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Lecture 10: Curl. Consider any vector field And a loop in one plane with vector area If is conservative then But in general And we introduce a new ( pseudo ) vector Of magnitude: Which depends on the orientation of the loop

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lecture 10 curl
Lecture 10: Curl
  • Consider any vector field
  • And a loop in one plane with vector area
  • If is conservative then
  • But in general
  • And we introduce a new (pseudo) vector
  • Of magnitude:
  • Which depends on the orientation of the loop
  • The direction normal to the direction of the plane (of area = ) which maximises the line integral

...is the direction of (will prove soon)

Can evaluate 3 components by taking areas with normals in xyz directions

curl is independent of small loop shape
Curl is independent of (small) loop shape
  • Sub-divide loop into N tiny rectangles of area
  • Summing line integrals, the interior contributions cancel, yielding a line integral over boundary C
  • Thus is same for large loop as for each rectangle
  • Hence, curl is independent of loop shape
  • But loop does need itself to be vanishingly small

area a

C

curl is a vector
Curl is a Vector
  • Imagine a loop of arbitrary direction
  • Make it out of three loops with normals in the directions
  • Because interior loops cancel, we get
  • Explains why line integral =

where  is angle between loop normal and direction of curl

in cartesian coordinates
In Cartesian Coordinates

4

  • Consider tiny rectangle in yz plane
  • Contributions from sides 1 and 2
  • Contributions from sides 3 and 4
  • Yielding in total

dz

z

1

2

3

dy

y

Az

y

slide5

Repeating this exercise for a rectangular loop in the zx plane yields

  • and in the xy plane yields
curl and rotation
Curl and Rotation

z

  • Consider a solid body rotating with angular velocity about the z axis
  • describes the velocity vector field
  • Evaluate curl from the definition
  • x-component:
  • Similarly for y-component
  • As only the z component is non-zero we see that and curl will be parallel

x

y

slide9

z component

  • In a rotating fluid, its easier to `swim round’ here than here
  • i.e. local evidence that the fluid is rotating

R2

d

R1

  • So magnitude of curl is proportional to any macroscopic rotation
  • In a more general case, curl is a function of position

Flow is said to be irrotational