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.
Graduate chapters and video availability01 · Read and understand
What you will learn
- Use an integral obstruction to distinguish closed from exact forms.
- Justify the conclusion "∮ω=2π, so ω is not globally exact" using the stated assumptions.
Before you start
Differential forms, exterior derivative, 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
On R² without the origin, analyze ω=(−y dx+x dy)/(x²+y²).
Why this math matters
Use an integral obstruction to distinguish closed from exact forms. 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 origin is excluded from the domain.
- The unit-circle orientation is counterclockwise.
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.
Find a closed differential form with nonzero circulation
PausedQuestion: Start with the question. Paused.
Question
Start with the question
On R² without the origin, analyze ω=(−y dx+x dy)/(x²+y²).
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
Writing P=−y/r² and Q=x/r² gives ∂Q/∂x=∂P/∂y
The equality holds away from the excluded origin, so dω=0.
Work through the mathematics
On the unit circle, x=cos t,y=sin t gives ω=dt
Substitute the curve and its differentials directly.
Check the conclusion
∮ω=2π, so ω is not globally exact
An exact form has zero integral around every closed loop; the hole prevents a global angle potential.
The result
∮ω=2π, so ω is not globally exact
An exact form has zero integral around every closed loop; the hole prevents a global angle potential.
Common mistakes to catch
- Closed does not imply globally exact on every domain.
- A locally defined angle is not automatically a single-valued global function.
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 integral on two counterclockwise turns?
Show a hint
Add the contributions of each traversal.
Reveal answer and explanation
4π
Circulation scales with the winding number.
Practice 2
Can ω be exact on a simply connected slit domain?
Show a hint
Choose a continuous branch of angle there.
Reveal answer and explanation
Yes
On such a domain a smooth angle function can satisfy dθ=ω.
Take the idea with you
Explain why local potential measurements may fail to assemble into a global potential.
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: Distinguish intrinsic curvature from visible bending
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