Vector integrals Line integrals Surface integrals Volume integrals Integral theorems

1. CALCULUS III CHAPTER 4: Vector integrals and integral theorems • Vector integrals • Line integrals • Surfaceintegrals • Volumeintegrals • Integral theorems • Thedivergencetheorem • Green’stheorem in theplane • Stoke’stheorem • Conservativefields and scalarpotentials • Vector potentials

2. VECTOR INTEGRALS

3. Line integral • Alsocalledpath integral (physics), contour integral, curve integral isan integral wherethefunctionisintegratedalong a curve r(t)instead of along a straight line (Riemann) • Thefunctionto be integrated can be either a scalar of a vectorfield • Ifwewanttointegrate a scalarfieldfalong a curve r(t), the line integral issimply • The line integral of a scalar field fover a curve C can be thought of as the area under the curve C along a surface z = f(x,y), described by the field.

4. Line integral of a scalarfieldover a curve

5. Line integral of vector fields: Simple integration of a vector Geometrically

6. Line integral of a vector field Line integral of a vector field

7. Line integral of a vector field

10. Interpretation of line integrals of vector fields: work / flow In general theworkissaidto be ‘pathdependent’ becausetheresult of the integral dependsonthe concrete shape of r. Do notconfusewithpathintegrationformulation of quantum mechanics (Feynman) (these are integrationover a space of paths)

11. Surfaceintegrals • The surface integral is a definite integral taken over a surface. • It can be thought of as the double integral analog of the line integral. • Given a surface, one may integrate over its scalar fields, and vector fields are surfaceintegrals of scalar fieldsoverplanesurfaces • Therefore, weneedtogeneralizethis concept: • Forcurvedsurfaces • For vector fields

12. Curvedsurfaces: areaelements

13. Surfaceintegrals of vector fields ( These can be thought as integration of scalarfieldover a surface: )

14. (integration of a vector fieldover a planesurface)

15. (integration of a vector fieldover a curvedsurface – a sphere)

16. Surfaceintegrals of vector fields: a general approach • Recallthat in general, a surface can be described in threeways • Theoptimaldescriptionwilldependonthe concrete surfaceto be described • Wewillthereforedevelopthreedifferentways of calculatingthesurface integral, dependingonthespecificdescription of thesurface (parametricform) (explicitform) (implicitform)

17. Surfaceintegrals of vector fieldsSurfacedescribed in parametricform (2 parameters)

18. Surfaceintegrals of vector fieldsSurfacedescribed in explicitform

19. Surfaceintegrals of vector fieldsSurfacedescribed in implicitform

20. Volumeintegrals • In thissectionwewillonlyconsiderintegrals of scalaror vector fieldsovervolumesdefined in , either in cartesianor in genericcurvilinearcoordinates. Where werecallthatthevolumentelementfor canonical curvilinearcoordinates CYLINDRICAL SPHERICAL

21. INTEGRAL THEOREMS

22. In theprecedingsectionswehavestudiedhowtocalculatetheintegrals of vector fieldsover curves (line integrals), surfaces, and volumes. • Itturnsoutthatthereexistrelationsbetweenthesekind of integrals in somecircumstances. • Theserelations are genericallygatheredunderthelabelintegral theorems. • Thesetheorems link theconcepts of line and surfaceintegralsthroughthedifferentialoperator

23. Thedivergencetheorem Statement • Thistheorem relates thesurface integral of a vector fieldwiththevolume integral of a scalarfieldconstructed as thedivergence of the vector field: • Thesurface S overwhichtheintegrationisperformedisindeedtheboundary of thevolume V • Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region.

24. Thedivergencetheorem Statement • Thistheoremalsorequiressomemathematicalconditions: - thevolume V must be compact and itsboundarysurfacemust be piecewisesmooth- the vector fieldFmust be continuouslydifferentiableontheneighborhood of V • This theorem is also called Gauss theorem or Ostrogradsky'stheorem, and is a special case of the more general Stoke’stheoremthatwewillsee in thenextsection • Thistheoremisveryimportant in physics (electromagnetism, fluid dynamics)

26. Thedivergencetheorem Statement Corollary(vector form of divergencetheorem)

27. Thedivergencetheorem Statement • Thistheoremisstated in . It has otherversions in lowerdimensions: • : the 1-dimensional version reduces tothefundamental theorem of calculus, that links theconcepts of derivative and integral of a scalarfield • : the 2-dimensional versioniscalledtheGreen’stheorem, that links the line integral of a vector fieldover a curve withthesurface integral over a planeregion. Let’sseethistheorem in more detail.

28. Green’stheorem • Green's theorem is also special case of the Stokes theorem that we will explain in the next section, when applied to a region in the xy-plane

30. Green’stheorem Corollary D

31. Stoke’stheorem • Thistheorem relates theline integral of a vector fieldwiththesurface integral of another vector field, constructed as thecurl of theformer:

33. Stoke’stheorem Corollary (vector form of Stokes theorem)

34. Someimportantapplications of divergence, Green and Stoke’stheorems Electromagnetism: Maxwell laws

35. Summarizingall of theabove in a general theorem (notexaminable) The integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e. • Fundamental theorem of calculus: f(x) dx is the exterior derivative of the 0-form, i.e. function, F: in other words, that dF = f dx(A closed interval [a, b] is a simple example of a one-dimensional manifold with boundary) • Divergencetheorem • Green’stheorem • Stokes theoremis a special case of the general Stokes theorem (with n = 2) once we identify a vector field with a 1 form using the metric on Euclidean three-space.

36. Conservativefields and scalarpotentials Nowthatwehavestudiedthegeneralities of integral theorems, wewillanalysesome concrete situations of specialinterest. IfFisconservative,

37. Conservativefields and scalarpotentials Physicalinterpretation of conservativefields • IfFisinterpreted as a forceappliedto a particle, thenifFisconservativethismeansthattheworkneededtotake a particlefrom position P to position Q isindependent of thepath • In otherwords, the net work in going round a pathtowhereonestarted (P=Q) iszero: energyisconserved. • ThegravitationalfieldF(r)isanexample of a conservativeforce. Itsassociatedscalarpotentialφ(r)is a scalarfieldcalledthepotentialenergy. • Usually, and withoutloss of generality, a minussignisintroduced: • toemphasizethatif a particleis moved in thedirection of thegravitationalfield, theparticledecreasesitspotentialenergy, and viceversa. • Energyconservation: * Theenergyweneedto use totake a bikerfrom B toA isstored as potentialenergy, and released in terms of kineticenergyas wedropitfrom A toB.* Thisenergyisindependent of theslope of thehill (pathindependence). A B B

39. Divergence-free fields and potentialvectors • A vector fieldFisdivergence-free iff • As thedivergence describes thepresences of sources and sinks of thefield, a divergence-free fieldmeansthatthe balance of sources and sinksisnull. • Example: themagneticfieldBisempiricallydivergence-free, and one of the Maxwell equationsis • Thissuggeststhatmagneticmonopoles(isolatedmagnetic ‘charges’, i.e. isolatedsourcesorsinks of magneticfields) do notexist (howeverstringtheories do predicttheirexistence, so it’scurrently a hottopic in particlephysics). Electric monopoles (charges) Magneticmonopoles

40. Divergence-free fields and potentialvectors Gauge transformation • Most fundamental physicaltheories are gauge invariant.