Vector Fields

A whole arrow at every point of the plane.

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

NameDate

Figures — use these to answer the problems

Every arrow points inward, so this is a sink. Release a point anywhere and it drains to the centre.

strength1

start x3

start y2

Figure 1

The same picture with the arrows turned sideways. Nothing drains anywhere; everything circles.

strength1

start x2

start y0

Figure 2

Straightforward practice. Get the method working first.

5 problems

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

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
F = ⟨3, 0⟩ everywhere. What is the y-component at any point?
Answer
F = ⟨2x, 2y⟩ is the gradient of x² + y². Is it a gradient field? 1 yes, 0 no.
Answer

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

4 problems

9
Match each field to what it does.
Draw a line from each item on the left to its match on the right.
- ⟨x, y⟩
- ⟨−x, −y⟩
- ⟨−y, x⟩
- Flows outward from the origin
- Flows inward to the origin
- Circles the origin
F = ⟨0, 0⟩ at every point. Does anything move? 1 yes, 0 no.
Answer
The Field Value
F = ⟨2x, y⟩. What is the x-component of F at the point (3, 5)?
Answer
The Classification
Arrows all point away from the origin. Enter 1 for source, 2 for sink.
Answer