Section 17.3 Vector Fields

1. Section 17.3Vector Fields

2. 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

4. A 2D vector field has the form: • A 3D vector field has the form:

5. 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