Lecture 10: Curl

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# Lecture 10: Curl - PowerPoint PPT Presentation

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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Presentation Transcript
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
• 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
• 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

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

• and in the xy plane yields
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

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