Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use Green's theorem with a consistent orientation.
- Justify the conclusion "The circulation is π" using the stated assumptions.
Before you start
Line integrals and partial derivatives.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For F=(−y/2,x/2), find counterclockwise circulation around the unit circle.
Why this math matters
Use Green's theorem with a consistent orientation. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The boundary is positively oriented and encloses the disk.
- The field is continuously differentiable nearby.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Turn boundary circulation into interior curl
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For F=(−y/2,x/2), find counterclockwise circulation around the unit circle.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
∂Q/∂x−∂P/∂y=1/2−(−1/2)=1
The scalar curl is constant throughout the disk.
Work through the mathematics
∮C P dx+Q dy=∫∫D1 dA
Green's theorem converts the boundary integral to an area integral.
Check the conclusion
The circulation is π
Counterclockwise orientation gives positive area; reversing the path would negate the answer.
The result
The circulation is π
Counterclockwise orientation gives positive area; reversing the path would negate the answer.
Common mistakes to catch
- Do not confuse circulation with outward flux.
- The curl formula's order determines its sign.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the circulation around a radius-two circle?
Show a hint
Integrate the same constant curl over its disk.
Reveal answer and explanation
4π
Disk area scales with radius squared.
Practice 2
What changes for clockwise orientation?
Show a hint
Reverse the boundary direction.
Reveal answer and explanation
The sign becomes negative
Orientation is part of the line-integral definition.
Take the idea with you
Estimate total circulation by adding small local rotational contributions over a region.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Compute outward flux by measuring sources inside
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.