𝑦(𝜕𝑥$
⋯ ⋯𝜕𝑦$𝜕𝑥(
⋮𝜕𝑦$𝜕𝑥$
https://www.math.duke.edu/education/ccp/materials/mvcalc/parasurfs/para1.html
DotProduct• �⃗�, 𝑏 ∈ 𝑅$, �⃗� 6 𝑏 = 𝑎7 ∗ 𝑏7 ∈ 𝑅.• Wegetthat�⃗� 6 𝑏 = �⃗� 𝑏 cos 𝜃,where𝜃 istheangle betweenthevectors.• Squarednormofvector: �⃗� ) = �⃗� 6 �⃗�.• Matrixmultiplicationresultó dotproductsofrowandcolumnvectors.
• Alternativenotation:�⃗� 6 𝑏 = �⃗�, 𝑏
DotProduct• Ageometricinterpretation:thepartof�⃗� whichisparallel toaunitvectorinthedirectionof𝑏.• Andviceversa!
• Projectedvector:𝑎∥ =>6?? 𝑏.
• Thepartof𝑏 orthogonal to�⃗� hasnoeffect!
CrossProduct
• Typicallydefinedonlyfor𝑅K.
• �⃗�×𝑏 = 𝑎N𝑏O − 𝑎O𝑏N, 𝑏Q𝑎O − 𝑏O𝑎Q, 𝑎Q𝑏N − 𝑎N ∈ 𝑅.
• Ormoregenerally:
�⃗�×𝑏 =𝑎Q 𝑎N 𝑎O𝑏Q 𝑏N 𝑏O𝑥R 𝑦R �̂�
CrossProduct
• Theresultvectorisorthogonal tobothvector• Direction:Right-handrule.• Normaltotheplanespannedbybothvectors.
• Itsmagnitude is �⃗�×𝑏 = �⃗� 𝑏 sin 𝜃.• Parallelvectorsó crossproductzero.
• Thepartof𝑏 parallel to�⃗� hasnoeffectonthecrossproduct!
BilinearMaps• Alsodenotedas“2-tensors”.
• 𝑀:𝑉×𝑉 → 𝑅,𝑀 𝑢, �⃗� = 𝑐.
• Taketwovectorsintoascalar.
• Symmetry:𝑀 𝑢, �⃗� = 𝑀 �⃗�, 𝑢
• Linearity:𝑀 𝑎𝑢 + 𝑏𝑤, �⃗� = 𝑎𝑀 𝑢, �⃗� + 𝑏𝑀 𝑤, �⃗� .• Thesamefor�⃗� forsymmetry.
• Canberepresentedby𝑛×𝑛 matrices:c=𝑢c𝑀�⃗�.
DirectionalDerivative
• Thechangeinfunction𝑢 inthe(unit)direction𝑑e:
𝛻f𝑢 = 𝛻𝑢, 𝑑
• Interpretation:“stretch”ofthefunctioninthisdirection: 𝛻f𝑢• Oftenusedsquared: 𝛻f𝑢 )
Example:JacobianandChangeofCoordinates• Supposechangeofcoordinates𝐺 𝑥(, 𝑥),⋯ , 𝑥$ = 𝑦(, 𝑦),⋯ , 𝑦$ .
• How does function 𝑓 𝑥(, 𝑥),⋯ , 𝑥$ transform?𝛻𝑓(𝑦) = 𝛻𝐺 6 𝛻𝑓(𝑥)
• Changeof length intransformed direction �⃗� = 𝛻𝐺 6 𝑢:�⃗� ) = 𝑢c 𝛻𝐺c 6 𝛻𝐺 𝑢.
• Where 𝛻𝐺c 6 𝛻𝐺 isasymmetricbilinearform.• Whichisalsoametric.
http://mathinsight.org/image/change_variable_area_transformation
VectorFieldsin3D
• Avector-valued functionassigningavectortoeachpointinspace:𝑓: 𝑅K → 𝑅K, 𝑓 �⃗� = �⃗�.• Physics:velocityfields,forcefields,advection,etc.• Specialvectorfields:• Constant• Rotational• Gradients ofscalarfunctions:�⃗� = 𝛻𝑔.
