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Mike, the Mathematician
Приєднався 22 січ 2023
Welcome to Mike the Mathematician's channel! Mike Dabkowski is a professor of Mathematics at the University of Michigan-Dearborn (he went to school there too!). Mike loves mathematics and loves sharing it with others. Mike did some graduate studies at Michigan State and earned a Ph.D. at the University of Wisconsin (On Wisconsin!).
This academic year, Mike is a visiting scholar at the University of Michigan- Ann Arbor (Go Blue!). (lsa.umich.edu/math/people/visiting/mgdabkow.html)
This academic year, Mike is a visiting scholar at the University of Michigan- Ann Arbor (Go Blue!). (lsa.umich.edu/math/people/visiting/mgdabkow.html)
The Mercator Projection
We consider the standard parameterization of the sphere. With a carefully chosen change of coordinates, we can reparameterize the sphere in a way that maps the parameter space to a strip in a conformal manner. By conformally mapping the sphere in this manner, we arrive at the Mercator projection which preserves angles, but distorts distances.
#mikethemathematician, #profdabkowski, #mikedabkowski, #differentialgeometry, #mercator
#mikethemathematician, #profdabkowski, #mikedabkowski, #differentialgeometry, #mercator
Переглядів: 75
Відео
The Psuedosphere
Переглядів 712 години тому
We consider a surface of revolution and its Gaussian curvature. Since the first fundamental form is of a particular type, the formula for the Gaussian curvature is the ratio of the second derivative of the rotated curve (in arclength parameter) relative to the function itself. We solve the differential equation for constant Gaussian curvature in the case of zero, one and negative one Gaussian c...
The Mobius Band is Not Orientable
Переглядів 894 години тому
We parameterize the Mobius Band. This is a surface which is formed by taking a rectangle and gluing the opposite ends together after a half twist. The resulting surface is not orientable. We find the normal vector to the surface along the medial circle, and show that it reverses orientation as you make a revolution around the unit circle. #mikethemathematician, #mikedabkowski, #profdabkowski, #...
Twisted Cubic Approximations of Curves
Переглядів 1917 годин тому
We consider a curve in arc-length parameter. We use the Frenet-Serret equations to find all of the derivatives of the curve in terms of curvature and torsion. After Taylor expanding the curve near a particular point, we use the Frenet-Serret equations to approximate the curve up to third order. This approximation shows that a curve is locally approximately a twisted cubic! #mikethemathematician...
Monge Approximation of a Surface
Переглядів 1569 годин тому
We consider a parameterized surface and compute the two principal curvatures. We then Taylor expand the parameterization around a point where the principal directions span the xy-plane. This can be accomplished by an orthogonal transformation. We dot the resulting expansion with the normal vector (which lies in the direction of the z-axis), to find an function that approximates the surface if w...
Surfaces that Only Have Umbilical Points
Переглядів 15212 годин тому
We show that if a surface is connected and only has umbilical points (points at which the principal curvatures are equal), then the surface must either be a plane or a sphere. In the case of a plane we know that both principal curvatures are zero and in the case of a sphere the principal curvatures are the reciprocal of the radius at every point on the sphere. #mikethemathematician, #profdabkow...
The Gaussian And Mean Curvatures of the Graph of a Function
Переглядів 16614 годин тому
We consider the graph of a function in three dimensional Euclidean space. With the standard parameterization we find the equations for the mean and Gaussian curvature of the surface. We find a necessary condition for the surface to have zero mean curvature, which is the minimal surface equation. #mikethemathematician, #mikedabkowski, #profdabkowski, #differentialgeometry
Gaussian Curvature as a Limit of Surface Areas
Переглядів 15116 годин тому
We consider the ratio of two surface area whose parameter space tends to zero (or the surface tends to a single point). If the numerator corresponds to the surface area generated by the unit normal vector parameterization of the surface and the denominator corresponds to the surface area generated by the given parameterization, then the limit of the ratios tends to the Gaussian curvature up to ...
The Gauss Equations for the Christoffel Symbols on a Surface
Переглядів 26319 годин тому
We consider a parameterization of a surface, then the second derivatives of the parameterization can be expressed in terms of the first derivatives and the unit normal vector. The coefficients of the normal vector in the linear expansion are the coefficients of the second fundamental form. The coefficients in terms of the first derivatives can be expressed by the Christoffel symbols. We prove t...
The Third Fundamental Form of a Surface
Переглядів 50221 годину тому
We define the third fundamental form of a surface. This is similar to the first fundamental form with the surface parameterization replaced with the unit normal vector. We relate the third fundamental form to the second and first fundamental forms. We use the Cayley-Hamilton Theorem to find an algebraic relationship between the first, second and third fundamental forms and the Gaussian and Mean...
