9'7 Divergence and Curl

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9.7 Divergence and Curl
Vector Fields
F(x,y) P(x,y) i Q(x,y) j F(x,y,z)
P(x,y,z) i Q(x,y,z) j R(x,y,z) k
Example 1 Graph the 2-dim vector field F(x,y)
-y i x j
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Draw vectors of the same length
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Divergence
Scalar
Example 2
IF
Find div F
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Curl
vector
Example 2
IF
Find curl F
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Curl
Cross product of the del operator and the vector F
Remarks 1) 2)
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WHY Vector Fields
The motion of a wind or fluid can be described by
a vector field. The concept of a force field
plays an important role in mechanics,
electricity, and magnetism.
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Physical Interpretations
Curl was introduced by Maxwell James Clerk
Maxwell (1831-1879) Scottish Physicist b.
Edinburgh, Scotland, June 13, 1831, d. Cambridge,
England, November 5, 1879 He published his first
scientific paper at age 14, entered the
University of Edinburgh at 16, and graduated from
Cambridge University.
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Physical Interpretations

• Curl is easily understood in connection with the
  flow of fluids. If a paddle device, such as shown
  in fig, is inserted in a flowing fluid, the the
  curl of the velocity field F is a measure of the
  tendency of the fluid to turn the device about
  its vertical axis w.
• If curl F 0 then flow of the fluid is said to
  be irrotational. Which means that it is free of
  vortices or whirlpools that would cause the
  paddle to rotate.
• Note irrotational does not mean that the fluid
  does not rotate.

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Physical Interpretations

• The volume of the fluid flowing through an
  element of surface area per unit time that is ,
  the flux of the vector field F through the area.
• The divergence of a velocity field F near a point
  p(x,y,z) is the flux per unit volume.
• If div F(p) gt 0 then p is said to be a source for
  F. since there is a net outward flow of fluid
  near p
• If div F(p) lt 0 then p is said to be a sink for
  F. since there is a net inward flow of fluid near
  p
• If div F(p) 0 then there are no sources or
  sinks near p.
• The divergence of a vector field can also be
  interpreted as a measure of the rate of change of
  the density of the fluid at a point.
• If div F 0 the fluid is said to be
  incompressible

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