what is the area of this Neumann Oval ?
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- Опубликовано: 19 окт 2024
- what is the area of this Neumann Oval ? We calculate the area of the Neumann oval, which is the image of the unit disk in R2 under the mapping z + z^2/2. For this we need a new way of changing variables which involves the derivative squared. This method uses Jacobians, double integrals, polar coordinates. At the end we explain why there is a |f’z|^2 term, but it basically follows from the Cauchy Riemann equations. This is a must see for complex analysis and multivariable calculus lovers!
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I am always fascinated when you as a expert start to rev up your matematik engine.
I understood maybe 3% of this whole section, purely because my matematik skill isn't on this level. Though I still find it enjoyable to watch.
Thanks so much!!
Can you tell me the name of a technique for integration which goes:
1)notice a part of the integrand as an antiderivative of a function.
2) that function turns out to be a geometric series
3) integration and summation order are changed( idk the rules for that)
Then the integral becomes easy
Never heard of it
Hey Peyam! Love your videos. I am just nearing the end of multivariable calculus and while watching this video I recognised many concepts with equivalents in MC. So my question is: Is multivariable calculus and complex analysis just different ways to describe the same ideas? are their applications/results similar?
Thank you!!
Fundamentally they are different but there are lots of topological connections, mainly because of the (x,y) to x + iy similarity
I am not a doctor in mathematics, I don't know too much, but complex space is like multivariable calculus but with multiplication and division defined as (x+iy)*(a+ib)=xa-yb+i(bx+ya)
@@drpeyam Okay, Thank you!! Love your content man keep doing what you're doing.
Thanks so much!!
So the area (of a circle) that is ~π^2 now has an area under the transformation ~π. How did we lose a factor of π? That is pretty amazing! I think the fundamental reason for this is worth exploring/detailing/creating a video for.
The area of a circle is ~π, the squaring is of the radius. No factor was harmed during this transformation.
Do the area of a Cassini oval please! :p Nice video btw!
@3:28: “... times a polar thing which is ARRR dr dθ”
- But where does the “ARRR” come from in polar coordinates?
- Well, polar coordinates are circular, so it’s because of the 𝝅-rate!
Nice 😂😂
but doesn’t this only work if the region is a bijection? it is not immediately obvious that the Neumann Oval is a non-singular transformation of the unit disk…
It should be a c1 diffeomorphism right?
I'm just watching this because I find it very interesting how smart some people really are haha
if anyone is interested, the equation for a neumann oval in cartesian coordinates is (x²+y²)²=a²(x²+y²)+4b²x²
Really cool 😎
what do you mean, this is the transformation of the unit disk under the mapping z+z^2/2. the unit disk is r*e^(it) for 0
It’s not a cardioid
You need to set r=1 rather than varying it from 0 to 1, that may help.
@@knivesoutcatchdamouse2137 i did and still got the same shape
Wait. You're from Berkeley? Or UCI? I know a scientist at UCI.
Both lol
@@drpeyam Rad
Beautiful
In the explanation of the Jacobian, there is a missing step at the end.
From the Cauchy-Rieman equations, we have that:
|det(Df)|=(uₓ)²+(uᵧ)²=(uₓ)²+(vₓ)²
And also it is known that:
f'(z)=df/dz=∂f/∂x=uₓ+ivₓ
So, f'(z)=uₓ+ivₓ and we can see that:
f'(z)*f̄'(z)=|f'(z)|=(uₓ)²+(vₓ)²=|det(Df)|
Missing step?
@@drpeyam Alsp why do you say that is usually all gibberish the Jacobian? It's not though at least mostly though? Can you clarify?
Hello there, I am keen to learn these sorts of problems, what do I need to learn to get there?
haha awesome! thanks a lot
it seems that the shape u presented does not math the mapping. it is a cardioid
I have a question. Is there any general equation for oval or egg shape. I tried to search a lot on google but they show general equation of ellipse? Even google confused between oval and ellipse 😅
Not sure about egg shape but for an oval that is symmetrical along the x and y axis u use the equation of ellipse
@@tanyuhur7055 the symmetric oval is an ellipse. It's like, as a circle is a special type of ellipse when major and minor are equal, a symmetric oval is same as an ellipse whose axis are symmetrical. But what about not a symmetrical oval. That would be like an egg shape, like small or squeezed on one side of ellipse.
@@mathevengers1131 I'll post the link. I hope it won't be removed
@@IoT_ most probably it's removed
@@mathevengers1131 RUclips deletes all of my comments
Hello again.....do not forget the Binomial theorem.....and sin and cos substitution for y and x in ( x + y)^n expansion ..........then because the function's is assumed a continuous differentiable for reducing and increasing powers expressed in the derivative and antiderivative representations......... captured to ...... n(x + y )^n-1 + 0 and (x + y)^n + z calculus (functional) transformation. The action of continuous differentiation terminates when the nth derivative of ( x + y)^n = 0 and (n-1)^th derivative of ( x + y)^n = const = Pn , given Pn is the n^th successor prime number P......and....?
?.... gosh.... almost forgot.....P is A Gaussian Prime ...and...GP = P mod(4) = 3 ....defines a set of grouped points intersecting the Cartesian plane at points calculated using Euler's amazing formula ..... one of many such number theoretic formulae....may one say you bring energy and humor to communicative mathematics