Arrows everywhere, added along a path

Mathematics · Calculus & Analysis · ages 20-21

Name ______________________ Date ____________

Mia sketches the constant field F(x, y) = (1, 0) with every arrow pointing left. Is Mia right?
Circle one: True False
Which integral adds up a field dotted with the direction of travel along a curve?
- The line integral of F dot dr
- The scalar line integral of f ds
- A double integral over a region
- An ordinary integral over an interval
What sits at each point of a vector field?
- An arrow showing flow there
- A single plain number
- A whole curve
What does dotting a field with your direction and adding it along a path measure?
- The length of the path
- The mass of the path
- How much the field helps or hinders the motion
The field is F(x, y) = (0, 1) and C is the segment from (4, 1) to (4, 6). What is the line integral of F along C?
Answer: ______________
The field is F(x, y) = (y, x). Walk the parabola y = x squared from (0, 0) to (1, 1) with r(t) = (t, t squared). Compute the work and type the result.
Answer: ______________
You integrate a field along a curve, then walk the same curve backwards. What happens to the result?
- It keeps its size but flips sign
- It stays exactly the same
- It becomes zero
The field is F(x, y) = (1, 0) and C is the straight segment from (0, 0) to (2, 5). What is the line integral of F along C?
Answer: ______________
A problem asks for the work done by a field along a path. A student reaches for f ds. What is the mistake?
- That tool always gives zero
- That tool ignores direction, so it cannot give work
- That tool only works on straight lines
Near some point, all the arrows loop around it in circles. What does the sketch tell you?
- The flow has a source there
- The flow is still there
- The flow spins there

Answer key

For grown-ups. Fold this page away before handing over the rest.

Arrows everywhere, added along a path W1-mt_VZxKEmsVgO-s1

False · Mia is wrong: positive x components point right, not left.
The line integral of F dot dr · Work and circulation use F dot dr; f ds ignores direction.
An arrow showing flow there · A field attaches one vector, one arrow, to every point.
How much the field helps or hinders the motion · That sum is work: positive when the field pushes with you.
5 · Only vertical motion counts: the segment rises 5, so the integral is 5.
1 · The dot product becomes 3t squared, and its integral from 0 to 1 is 1.
It keeps its size but flips sign · Reversing flips every tangent, so the whole vector integral negates.
2 · Only the x component of motion counts: the segment advances 2 in x, so the integral is 2.
That tool ignores direction, so it cannot give work · Work needs the tangent direction, which only F dot dr uses.
The flow spins there · Looping arrows mark a swirl, the rotation pattern in the sketch.