Uploaded December 2025 | Updated September 2026, 2 weeks ago
In this video, I go over the definition of curvature, which describes how quickly a curve changes direction at a given point. It is defined as the magnitude of the derivative of the unit tangent vector in terms of its arc length, so that it is independent of any specific parameterization. I also rewrite this definition to obtain curvature, indicated by the Greek letter Kappa (κ), as a function of the parameter t, as this is often easier to compute. I illustrate this with an example on obtaining the curvature of a circle, which turns out to be equal to 1 divided by its radius. Note that the curvature of a straight line is equal to zero since the unit tangent vector would be zero in that case.
#math #vectors #calculus #arclength #curvature
Timestamps:
- Curvature: Smooth parameterizations are continuous and the tangent vector is not equal to 0 – 0:00
- Recap on Unit Tangent Vector, T – 1:13
- Graph of T, which changes quickly when C bends sharply – 2:46
- Definition of curvature as the magnitude of the derivative of T in terms of arc length – 5:00
- Curvature is easier to compute when written in terms of t – 7:30
- Curvature = magnitude of derivative of T divided by derivative of the vector function – 11:35
- Example 3: Curvature of a circle is 1/radius – 13:52
- Obtaining parametric equations of a circle – 15:10
- Curvature of a circle with radius a is 1/a: Large circles have small curvature – 24:05
- Curvature of a straight line is zero – 25:00
Notes and playlists:
- Summary: inleo.io/threads/view/mes/re-leothreads-sisojcak
- Notes: peakd.com/@mes/koxharrd
- Sections playlist: youtube.com/playlist?list=PLai3U8-WIK0Gd5OieYiZMaSKw1QfyHbiG
- Vector Functions playlist: youtube.com/playlist?list=PLai3U8-WIK0HQl7xTQBS_O8Te8PcpNm4U .
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In this video, I go over the definition of curvature, which describes how quickly a curve changes direction at a given point. It is defined as the magnitude of the derivative of the unit tangent vector in terms of its arc length, so that it is independent of any specific parameterization. I also rewrite this definition to obtain curvature, indicated by the Greek letter Kappa (κ), as a function of the parameter t, as this is often easier to compute. I illustrate this with an example on obtaining the curvature of a circle, which turns out to be equal to 1 divided by its radius. Note that the curvature of a straight line is equal to zero since the unit tangent vector would be zero in that case.
#math #vectors #calculus #arclength #curvature
Timestamps:
- Curvature: Smooth parameterizations are continuous and the tangent vector is not equal to 0 – 0:00
- Recap on Unit Tangent Vector, T – 1:13
- Graph of T, which changes quickly when C bends sharply – 2:46
- Definition of curvature as the magnitude of the derivative of T in terms of arc length – 5:00
- Curvature is easier to compute when written in terms of t – 7:30
- Curvature = magnitude of derivative of T divided by derivative of the vector function – 11:35
- Example 3: Curvature of a circle is 1/radius – 13:52
- Obtaining parametric equations of a circle – 15:10
- Curvature of a circle with radius a is 1/a: Large circles have small curvature – 24:05
- Curvature of a straight line is zero – 25:00
Notes and playlists:
- Summary: inleo.io/threads/view/mes/re-leothreads-sisojcak
- Notes: peakd.com/@mes/koxharrd
- Sections playlist: youtube.com/playlist?list=PLai3U8-WIK0Gd5OieYiZMaSKw1QfyHbiG
- Vector Functions playlist: youtube.com/playlist?list=PLai3U8-WIK0HQl7xTQBS_O8Te8PcpNm4U .
------------------------------------------------------
Become a MES Super Fan! youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join
DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate
SUBSCRIBE via EMAIL: https://mes.fm/subscribe
MES Links: https://mes.fm/links
MES Truth: https://mes.fm/truth
Official Website: https://MES.fm
Hive: peakd.com/@mes
Email me: contact@mes.fm
Free Calculators: https://mes.fm/calculators
BMI Calculator: https://bmicalculator.mes.fm
Grade Calculator: https://gradecalculator.mes.fm
Mortgage Calculator: https://mortgagecalculator.mes.fm
Percentage Calculator: https://percentagecalculator.mes.fm
Free Online Tools: https://mes.fm/tools
iPhone and Android Apps: https://mes.fm/mobile-apps
