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.