Mu Prime Math
Mu Prime Math
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Does the Gaussian integral trick work for other functions?
Proof for Riemann-integrable functions:
Robert J. MacG. Dawson. “On a “Singular” Integration Technique of Poisson”. American Mathematical Monthly, 2005. cs.smu.ca/~dawson/Poisson.pdf
We can compute the integral of e^(-x^2) using a very cool trick that lets us switch to polar coordinates and use the Jacobian for a u-sub. But does this technique work for any other function? In this video we turn that problem into a differential equation and find all of the solutions!
Mariano Suarez-Alvarez's proof: math.stackexchange.com/a/4517530/713547
0:00 The Functional Equation
4:23 The Differential Equation
7:14 Solving
12:34 Additional Notes
Calculus Problems playlist: ua-cam.com/play/PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao.html
Subscribe to see more new math videos!
Music: C418 - Pr Department
Переглядів: 3 510

Відео

Proof: Uniqueness of the Tensor Product
Переглядів 2,1 тис.Рік тому
Universal property introduction: ua-cam.com/video/vZzZhdLC_YQ/v-deo.html This video proves the uniqueness of the tensor product of vector spaces (or modules over a commutative ring). This uses the universal property of the tensor product to prove the existence of an isomorphism (linear bijection) between any two "tensor products". Tensor Products playlist: ua-cam.com/play/PLug5ZIRrShJHCtzgzZyRq...
Tensor Product Basis With the Universal Property
Переглядів 3 тис.Рік тому
Tensor product universal property explanation: ua-cam.com/video/vZzZhdLC_YQ/v-deo.html Generating set proof: ua-cam.com/video/KnSZBjnd_74/v-deo.html timestamp 23:57 If we have a basis for each of two vector spaces (or modules over a commutative ring) V and W, then we can use that to form a basis for the tensor product V⊗W. The proof uses the universal property of the tensor product, which conne...
Complete Derivation: Universal Property of the Tensor Product
Переглядів 7 тис.Рік тому
Previous tensor product video: ua-cam.com/video/KnSZBjnd_74/v-deo.html The universal property of the tensor product is one of the most important tools for handling tensor products. It gives us a way to define functions on the tensor product using bilinear maps. However, the statement of the universal property can be confusing if it is presented without background. This video is an explanation o...
How to Define Homomorphisms on Quotient Groups
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One way to define a function on a quotient group is to define it in terms of the coset representatives. However, this approach runs into problems of well-definedness. This video explains how we can address that problem. 0:00 Problem Introduction 5:42 Solution 10:34 Non-example: Integers mod 5 11:13 Example: Alternating Group Group Theory playlist: ua-cam.com/play/PLug5ZIRrShJHDvvls4OtoBHi6cNnTZ...
Why can we change lim n→∞ to lim x→∞?
Переглядів 2,2 тис.Рік тому
When computing the limit of a sequence, it's often useful to consider a limit of real numbers so that we can do things like take derivatives. But why is this allowed in the first place? Why can we change a limit of integers to a limit of real numbers? This video gives an explanation. Calculus Problems playlist: ua-cam.com/play/PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao.html Subscribe to see more new ma...
A Natural Proof of the First Isomorphism Theorem (Group Theory)
Переглядів 6 тис.Рік тому
The first isomorphism theorem is one of the most important theorems in group theory, but the standard proof may seem artificial, like every step of the proof is set up knowing that we're trying to create an isomorphism. In this video, we show an alternate proof with no such tricks using the preimage map of a group homomorphism. Group theory playlist: ua-cam.com/play/PLug5ZIRrShJHDvvls4OtoBHi6cN...
Integral Formula for Natural Log (without knowing the derivative)
Переглядів 3,1 тис.Рік тому
This video proves that the natural log equals the integral from 1 to x of 1/t dt under the assumption that ln(x) is the inverse function to the exponential e^x. We can do this without already knowing the derivative of the natural log! More details on why the integral is the inverse of e^x: We proved in the video that any right inverse to e^x must equal that integral. However, we didn't prove th...
Supremum, Infimum: Definition and Explanation
Переглядів 3,7 тис.2 роки тому
Not all sets of real numbers have a maximum and a minimum. In this video, we introduce the supremum and infimum as a generalization of max and min. Sup and inf appear everywhere in analysis. Subscribe to see more new math videos! Music: OcularNebula - The Lopez
Topology Definitions: Connected and Path-Connected
Переглядів 3,3 тис.2 роки тому