Rodrigues Theorem on Lines of Curvature
Переглядів 188День тому
We prove the Rodrigues Theorem for lines of curvature on a surface. We say that the curve is a line of curvature on a surface if its tangent vector is always a principal direction. We prove that the tangent vector can is a principal direction if and only if it is proportional to the derivative of the normal vector to the surface along the curve. #mikethemathematician, #mikedabkowski, #profdabko...
The Gaussian Curvature of Ruled Surfaces is Non-Positive
Переглядів 160День тому
We have previously defined the Gaussian curvature of a surface (and saw that a cylinder had zero Gaussian curvature). We will show that all ruled surfaces have non-positive Gaussian curvature. #mikethemathematician, #mikedabkowski, #profdabkowski, #differentialgeometry
The Gaussian and Mean Curvatures of a Surface
Переглядів 209День тому
We consider the connections between the first and second fundamental forms of a surface, and how these forms can help determine notions of curvature. The Weingarten matrix has two real eigenvalues. The product of these eigenvalues (or determinant of the Weingarten matrix) will be the Gaussian curvature, and the arithmetic mean of the eigenvalues (half of the trace of the Weingarten matrix) will...
The Weingarten Matrix
Переглядів 223День тому
The Weingarten matrix is the matrix representation of the Weingarten map (or shape operator). It stores the coefficients of the partial derivatives of the unit normal vector with respect to the standard basis of the tangent plane (r_u and r_v). The eigenvalues of this matrix are identical to the constants for which the first and second fundamental form are proportional, which are the principal ...
Principal Curvatures of Planes, Spheres and Cylinders
Переглядів 14814 днів тому
We have previously discussed the ideas of principal curvatures. These are the constants that exist when the first and second fundamental form are proportional. We compute the principal curvatures of a plane, cylinder and a sphere and see several examples of what the principal curvatures tell us about the surface. #mikethemathematician, #profdabkowski, #mikedabkowski, #differentialgeometry
Euler's Theorem For Principal Curvatures
Переглядів 30314 днів тому
Euler's Theorem For Principal Curvatures
Geodesic Torsion and Frenet-Serret on a Surface
Переглядів 49014 днів тому
Geodesic Torsion and Frenet-Serret on a Surface
The Cornu Spiral and Planar Curves with Given Curvature
Переглядів 14721 день тому
The Cornu Spiral and Planar Curves with Given Curvature
Schur's Theorem on Riemannian Curvature
Переглядів 71621 день тому
Schur's Theorem on Riemannian Curvature
Conformally Flat Two Dimensional Spaces
Переглядів 16421 день тому
Conformally Flat Two Dimensional Spaces
The Riemann-Christoffel Tensor of a Two Dimensional Manifold
Переглядів 28821 день тому
The Riemann-Christoffel Tensor of a Two Dimensional Manifold
Independent Components of the Riemann-Christoffel Tensor
Переглядів 31521 день тому
Independent Components of the Riemann-Christoffel Tensor
First and Second Fundamental Forms of Surfaces of Revolution Source
Переглядів 30028 днів тому
First and Second Fundamental Forms of Surfaces of Revolution Source
Angles Between Curves on Surfaces Source
Переглядів 21828 днів тому
Angles Between Curves on Surfaces Source
The Transformation Law for the First Fundamental Form and Normal Vector
Переглядів 21028 днів тому
The Transformation Law for the First Fundamental Form and Normal Vector
5:23 Why is that Riemann's sum? Aren't t*_k and t**_k different variables?
First
poggers vid
How do you write like that? Is it some editing?
Professor, this question is unrelated to the lecture, but may I ask something? I'm dying to know now. When deriving the formula for the area of a surface of revolution, we approximate x_k+1 = x_k = c_k, and then we get the Riemann sum. Could you explain why we can approximate the value inside the limit of sigma?
Thanks King
My pleasure!
i hate that d/ds part, god. i fail everytime when this kind of question appear in my exam.
simple solution and excellent explanation
Thank you!
I always watch your videos to get a concise summary on concepts I learnt in class. Plus I find the sound of the marker scribbling on the window satisfying.
Thank you so much!
Thanks so much. I was struggling with order statistics and your videos really helped
Thanks for the clear explanation and nice quality of video
Thanks for the breakdown! I have a quick question: I have a SafePal wallet with USDT, and I have the seed phrase. (alarm fetch churn bridge exercise tape speak race clerk couch crater letter). Could you explain how to move them to Binance?
Hi, is there a mistake at 3:57 in the summation of omega n is XT = i not XT = j?
fantastic sir
thank you!
