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## Multivariable calculus

### Course: Multivariable calculus>Unit 2

Lesson 6: Curvature

# Curvature intuition

An introduction to curvature, the radius of curvature, and how you can think about each one geometrically. Created by Grant Sanderson.

## Want to join the conversation?

• Is this the same type of curvature physicists mean, when they talk about the curvature of space (space-time)? If so, they talk about negative curvature, which seems impossible with the definition: k = 1 / R, with R positive (and also with k = || dT/ds ||). Great video! :)
• Not really. Without getting too much into it, "physical" curved space is modeled using a non euclidean, topological, metric space. In a non euclidean space the Pythagorean theorem does not hold - which intuitively could be described as a space where the shortest path between two points isn't a straight line, but a curved one. General relativity asserts that the only reason we think we live in a euclidean space is because the mass around us bends the very fabric of space (and time) around us.

I guess you could look at it as some kind of generalization of this for Riemannian manifolds, but it's not exactly analogues. Sorry if this sounds a bit vague, but this is a really confusing concept as is (not to mention one I'm not overly familiar with!), so trying to describe it without diving too much into advanced math is a bit hard.
• am I the only one who did not understand his car analogy at all? what does he mean the car is turning? if the road is the curve wouldn't he just be going straight on the road and its like he's going up or downhill? So confused
• The graph is like a bird's eye view, looking down from above. He's drawing curved line on the road and you are driving tracing that curved line. It's not uphill or downhill.

He is driving and trying to trace out the curved line so he has to turn the steering wheel. Since the steering wheel is turned and if it get stuck at that turned position, the car would be curving and it wouldn't be going straight on the road. He mentions that steering wheel isn't straight and you get stuck in that state at .
• Is he the person from the youtube channel 3Blue1Brown?
• How do we know that the car traces out a circle?
• If you steer the handle and kept it at the same place and you push down the gas pedal and kept it at the same speed, the car will trace out a circle. You can try it at the big open place or by using a remote control car. The curve angle, or how much it curves, is going be be consistent because the steering wheel is fixed at certain steered angle. Constant speed and constant curve angle traces out a circle.
• Can the curvature be calculated by the absolute value of the second derivative, or concavity?
• I am definitely a bit late, but I looked it up and it seems one definition of curvature is that if you have a unit tangent vector on a curve, the derivative of that tangent vector with respect to time (as the vector moves along the curve) is the curvature. So in a way, I think the second derivative notion is correct.
(1 vote)
• What the? I didn't get this at all! Why is the center of the green circle moving? The blue curve is a circle, right? And at a certain point of the center of the green circle, the 2 circles (curves) seem to coincide...? somewhere around .
• The blue curve is actually not a circle, otherwise the curvature would be the same everywhere and the green circle would stay the same.
The two curves almost coincide when the red point ist at the top but there's still a noticable difference where the blue curve is close to the x-axis.

The center of the green circle is moving because the curvature of the blue curve isn't the same everywhere, so the radius of the green circle changes. SInce its edge always touches the red point, its center has to move.
• How do you determine positive curvature and negative curvature?