Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compute curvature and torsion for a circular helix.
- Justify the conclusion "τ=((r′×r″)·r‴)/|r′×r″|²=1/2" using the stated assumptions.
Before you start
Cross products and three derivatives.
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 r(t)=(cos t,sin t,t), find curvature κ and torsion τ.
Why this math matters
Compute curvature and torsion for a circular helix. 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 parameter t is real and uses radians.
- The standard right-handed orientation sets the sign convention.
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.
Separate spatial twisting from bending
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For r(t)=(cos t,sin t,t), find curvature κ and torsion τ.
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
r′=(−sin t,cos t,1), r″=(−cos t,−sin t,0)
These give speed √2 and cross-product magnitude √2.
Work through the mathematics
κ=|r′×r″|/|r′|³=1/2
Curvature measures tangent turning per arc length.
Check the conclusion
τ=((r′×r″)·r‴)/|r′×r″|²=1/2
The numerator is one and the denominator two; torsion records departure from a single plane.
The result
τ=((r′×r″)·r‴)/|r′×r″|²=1/2
The numerator is one and the denominator two; torsion records departure from a single plane.
Common mistakes to catch
- Torsion and curvature measure different changes.
- Changing a helix's handedness can change torsion sign.
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 torsion for a circle in the xy-plane?
Show a hint
Its third derivative remains in that plane.
Reveal answer and explanation
Zero
The scalar triple product vanishes.
Practice 2
Why must the cross product be nonzero for this torsion formula?
Show a hint
Inspect the denominator.
Reveal answer and explanation
Otherwise the Frenet normal and binormal can be undefined
The formula assumes nonzero curvature.
Take the idea with you
Compare a coiled spring with a planar curved wire.
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: Find shortest spherical travel through a central angle
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