Title: Vector Calculus (Chapter 13)
1Vector Calculus(Chapter 13)
2Vector Calculus
Chapter
13
13.2-13.3
13.4
13.1
ScalarFields, 2D VectorFields
Gradient VectorFields
LineIntegrals
GreensTheorem
3F(x,y)ltP(x,y),Q(x,y)gt
4Scalar 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)
5Scalar Fields Visualization of zV(x,y)
- Scalar potential function for a dipole V(x,y)
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7Maple commands
8Scalar 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)
9The Gradient defines a Vector Field (the force
field)
10Arrow Diagram for Vector Field
11Direction Field (magnitude1)
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13Equipotential 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)
142D vector field visualization of the flow field
past an air foil using arrows