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Vector Calculus (Chapter 13)

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(Chapter 13) Vector Calculus F(x,y)= Scalar Fields and Vector Fields The simplest possible physical field is the scalar field. It represents a function ... – PowerPoint PPT presentation

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Title: Vector Calculus (Chapter 13)


1
Vector Calculus(Chapter 13)
2
Vector Calculus
Chapter
13
13.2-13.3
13.4
13.1
ScalarFields, 2D VectorFields
Gradient VectorFields
LineIntegrals
GreensTheorem
3
F(x,y)ltP(x,y),Q(x,y)gt
4
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)

5
Scalar Fields Visualization of zV(x,y)
  • Scalar potential function for a dipole V(x,y)

6
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7
Maple commands
8
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)

9
The Gradient defines a Vector Field (the force
field)
10
Arrow Diagram for Vector Field
11
Direction Field (magnitude1)
12
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13
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)
14
2D vector field visualization of the flow field
past an air foil using arrows
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