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Graduate chapters and video availability01 · Read and understand
What you will learn
- Identify a weak derivative without requiring a classical derivative everywhere.
- Justify the conclusion "The weak derivative is sign(x) almost everywhere" using the stated assumptions.
Before you start
Integration by parts and smooth compactly supported test functions.
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
Find the weak derivative of u(x)=|x| on (−1,1).
Why this math matters
Identify a weak derivative without requiring a classical derivative everywhere. 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:
- Test functions are smooth with compact support in (−1,1).
- Weak derivatives are interpreted as locally integrable functions when possible.
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.
Differentiate a corner through integration against tests
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the weak derivative of u(x)=|x| on (−1,1).
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
Split ∫|x|φ′(x)dx at zero
On each side the function is classically smooth.
Work through the mathematics
Integration by parts gives ∫uφ′=−∫sign(x)φ
The boundary contributions at zero vanish because u(0)=0, and endpoint terms vanish by compact support.
Check the conclusion
The weak derivative is sign(x) almost everywhere
The corner is allowed in W¹,∞; the derivative's value at the single point zero does not change the weak identity.
The result
The weak derivative is sign(x) almost everywhere
The corner is allowed in W¹,∞; the derivative's value at the single point zero does not change the weak identity.
Common mistakes to catch
- A Dirac distribution is not an ordinary pointwise function.
- A weak derivative need not agree with a classical derivative at every point.
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 second distributional derivative of |x|?
Show a hint
Differentiate the jump in sign.
Reveal answer and explanation
2δ₀
The jump size two creates twice the Dirac distribution.
Practice 2
Does changing sign(0) affect the weak derivative class?
Show a hint
Singletons have Lebesgue measure zero.
Reveal answer and explanation
No
Sobolev derivatives are identified up to almost-everywhere equality.
Take the idea with you
Explain why piecewise-linear approximations can fit naturally into variational PDE spaces.
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.
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