Vector Fields

A whole arrow at every point of the plane.

12 problems · about 22 minutes · N-VM.A.1

What this lesson teaches

The student reads a vector field and identifies sources, sinks and rotation.

A vector field assigns a vector to every point, so it is drawn as a field of arrows.
Arrows flowing outward mark a source; arrows flowing inward mark a sink.
Some fields are gradients of a surface, and a purely rotating field never is.

Straightforward practice. Get the method working first.

5 problems

F = ⟨y, x⟩ at (2, 7). What is the x-component?
Answer 7
Why 7.
What does a vector field assign to each point?
Answer A vector, with both a direction and a size.
Why A whole vector.
F = ⟨x, y⟩ at (4, 1). What is the x-component?
Answer 4
Why It equals x.
F = ⟨−y, x⟩ at (1, 0). What is the y-component?
Answer 1
Why The y-component is x.
Arrows converging on a point. Enter 1 for source, 2 for sink.
Answer 2
Why Everything drains in.

The same ideas with more to keep track of.

3 problems

F = ⟨−y, x⟩. Does a paddle wheel at the origin spin? 1 yes, 0 no.
Answer 1
Why The field circulates.
F = ⟨3, 0⟩ everywhere. What is the y-component at any point?
Answer 0
Why A constant field.
F = ⟨2x, 2y⟩ is the gradient of x² + y². Is it a gradient field? 1 yes, 0 no.
Answer 1
Why A potential function exists.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each field to what it does.
Answer ⟨x, y⟩ → Flows outward from the origin; ⟨−x, −y⟩ → Flows inward to the origin; ⟨−y, x⟩ → Circles the origin
Why The minus signs decide inward against outward.
F = ⟨0, 0⟩ at every point. Does anything move? 1 yes, 0 no.
Answer 0
Why No arrows, no motion.
The Field Value: F = ⟨2x, y⟩. What is the x-component of F at the point (3, 5)?
Answer 6
Why 6.
The Classification: Arrows all point away from the origin. Enter 1 for source, 2 for sink.
Answer 1
Why A source.