The Main Intuitive Idea of Calculus (applied to curves in Differential Geometry)
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- čas přidán 22. 05. 2024
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Your videos are so good, they are fire, please keep making them!!!!!!!!!!!!!!!!!
I did ok at math in college, but if it wasn't for extremely committed (internet) math educators like you I would have forgotten what I learned and became uninterested in understanding math from a perspective of enriching my intelligence and expanding my understanding of how fascinatingly beautiful this universe is.
Awesome stuff, thank you for recording these.
Thank you - it means a lot!
The Weierstrass function would like a word!
And it would have a point!
He did say smooth curve 0:25
@@drslyonethe weierstrass function is smooth everywhere and differentiable nowhere. That’s exactly the counter example to the intuition of this video about differentiation.
@@austin8179 Well my textbook defines a vector-valued function to be smooth if it has a continuous first derivative, and ||r'(t)|| is never 0.
@@drslyonethe weierstrass function is real valued. If you want some more info, you can check this link : en.wikipedia.org/wiki/Weierstrass_function. In general, continuity does not imply differentiability and it is easy to show example of continuous functions that are not differentiable on a finite set of points (example the absolute value of x is not differentiable in 0 though continuous in 0), and as the weierstrass function shows, there are even ways to find functions with infinite non differentiable values.
When I was in high school we defined differentiability this way:
A function f is differentiable in x₀ if there exists a number a and a function E(h) →0 for h →0 (called an epsilon function) such that
f(x₀+h) = f(x₀) + a·h + E(h)·h
a is then called the differential quotient.
This definition emphasises the quasi linearity, I think.
Ex 1. f(x) = x²
(x₀ + h)² = x₀² + 2x₀·h + h·h
f'(x₀) = 2x₀
Ex 2. f(x) = x³
(x₀ + h)³ = x₀³ + 3x₀²·h + (3x₀h + h²)·h
f'(x₀) = 3x₀²
Yes!
Very good 🎉
To maintain the axiom of RS = LS it should be reinforced that d/dx y or f(x) is literally y’ and f’(x) respectively.
d/dx F(x) = f(x) = int{ f’(x) dx
F(x) = int{ f(x) dx = int{ f’(x) dx^2
d^2/dx^2 F(x) = d/dx f(x) = f’(x)
To further that notion, as above, the integral sign and dx are just brackets and uSub is a change of base for the brackets.
When d^n y/dx^2 or partial integral shows up, with any axis or uSub variation for y and x, it is simply a variable placeholder for the respective solution in that axis.
In Algebraic Calculus, there is yet another way of looking at the derivative of ordinary curves where y = x^n. In this perspective the primary object is the invariant, not the the limit.
The other perspective is, if you zoom out far enough does a straight line become curved? This is important when considering the context of a function. An example that I'm sure you're familiar with is the length of the shoreline around England. As you take smaller and smaller "straight line approximations" and add them up, the length approaches infinity. But if you choose that "infinite" length to be a reference length it is one unit long. Not straight? Sure, But it exists within a finite embedding space, Earth. Which also exists with a finite embedding space, our solar system .. etc.. etc.... It's not a context fee infinite.
I.e., there is no way to define a "whole" line except arbitrarily. And because of this you can use dx or 1 interchangeably in the context of calculus. The concept of "small" depends on a concept of "large" . I.e., relative.
You started good, then got a little delusional and made a conclusion about something that it is obvious by itself. Yeah, "small" is relative. You need to invoke some fractal infinite length to figure it out that???
Sir you say that if we zoom curve it become straight line then why at the next point of curve differentiation is different because it should be constant for the straight line?
I think that you are confusing fictious (infinitesimal) and actual segments of the curve. The teacher should not have assigned real value to the fictious straight line segment of the curve.
In the zoomed in area the derivative would be the same (to a certain level of precision). If in the zoomed in region the curve is "flat" and just a straight line, then the derivative is a constant. So the derivative is a constant across that zoomed in region.
@@alizuto If derivative is constant in the zoomed region and that zoomed region lie on curve then why derivative of curve is changing at every point in this region?
This is the problem with this explanation and the benefit of the formal approach. In the formal approach, the line is never straight, no matter how much you zoom in. But it "looks straight" upto a precision. Basically you can ask for a particular level of precision, and we can zoom in until it is "straight" upto that level of precision. If the curve is "smooth enough" the value of the derivative near the point will indeed be close to the original point, but again upto some precision. You can make these things very very precise, and a lot of things will depend on how smooth the function is. There are classes of functions like C1, C2, ..., Cinfinity, Comega. Which stand for continuous, continuously differentiable, continuously second differentiable, ..., infinitely differentiable, real analytic. Read more here : en.m.wikipedia.org/wiki/Smoothness.
Basically, you can't have both. You can be either very precise and formal, and you will get exact results, theorems, formulas and solution values. Or you can follow this professor's approach. He is just giving an intuition for things, which is also very very important.
If we consider a circle, a tangent line only touches the circle at one point. And the slope at every point is different. So zooming in won't produce a literal line, if measured with infinite precision. But
if you specify first a tolerance, say even the size of a pixel, then we can (in theory) zoom in enough so the curve will match the tangent line within that tolerance.
Thanks, Sir. ΞασΞ
Does this stuff still follow in higher dimensions?
Here's a video that attempts to partially answer that question: czcams.com/video/s6hvx7VPu3E/video.html
The most important idea is you don't need limits. dx = [[0 1] [0 0]], a nilpotent matrix. Calculate f(x + dx). x is [[x 0] [0 x]] of course
From where did you learn that?
Is that really intuitive? I like limits. They make sense. They are precise, and give exact answers. They are also somewhat intuitive (upto a precision, 😁)
@@user-dx5oo7jm4o Fermat? Nilpotents ... oh, Smooth Infinitesimal Analysis. SDG also
@@akashpremrajan9285 Summing an infinite series upsets some people, because you can't do that, really.
It is nice to know that there is a perfectly finite way of constructing the same answer.
I think infinity has helped math a lot in its history. All mathematicians of history have used infinity one way or another for the most part. As long as we do it carefully, what is the problem with limits of infinite sequences and sums of infinite series, I don’t get it?!?