Awesome proof! Thank you so much!
I didn’t understand and was going insane until I found your video. You cleared up all my doubts! Thank you so much!
Gauss would be proud! 😊
Insanely Neat!!!!!
I am glad that you enjoyed it!
Thanks for the video. At 3:55, the "=" should have been >=. This is because the Omega_k sets don't form a partition of Omega, but a subset of Omega. There is still a set where even |S1| < lambda.
Happy new year sir
Happy New Year! I have some god ideas for videos in 2025! We have a long way to go in differential geometry still! I am also going to try to make more shorts and do some live streams this year!
Happy new year, Mike! 🎉😊
Happy New Year! I have a lot of great videos lined up! I am also happy to take any requests!
Although this may not be directly related to the video, I am curious about mathematical proofs. would be quite interesting!
I will absolutely make a playlist something like "Intro to Proofs"
Thanks
@@aryabartarout5697 my pleasure
Having K(x) = k1 x k2 You assume the convexity factor (LN - M^2) non-positive provided the radius (1/(EG - F^2)) is positive Meaning the sub-ruled surface described by the directrix c(A,B) d(A,B) yields a given curvature K(A,B) Having the initial ruled-surface described by sum(Kn) Why would sum(Kn) necessarily be non-positive ? Simple cone Sphere Quarter-sphere Escalated cumulative quarter-sphere Homothetic geometry based surfaces Indeed for random ruled surfaces sum(Kn) is non-positive Best Regards
Today I completed this course. This course is very comprehensive. Thanks sir.
Thanks
I'm confused with the example, where you got a_(k+1)/a_k.
Mike rules! (This one was almost too easy.,) 😅
Thanks! Sometimes an easy example helps solidify a definition!
thats cool, but is the graphical way is the only way? and is it formal?
There is another way. You can use the xyz-simple method. I’d recommend watching professor Leonard’s videos on the topic to learn it. It will allow you to do these without doing a 3D graph. The idea is to represent the triple integral as the double integral of a single integral and vice versa and go from there. So if I know it’s z-simple for example, I just find two functions of z to bound it by. I then set z to zero and graph a region on the xy plane to figure out the two remaining bounds. I never needed to do the whole 3D graph this way.
This distribution is heavily used to model stock prices in option pricing models. Thanks professor for explaining this.
My pleasure!
Hey! Your videos are great but I'd like to suggest something. Can you please arrange or make playlists in such a manner that any video we want to watch is preceded by the videos that cover the pre-requisites required to understand that video? It'd be really great to have.
Yes, it is pretty close to the correct order! I will work on this!
Hard to believe there is so much math! 😊
Ohh my, it is even more vast than I have put out there!
Nice we can screenshot then print 1 idea on a single paper sheet
Absolutely! Good idea!
This stuff is amazing. 😊
Thanks so much! More on the way!
dg videos are rare on youtube with this sort of clearity
Thank you so much! More coming! DG is beautiful because there are so many PDEs embedded in these questions. I have worked on some problems in Kahler geometry which were PDE rich!
Nice session! I learned a lot just by seeing the process and your energy and annunciation are very good.
Thanks for watching!
Great video! Thank you. Glad I have discovered your channel.
Thank you so much!
Thanks for the class
@@manuelangeldomingueztoribi6516 my pleasure! I will add to this playlist in the future! I produce videos based on the classes that I am currently teaching, so we are on a differential geometry journey right now!
Great videos!
Thank you so much!
These videos are excellent 😊
Thank you so much!
Are these videos in the intended order?
Thank you for the lecture!
Professor, may I ask you a favor? I noticed that you have a video on the proof of the Cauchy product. Could you please make a video on the proof of the Cauchy product of power series? I’m a Korean student studying Calculus, and I watch your videos frequently!
Sure thing! Thankfully it already exists! ua-cam.com/video/yaISMOyFpdE/v-deo.html
@@mikethemathematician Oh, I’m sorry, Professor. This proof is equivalent to the proof in the video you've already uploaded. Thank you for the lecture!
Can u tell me how to do for oi spurt???
Nice!
fantastic, btw suggest the book for the notes , i have andrew pressley's dg book
Prof Mike, I am thinking it shd be an annuity immediate rather than the due.
@@collinsanokye6857 you are right! I misspoke!
Amazing video! I assume one can derive the equivalent Frenet-Serret form using the unit-speed parameterisation? Could we also do the same for the Bishop/ Parallel Transport frame? Coming at this as an Engineering PhD student so don't have the formal differential geometry background! Looking forward to the next videos!
Bro thanks so much this video has helped so much
and for alpha less than 0?