Vector Calculus (Chapter 13)

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Vector Calculus(Chapter 13)
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Vector Calculus
Chapter
13
13.2-13.3
13.4
13.1
ScalarFields, 2D VectorFields
Gradient VectorFields
LineIntegrals
GreensTheorem
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F(x,y)ltP(x,y),Q(x,y)gt
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Scalar Fields and Vector Fields

• The simplest possible physical field is the
  scalar field.
• It represents a function depending on the
  position in space. A scalar field is
  characterized at each point in space by a single
  number.
• Examples of scalar fields
• temperature, gravitational potential,
  electrostatic potential (voltage)

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Scalar Fields Visualization of zV(x,y)

• Scalar potential function for a dipole V(x,y)

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Maple commands
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Scalar Fields and Equipotential Lines

• The level curves or contours of the function
  zV(x,y) are the equipotential lines of the
  scalar potential field V(x,y)

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The Gradient defines a Vector Field (the force
field)
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Arrow Diagram for Vector Field
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Direction Field (magnitude1)
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Equipotential surfaces are orthogonal to the
electric force field
Notice the force field is directed towards
placeswhere the potential V is lower, e.g.,
where thecharge is negative - at(0.25,0).But
mathematically,the gradient points in the
opposite direction (greatest ascent) which is
why f-Vand Fgrad(f)grad(-V)
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2D vector field visualization of the flow field
past an air foil using arrows