Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Find a potential and verify path independence.
- Justify the conclusion "The work is 12" using the stated assumptions.
Before you start
Gradients 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=(2xy,x²), compute work from (0,0) to (2,3).
Why this math matters
Find a potential and verify path independence. 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 defined smoothly on all of R².
- Paths are piecewise smooth.
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.
Replace a path integral with endpoint values
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For F=(2xy,x²), compute work from (0,0) to (2,3).
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
φ(x,y)=x²y has ∇φ=(2xy,x²)
Constructing an explicit potential is stronger than guessing from a picture.
Work through the mathematics
∫C F·dr=φ(2,3)−φ(0,0)
The gradient theorem applies to every piecewise smooth joining path.
Check the conclusion
The work is 12
The endpoints determine this conservative work, regardless of the route.
The result
The work is 12
The endpoints determine this conservative work, regardless of the route.
Common mistakes to catch
- Path independence is a property of the field and domain together.
- A line integral of a vector field is not generally just path length.
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 work around a closed loop?
Show a hint
Start and end at the same potential value.
Reveal answer and explanation
Zero
The endpoint difference vanishes.
Practice 2
Does zero curl on a domain with a hole always provide a global potential?
Show a hint
Recall circulation obstructions.
Reveal answer and explanation
No
Domain topology matters; an explicit potential or sufficient domain assumptions are needed.
Take the idea with you
Compare conservative work with a force law whose work depends on the route.
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: Turn boundary circulation into interior curl
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