if anyone is interested, the equation for a neumann oval in cartesian coordinates is (x²+y²)²=a²(x²+y²)+4b²x²

@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!

@drpeyam

2 жыл бұрын

Nice 😂😂

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.

@dlevi67

2 жыл бұрын

The area of a circle is ~π, the squaring is of the radius. No factor was harmed during this transformation.

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.

@drpeyam

2 жыл бұрын

Thanks so much!!

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)|

@drpeyam

2 жыл бұрын

Missing step?

@leif1075

2 жыл бұрын

@@drpeyam Alsp why do you say that is usually all gibberish the Jacobian? It's not though at least mostly though? Can you clarify?

Beautiful

Really cool 😎

I'm just watching this because I find it very interesting how smart some people really are haha

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…

@Test-zd4mp

2 жыл бұрын

It should be a c1 diffeomorphism right?

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

@drpeyam

2 жыл бұрын

Never heard of it

Do the area of a Cassini oval please! :p Nice video btw!

haha awesome! thanks a lot

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!!

@drpeyam

2 жыл бұрын

Fundamentally they are different but there are lots of topological connections, mainly because of the (x,y) to x + iy similarity

@Noam_.Menashe

2 жыл бұрын

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)

@proxagonal5954

2 жыл бұрын

@@drpeyam Okay, Thank you!! Love your content man keep doing what you're doing.

@drpeyam

2 жыл бұрын

Thanks so much!!

Hello there, I am keen to learn these sorts of problems, what do I need to learn to get there?

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

@drpeyam

2 жыл бұрын

It’s not a cardioid

@knivesoutcatchdamouse2137

2 жыл бұрын

You need to set r=1 rather than varying it from 0 to 1, that may help.

@aneeshsrinivas9088

2 жыл бұрын

@@knivesoutcatchdamouse2137 i did and still got the same shape

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 😅

@tanyuhur7055

2 жыл бұрын

Not sure about egg shape but for an oval that is symmetrical along the x and y axis u use the equation of ellipse

@mathevengers1131

2 жыл бұрын

@@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.

@IoT_

2 жыл бұрын

@@mathevengers1131 I'll post the link. I hope it won't be removed

@mathevengers1131

2 жыл бұрын

@@IoT_ most probably it's removed

@IoT_

2 жыл бұрын

@@mathevengers1131 KZread deletes all of my comments

Wait. You're from Berkeley? Or UCI? I know a scientist at UCI.

@drpeyam

2 жыл бұрын

Both lol

@silvermica

2 жыл бұрын

@@drpeyam Rad

it seems that the shape u presented does not math the mapping. it is a cardioid

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....?

@abdonecbishop

2 жыл бұрын

?.... 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