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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Apply the divergence theorem to a closed surface.
- Justify the conclusion "Flux=3·(4π/3)=4π" using the stated assumptions.
Before you start
Divergence and volume 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
Find the outward flux of F=(x,y,z) across the unit sphere.
Why this math matters
Apply the divergence theorem to a closed surface. 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 surface is closed and outward oriented.
- The field is smooth throughout the enclosed volume.
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.
Compute outward flux by measuring sources inside
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the outward flux of F=(x,y,z) across the unit sphere.
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
∇·F=1+1+1=3
Each coordinate component contributes one unit of local expansion.
Work through the mathematics
∮S F·n dS=∫∫∫B3 dV
The divergence theorem applies to the sphere's enclosed ball.
Check the conclusion
Flux=3·(4π/3)=4π
On the unit sphere F equals the unit outward normal, so area gives the same answer.
The result
Flux=3·(4π/3)=4π
On the unit sphere F equals the unit outward normal, so area gives the same answer.
Common mistakes to catch
- A singularity inside the volume invalidates a naive smooth-field application.
- Surface area scaling alone misses the field's changing magnitude.
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 flux across a radius-R sphere?
Show a hint
Integrate divergence over its ball.
Reveal answer and explanation
4πR³
The volume contribution is 3·4πR³/3.
Practice 2
Can the theorem be used on an open hemisphere alone without another boundary?
Show a hint
A volume needs a closed boundary.
Reveal answer and explanation
Not directly
Add the base disk and account for its flux.
Take the idea with you
Connect total boundary outflow to local expansion inside a control volume.
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: Match curl flux with circulation around an edge
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