Lecture 11 stokes theorem
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Lecture 11: Stokes Theorem. Consider a surface S , embedded in a vector field Assume it is bounded by a rim (not necessarily planar) For each small loop For whole loop (given that all interior boundaries cancel in the normal way).

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Lecture 11: Stokes Theorem

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Lecture 11 stokes theorem

Lecture 11: Stokes Theorem

  • Consider a surface S, embedded in a vector field

  • Assume it is bounded by a rim (not necessarily planar)

  • For each small loop

  • For whole loop (given that all interior boundaries cancel in the normal way)

OUTER RIM SURFACE INTEGRAL OVER ANY

SURFACE WHICH SPANS RIM


Consequences

Consequences

  • Curl is incompressible

  • Consider two surfaces (S1 + S2) with same rim

    must be the the same for both surfaces (by Stokes Theorem)

  • For closed surface formed by

  • Also for a conservative field

using the divergence theorem

by Stokes Theorem

CONSERVATIVE = IRROTATIONAL

  • These are examples (see Lecture 12) of second-order differential operators in vector calculus


Example

Example

  • So this is a conservative field, so we should be able to find a potential 

  • All can be made consistent if


Another example

Another Example

z

(0,0,1)

C2

quarter circle

C1

C3

y

(0,1,0)

x

  • Check via direct integration


Physical example

Physical Example

  • Magnetic Field due to a steady current I

  • “Ampere’s Law”

  • Since

  • obeys a continuity equation with no sources or sinks (see Lecture 13)

applying Stoke’s Theorem

units: A m-2

Differential form of Ampere’s Law


Lecture 11 stokes theorem

  • Now make the wire ‘fat’ (of radius a), carrying a total current I

  • Inside the wire

  • Outside the wire

  • Inside, has a similar field to in solid-body rotation, where

z axis along the wire, r is radial distance from the z axis

|B|

a r


Lecture 11 stokes theorem

  • Outside wire: z component from loops in the xy plane

  • z component is zero, because

  • and x and y components are also zero

    in the yz and xz planes

  • Therefore

R2

d

R1

  • Note that region where field is conservative ( ) is not simply connected: loop cannot be shrunk to zero without going inside wire (where field is not conservative).


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