Transcript Section 17.3 Vector Fields

Section 17.3
Vector Fields
Intro to Vector Fields
• A vector field is a function that assigns a vector to
each point in the plane or in 3-space
• The input for a vector field is a position vector
• We have seen vector fields before
– The gradient of a function f(x,y) is a vector field
– At each point (x,y) the gradient vector points in the
direction of maximum rate of increase of f
• Let’s take a look at the plot on page 847
– We have a velocity vector field for a part of the Gulf
stream
– Each vector shows the velocity of the current at that
point
• A 2D vector field has the form:
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F (r )  f (r ), g (r )  where r  x, y 
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or F ( x, y )  f ( x, y ), g ( x, y ) 
• A 3D vector field has the form:
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F (r )  f (r ), g (r ), h(r )  where r  x, y, z 
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or F ( x, y, z )  f ( x, y, z ), g ( x, y, z ), h( x, y, z ) 
• We have seen a velocity vector field
• Another physical quantity typically represented
by a vector is force
• For example, gravity is a force that pulls all
masses towards the center of the earth, this is
creates a vector field (or a force field)
• Let’s see how we can visualize a vector field
given a formula
• What does the vector field in 2 space look like for
 
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F (r ) where r  y, x  or r  yi  xj
• Let’s take a look with Maple