Section 17.3Vector 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:

• A 3D vector field has the form:

),(),,(),(or

,where)(),()(

yxgyxfyxF

yxrrgrfrF

),,(),,,(),,,(),,(or

,,where)(),(),()(

zyxhzyxgzyxfzyxF

zyxrrhrgrfrF

• 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

• Let’s take a look with Maple

jxiyrxyrrF

or,where)(