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Graduate chapters and video availability01 · Read and understand
What you will learn
- Compute the divergence-free projection of a single nonzero Fourier mode.
- Justify the conclusion "k·Pkv=0" using the stated assumptions.
Before you start
Fourier modes, dot products, and orthogonal projection.
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 wavevector k=(1,1,0) and coefficient v=(2,0,1), compute Pkv=v−k(k·v)/|k|².
Why this math matters
Compute the divergence-free projection of a single nonzero Fourier mode. 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 calculation uses a nonzero Fourier wavevector.
- The projection is the Euclidean orthogonal projection in coefficient space.
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.
Remove the longitudinal component of a Fourier velocity
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For wavevector k=(1,1,0) and coefficient v=(2,0,1), compute Pkv=v−k(k·v)/|k|².
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
k·v=2 and |k|²=2
The longitudinal coefficient is the component parallel to the wavevector.
Work through the mathematics
Pkv=(2,0,1)−(1,1,0)=(1,−1,1)
Subtracting the gradient-like direction leaves a transverse coefficient.
Check the conclusion
k·Pkv=0
The projected Fourier mode is divergence-free because Fourier divergence is multiplication by ik·.
The result
k·Pkv=0
The projected Fourier mode is divergence-free because Fourier divergence is multiplication by ik·.
Common mistakes to catch
- A Fourier-space projection is not componentwise deletion in physical coordinates.
- Zero frequency needs an explicit convention.
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 happens to a coefficient parallel to k?
Show a hint
The projection subtracts it entirely.
Reveal answer and explanation
It becomes zero
Pure longitudinal components lie in the projection kernel.
Practice 2
Why treat k=0 separately?
Show a hint
The denominator would vanish.
Reveal answer and explanation
The formula is undefined there
A constant velocity mode is already divergence-free and is usually retained by convention.
Take the idea with you
Explain how a spectral solver can enforce incompressibility mode by mode.
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: Recover pressure as a constraint-enforcing field
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