Connectedness is a key idea in topology and metric spaces that describes whether a topological space can be separated into two components. This video explains the open set definition of connectedness and describes the topologist's sine curve as a non-example that motivates the definition of path-connectedness. Timestamps: 0:00 connectedness 9:59 path-connectedness Subscribe to see more new math...
Does 1+2+3+...=-1/12?
Переглядів 3,5 тис.2 роки тому
The answer to the question in the title is that it depends on how you define the infinite sum function. Until the infinite sum function is specified, the question is not well-defined; it's the equivalent of asking "what is f(5)?" without defining the function f. The sum of all natural numbers, and the sums of divergent series more generally, are often debated because it seems incoherent to say ...
Proof: Orthogonal Matrices Satisfy A^TA=I
Переглядів 12 тис.2 роки тому
One way to characterize orthogonal matrices is to say that a matrix orthogonal if and only if A transpose times A is the identity matrix. In this video, we prove this result using basic matrix calculations and the definition of orthonormal vectors. Learning Linear Algebra playlist: ua-cam.com/play/PLug5ZIRrShJHNCfEiX6l5CKbljWayGEcs.html Subscribe to see more new math videos! Music: C418 - Pr De...
My favorite proof of the n choose k formula!
Переглядів 37 тис.2 роки тому
The binomial coefficient shows up in a lot of places, so the formula for n choose k is very important. In this video we give a cool combinatorial explanation of that formula! Challenge Problems playlist: ua-cam.com/play/PLug5ZIRrShJGkzGsXMYQt8bi5ImYtiEMM.html Subscribe to see more new math videos! Music: OcularNebula - The Lopez
Epsilon-Delta proofs: Can't we make the limit equal anything?
Переглядів 4,7 тис.2 роки тому
Epsilon-Delta definition explanation: ua-cam.com/video/JbbRaiXI6yw/v-deo.html Proofs that use the delta-epsilon definition of the limit can be confusing because it seems like we can prove that the limit is anything we want if we pick the right value of delta. In this video we prove that limits are unique and go over some examples of disproving limits! Calculus Problems playlist: ua-cam.com/play...
Why can't we group the terms in 1-1+1-1+... ?
Переглядів 2,6 тис.2 роки тому
We can group the terms in the divergent infinite sum 1-1 1-1 ... to get a value of zero or a value of one, even though the series doesn't converge. This video gives an explanation of why grouping terms in Grandi's series doesn't give the correct answer. Calculus Problems playlist: ua-cam.com/play/PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao.html 0:00 Why the sum diverges 7:33 Why grouping fails 12:57 Whe...
When can we switch the limit and function?
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When can we switch the limit and function?
Topology Definitions: Closure, Boundary, Interior
Переглядів 20 тис.2 роки тому
Topology Definitions: Closure, Boundary, Interior
Why can we do this to find inverse functions?
Переглядів 2,1 тис.2 роки тому
Why can we do this to find inverse functions?
Bijective Functions Have a Two-Sided Inverse
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Bijective Functions Have a Two-Sided Inverse
Proof: Two-Sided Inverse Functions are Unique
Переглядів 2,5 тис.2 роки тому
Proof: Two-Sided Inverse Functions are Unique
Sum of natural numbers equals n+1 choose 2!
Переглядів 4,9 тис.2 роки тому
Sum of natural numbers equals n 1 choose 2!
Representation Theory: Irreducible Characters Are an Orthonormal Basis
Переглядів 1,3 тис.2 роки тому
Representation Theory: Irreducible Characters Are an Orthonormal Basis
Matrix Invertibility With F[x]-Modules
Переглядів 1,1 тис.2 роки тому
Matrix Invertibility With F[x]-Modules
Difference Between Normalizer, Centralizer, and Stabilizer
Переглядів 7 тис.2 роки тому
Difference Between Normalizer, Centralizer, and Stabilizer
A Concrete Introduction to Tensor Products
Переглядів 45 тис.2 роки тому
A Concrete Introduction to Tensor Products
Euler's Identity With a Limit!
Переглядів 6 тис.2 роки тому
Euler's Identity With a Limit!
Why Are Odd Numbers Not Even?
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Why Are Odd Numbers Not Even?
The Cayley-Hamilton Theorem is Easy with F[x]-Modules
Переглядів 2,2 тис.3 роки тому
The Cayley-Hamilton Theorem is Easy with F[x]-Modules
How the Matrix Characteristic Polynomial is Connected to F[x]-Modules
Переглядів 1,4 тис.3 роки тому
How the Matrix Characteristic Polynomial is Connected to F[x]-Modules
F[x]-Module Derivation of Rational and Jordan Canonical Forms
Переглядів 2,9 тис.3 роки тому
F[x]-Module Derivation of Rational and Jordan Canonical Forms

КОМЕНТАРІ

  • @danw6406
    @danw6406 11 годин тому

    Legend

  • @sportivescientist
    @sportivescientist 12 годин тому

    thankyou my lord ♥

  • @marcschmidtpujol550
    @marcschmidtpujol550 День тому

    Great explanation!