http://vis.cs.brown.edu/results/images/Laidlaw-2001-QCE.011.html
IntegrationoveraCurve
• Givenacurve𝐶 𝑡 = 𝑥 𝑡 ,𝑦 𝑡 , 𝑧(𝑡) , 𝑡 ∈ [𝑡o, 𝑡(].• Andavectorfield�⃗�(𝑥, 𝑦, 𝑧)• Theintegrationofthefieldonthecurveisdefinedas:
q �⃗� 6 𝑑𝐶r
= q �⃗� 6𝑑𝑥𝑑𝑡,𝑑𝑦𝑑𝑡,𝑑𝑧𝑑𝑡
𝑑𝑡st
su
ConservativeVectorFields
• Avectorfield�⃗� isconservative ifthereisascalar function𝜑 sothatforeverycurve𝐶 𝑡 , 𝑡 ∈ [𝑡o, 𝑡(]:
q �⃗� 6 𝑑𝐶r
= 𝜑 𝑡( − 𝜑 𝑡o
• Equivalently:if�⃗� = 𝛻𝜑.• Theintegralisthenpathindependent.
ConservativeVectorFields
• Physicalinterpretation:thevectorfield�⃗� istheresultofapotential𝜑.• Example:thework(potentialenergy)𝑊 donebygravityforce�⃗� =𝛻𝑊 isonlydependentoftheheightgained\lost.• Corollary:theintegralofaconservativevectorfieldoveraclosedcurve iszero!
TheCurl(Rotor)Operator• Definition:𝛻×�⃗� = x
xQ⁄ , x xNz , x xO⁄ ×�⃗�.• Producesavectorfieldfromavectorfield.• Geometricintuition:𝛻×�⃗� encodeslocalrotation(vorticity)thatthevectorfield(asaforce)induceslocallyonthepoint.• Direction:therotationaxis𝑛R.
• Integraldefinition:
𝛻×�⃗� 6 𝑛R = lim|→o
1𝐴� 𝑑�⃗� 6 𝑑𝐶r
• 𝐶 isaninfinitesemalcurvearoundthepoint• 𝐴 isthearea itencompasses.
http://www.chabotcollege.edu/faculty/shildreth/physics/gifs/curl.gif
Irrotational Fields
• Fieldswhere𝛻×�⃗� = 0.• AlsodenotedCurl-free.
• Conservativefields=>irrotational.• asforeveryscalar𝜑:
𝛻×𝛻𝜑 = 0
• Itisevidentfromtheintegraldefinition: lim|→o
(| ∮ 𝑑�⃗� 6 𝑑𝐶r .
• Isirrotational =>Conservativefieldsalsocorrect?• Only(andalways)forsimply-connecteddomains!
Divergence
• Definition:𝛻 6 �⃗� = xxQ⁄ , x xNz , x xO⁄ 6 �⃗�.
• Producesascalarvaluefromavectorfield.• Geometricintuition:𝛻 6 �⃗� encodeslocalchangeindensityinducedbyvectorfieldasaflux.• Integraldefinition:
𝛻 6 �⃗� = lim�→ �
1𝑉� �⃗� 6 𝑛R�(�)
• 𝑆(𝑉) isthesurfaceofaninfinitesimalvolumearoundthepoint.• 𝑛R istheoutwardlocalnormal.
http://magician.ucsd.edu/essentials/WebBookse8.html
Laplacian
• Thedivergenceofthegradientofascalarfield:∆𝜑 = 𝛻)𝜑 = 𝛻 6 𝛻𝜑 .
• Producesascalarvaluefroma scalarfield.• Geometricintuition:Measuringhowmuchafunctionisdiffused orsimilartotheaverageofitssurrounding.• Foundinheatandwaveequations.• Usedextensivelyinsignalprocessing,e.g.fordenoising.
StokesTheorem
• Amoregeneralformoftheideaof“conservativefields”• Themoderndefinition:
q 𝑑𝑤�
= q 𝑤x�
• Geometricinterpretation:Integratingthedifferentialofafieldinsideadomainó integratingthefieldontheboundary.
StokesTheorem
• Generalizesmanyclassicalresults.
• Integratingalongacurve:∫ 𝛻𝜑 6 𝑑𝐶r = 𝜑 𝑡( − 𝜑 𝑡o .• Specialcase:Fundamentaltheoryofcalculus:∫ 𝐹� 𝑥 𝑑𝑥Qt
Qu= 𝐹 𝑥( − 𝐹(𝑥o).
• Kelvin-Stokes Theorem:
∮ 𝑣 6x� 𝑑𝐶 = ∬ 𝛻×𝑣 𝑑𝑆� .• Divergence theorem:
� 𝛻 6 𝑣 𝑑𝑉�
= � 𝑣 6 𝑛R 𝑑𝑆x�
• …andmanysimilarmore.