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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use Stokes' theorem with compatible boundary orientation.
- Justify the conclusion "∮C F·dr=∫∫disk1 dA=π" using the stated assumptions.
Before you start
Curl, surface normals, and line integrals.
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/2,0), compute circulation around the unit circle viewed counterclockwise from above.
Why this math matters
Use Stokes' theorem with compatible boundary 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 field is smooth near the chosen surface.
- Boundary and normal orientations follow the right-hand rule.
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.
Match curl flux with circulation around an edge
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For F=(−y/2,x/2,0), compute circulation around the unit circle viewed counterclockwise from above.
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
∇×F=(0,0,1)
The field's rotation points upward.
Work through the mathematics
Use the unit disk with normal n=(0,0,1)
The right-hand rule matches the stated counterclockwise boundary.
Check the conclusion
∮C F·dr=∫∫disk1 dA=π
Stokes' theorem equates the boundary circulation with curl flux through the spanning disk.
The result
∮C F·dr=∫∫disk1 dA=π
Stokes' theorem equates the boundary circulation with curl flux through the spanning disk.
Common mistakes to catch
- A normal cannot be reversed while leaving the theorem's boundary orientation unchanged.
- Curl flux is distinct from flux of the original field.
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 if the disk normal points downward?
Show a hint
Reverse the compatible boundary direction too.
Reveal answer and explanation
Both oriented integrals change sign
Consistent orientation preserves the theorem.
Practice 2
Can another smooth spanning surface give a different answer here?
Show a hint
The same boundary and smooth global field are used.
Reveal answer and explanation
No
Stokes' theorem fixes the circulation through the shared boundary.
Take the idea with you
Choose a simpler spanning surface to evaluate a difficult-looking circulation integral.
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: Solve a linear ODE by building a product derivative
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