Tip:
Highlight text to annotate it
X
And well, let me talk now about curvature.
Because last time we discussed what the curvature is about ...
And geometrically ...
So we talked about the curvature from geometric point of view.
Geometrically we see that the curve at this point is curved less than the curve at that point.
And ...
What we can observe is the tangent line.
And what we pretend to do is ...
We pretend to follow the curve ...
... and observe our tangent line changes.
And as we follow the curve here see how the tangent line changes.
And then what we see is that tangent line turns.
And we measure this angle.
And then we relate that angle to the distance travelled.
And we see that we have to travel a lot ...
... to make that change in the angle.
And we have to travel just a little bit ...
... to make a comparable change of the angle.
And that makes the difference between the curvature.
So we have to look at the ratio of the ...
... of angle ...
... between the tangent lines ...
... over the distance ...
... travelled.
And of course we have to take the limit of that.
That was the fundamental observation ...
... that we don't just take one step ...
... measure these numbers and take the ratio.
We take limit as ...
As what? As this distance goes to zero.
And our question is how to understand that.
And for us ...
... the way of trying to understand something I'm promoting ...
... so far is by taking a different point of view on the same thing.
Because we are in geometry now ...
... we are trying to understand things geometrically.
And geometrically we see everybody ...
... is easily convinced that the curve is curved more here than there.
So there is probably nothing else we can extract from purely geometric point of view.
And the better understanding we can get is by relating this point of view to a different point of view.