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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compute divergence to distinguish volume preservation from motionlessness.
- Justify the conclusion "The field is divergence-free although it is not zero" using the stated assumptions.
Before you start
Partial derivatives and vector fields.
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
Is the velocity field u(x,y,z)=(x,−y,0) incompressible?
Why this math matters
Compute divergence to distinguish volume preservation from motionlessness. 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 velocity field is smooth.
- Incompressibility means zero velocity divergence in this constant-density model.
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.
Test incompressibility without requiring zero velocity
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Is the velocity field u(x,y,z)=(x,−y,0) incompressible?
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
∇·u=∂ₓx+∂ᵧ(−y)+∂z0
Divergence adds directional expansion rates.
Work through the mathematics
∇·u=1−1+0=0
Expansion in one direction is balanced by contraction in another.
Check the conclusion
The field is divergence-free although it is not zero
A local fluid element can deform while preserving its volume in this ideal smooth flow.
The result
The field is divergence-free although it is not zero
A local fluid element can deform while preserving its volume in this ideal smooth flow.
Common mistakes to catch
- Incompressible does not mean undeformed or stationary.
- Divergence uses matching component derivatives, not every partial derivative.
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
Find the divergence of v=(x,y,z).
Show a hint
Differentiate each matching component.
Reveal answer and explanation
Three
All three coordinate directions expand rather than cancel.
Practice 2
Do particles in the first field remain fixed?
Show a hint
Solve x′=x,y′=−y.
Reveal answer and explanation
Generally no: x=x₀exp(t),y=y₀exp(−t)
Their positions change while the local volume scaling factors multiply to one.
Take the idea with you
Track a small rectangular fluid element as one side stretches and another contracts.
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: Separate change at a point from change along a moving particle
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