Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Graduate chapters and video availability01 · Read and understand
What you will learn
- Compute a cylinder's metric and compare its two principal curvatures.
- Justify the conclusion "The principal curvatures have magnitudes 1/R and 0, so K=0" using the stated assumptions.
Before you start
Surface parameterizations and first fundamental forms.
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 X(u,v)=(R cos u,R sin u,v), what is the Gaussian curvature?
Why this math matters
Compute a cylinder's metric and compare its two principal curvatures. 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:
- R is positive.
- The discussion is local and ignores overlap at a seam.
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.
Distinguish intrinsic curvature from visible bending
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For X(u,v)=(R cos u,R sin u,v), what is the Gaussian curvature?
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
Xᵤ·Xᵤ=R², Xᵥ·Xᵥ=1, Xᵤ·Xᵥ=0
The metric is ds²=R²du²+dv².
Work through the mathematics
With s=Ru, the metric becomes ds²+dv²
Locally unrolling the cylinder gives the flat plane metric.
Check the conclusion
The principal curvatures have magnitudes 1/R and 0, so K=0
Extrinsic bending in one direction does not create intrinsic Gaussian curvature.
The result
The principal curvatures have magnitudes 1/R and 0, so K=0
Extrinsic bending in one direction does not create intrinsic Gaussian curvature.
Common mistakes to catch
- Mean curvature and Gaussian curvature are different quantities.
- Zero Gaussian curvature does not mean a surface lies in a plane in space.
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
Is a sphere intrinsically flat?
Show a hint
Its two principal curvatures have nonzero equal magnitude.
Reveal answer and explanation
No; K=1/R²
Their product is positive.
Practice 2
Can a paper rectangle wrap onto a cylinder locally without stretching?
Show a hint
Compare the local metrics.
Reveal answer and explanation
Yes in the ideal surface model
The coordinate change s=Ru preserves lengths on the developed sheet.
Take the idea with you
Explain the difference between rolling a sheet into a cylinder and stretching it onto a sphere.
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: Specify which events a probability model can distinguish
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.