Teaching Calculus

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  • Nikolay Brodskiy
  • Alexander Brodsky

Part 05 (Transcript)

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So let us look at ...

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... geometry of a surface.

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Because algebraically Taylor series suggests that any function can be linearized ...

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... by dropping all the higher terms of Taylor series. And then you can study that function algebraically.

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That linear function.

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What about geometry of that?

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What if you take ...

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... a graph of a function in three-space?

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So this surface ...

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... that looks like a saddle, right? So ...

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It's curved one way ...

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... and the other way.

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Er ...

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So why is it that ...

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... localization -- zooming in at a point gives you a plane.

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Well you just have to zoom in. And what I am doing is ...

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... is honest recalculating of the same graph, of the same function ...

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... just changing the scale and zooming in to that point.

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And from faraway that point looks curved.

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But if you zoom in ...

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... then you see how this surface, this pink surface ...

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... becomes flatter and flatter.

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And now those curves -- blue and green curves ...

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... they are almost straight.

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And the pink surface containing those curves ...

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We don't even distinguish ...

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Well you don't even see the difference between the plane and ...

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... that [...] curve. So ...

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So that's an example of ...

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... geometric process of zooming in.

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And no matter where we zoom in we expect the surface to look like a plane locally.

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So that's the ...

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... simple justification of why we look at ...

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... planes as linearizations of surfaces.

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And of course not all the surfaces can be linearized.

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If you look at special examples like a cone ...

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... then looking at the vertex of the cone and zooming in ...

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... no matter how you zoom in you'll see exactly the same cone.

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Right? The angle of the cone will never change.

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It'll stay the same.

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And you zoom in, and zoom in and you will always see this cone.

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And this cone at that point will never become a plane.

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So that is ...

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... the simplest example of a surface that cannot be linearized.

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Well of course that surface defines a function.

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Right? If you put that surface in that x, y, z ...

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... space in this way ...

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... [...] of course for every x, y on x, y plane ...

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... you will have some point on the cone above it.

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And that defines a function.

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And the typical example would be ...

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Er ...

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... the polar distance actually.

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The distance from the origin.

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So z equals ...

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... the magnitude of the vector x, y.

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Or in other words the square root of x squared plus y squared

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So that function ...

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... that algebraic function ...

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... has a graph that looks like a cone. And it's not linearizable at zero.

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So we'll try to stay away from polar coordinates for as long as possible.

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And that's exactly the reason.

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And the ...

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... even in er ...

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... the variables x and y ...

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... the function y equals absolute value of x is not good.

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Right? Because the graph of that function on the plane has that corner.

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And we have to stay away from that studying calculus.

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Keeping in mind that yes you can study that function using calculus but there are some ...

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... exceptional things you can do ... well you have to do.

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