Teaching Calculus

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  • Nikolay Brodskiy
  • Alexander Brodsky

Part 03 (Transcript)

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The next step in studying 3-dimentional curves ...

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Right? We studied curvature.

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And now we can look at ...

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... look back at what we studied about curvature and how we studied it.

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And apply that to the new concept of torsion.

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Just to begin with to determine what it is.

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And how to measure. Because I'd like to measure it.

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Well, especially when it is so related to ...

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... well, to how people fly, right?

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And how to build those jets.

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So what is it?

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Er ... remember what we thought about the curvature.

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We thought about the tangent line changing.

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And then as it changes ...

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... it changes direction.

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Amd we measured the angle between the adjacent tangent lines. And we ...

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... related that angle to the distance travelled.

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And that rate of change was the curvature.

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Now in 3D it is not only that the tangent line changes.

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Which determines the curvature again.

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But also the osculating plane is changing. Right?

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So, if the osculating plane changes ...

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... then the position of that plane at one moment ...

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... and position at the other moment ...

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Those two are determined by ...

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... are different by an angle again.

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Right? Those two different planes have some angle between them.

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And then ...

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What is it that makes sence to measure?

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Isn't it that we measure the angle between those two planes?

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And relate it to the distance we travelled.

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And then we say 'the greater that ratio is ...

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... the greater the torsion is.'

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Because the faster we change the plane ...

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... the greater that ratio is supposed to be.

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Right? So the torsion is ...

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Our first simplistic idea is ...

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... torsion to be of course limit ...

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Right? As distance travelled ...

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... going to zero ...

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... well, the angle ...

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... er ... between osculating planes ...

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... related to the distance travelled.

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And here I'm taking the purely geometric point of view.

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There is no physics involved.

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But when you see the fighter jet flying you first think about physics probably. Right?

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