  • @user-ou1yw4cg6y
    @user-ou1yw4cg6y 4 дні тому

    Thank you for the delicate demonstration

  • @jasonzechengpan4491
    @jasonzechengpan4491 5 днів тому

    Thanks! Extremely clear and concise explanation.

  • @ryanblais6208
    @ryanblais6208 5 днів тому

    What is the music that is playing at the end of the video? It's interesting.

  • @NonameBozo88
    @NonameBozo88 7 днів тому

    Please don't use your finger to wipe the board, oil from the skin ruins the pen and the board

  • @user-yh8vn3vx4z
    @user-yh8vn3vx4z 8 днів тому

    Great explanation! Thank you

  • @FelipeHenrique-yq3bu
    @FelipeHenrique-yq3bu 9 днів тому

    I got the eigenvector associated to the eigenvalue -2 equal to (1, -4/3), is this okay?

    • @MuPrimeMath
      @MuPrimeMath 9 днів тому

      Yes. Any scalar multiple of an eigenvector is also an eigenvector with the same eigenvalue.

  • @GokuTheSuperSaiyan1
    @GokuTheSuperSaiyan1 10 днів тому

    I don't follow this last equality that : a x (b sin(theta) ) = a x b , where a and b are vectors. This is sloppy. Because theta is between 0 and 180 degrees, sin(theta) is between 0 and 1. This means that the vector "b sin(theta) " is in the exact same direction as b but is shorter. This means that the area of the parallelogram formed with sides "a" and "b sin(theta)" is smaller than the area of a parallelogram with sides "a" and "b".

    • @MuPrimeMath
      @MuPrimeMath 10 днів тому

      The vector over b in b sin θ at 5:40 is a typo. The intent was to say that the length of that orthogonal vector equals the length of b times sin θ. The cross product a × b equals the cross product of a with that orthogonal component.

  • @bintangagams.7941
    @bintangagams.7941 10 днів тому

    Hey, thanks for the thorough explanations, it actually makes sense. One question, how do you get the exact error of taylor series in the integral form? could you perhaps send me the video link or any paper for the derivation of the equation? thanks!

  • @user-ke5pi4kn9d
    @user-ke5pi4kn9d 12 днів тому

    고정된 상수값을 원으로 생각하니 이해가바로되네요!

  • @ciiil8802
    @ciiil8802 12 днів тому

    Differentiating Integration of Integral

  • @weo9473
    @weo9473 12 днів тому

    Feels like he's a right handed and trying to be left handed.

  • @sahelexol7541
    @sahelexol7541 13 днів тому

    English is not my first language but bro was amazing

  • @zlodevil426
    @zlodevil426 14 днів тому

    I think we can add negative terms like e^-x, e^(-x/2), e^(-x/3), … to the sum to make it converge on the positive side as far as it does on the negative side, even with a finite number of terms

  • @RomanWhite-xl5pz
    @RomanWhite-xl5pz 15 днів тому

    u just made this make sence thank you thank you

  • @prakharpandey8968
    @prakharpandey8968 15 днів тому

    Sir please explain the base system and conversion btw different bases

  • @umj369gmailcom
    @umj369gmailcom 16 днів тому

    Amazing. Masha allah. Can you explain how your thought process work or how you approach problems. So we can follow it and get some good GPA

  • @strippins
    @strippins 18 днів тому

    I spent four years doing a physics degree starting in 2003. UA-cam existed since 2005 and this sort of content was certainly not available until after I finished. I always found the unengaging lectures difficult to follow, printed lecture notes missing insight and text books impossibly heavy. I wonder how much more I could have got out of that education had content like this been around to enhance conceptual understanding .

  • @prakharpandey8968
    @prakharpandey8968 18 днів тому

    I could not understand injection mod n

  • @prakharpandey8968
    @prakharpandey8968 18 днів тому

    9:23 r subi is not a divisor of a

  • @ClumpypooCP
    @ClumpypooCP 18 днів тому

    any more advanced algebra videos you can do ... please do! These videos are great

  • @prakharpandey8968
    @prakharpandey8968 18 днів тому

    thankU love from INDIA

  • @mnqobimsizi4328
    @mnqobimsizi4328 19 днів тому

    LEFT HANDED AND BEING A MATHEMATICIAN IS TRULY A GIFT

  • @jonbielli9380
    @jonbielli9380 21 день тому

    Thank you for posting this video! I've been searching for the geometric Interpretation of determinants for 3x3 matrices and this video nails it.

  • @Junyan-ft5vn
    @Junyan-ft5vn 22 дні тому

    Thank you!!!

  • @user-wl9dv2gx3p
    @user-wl9dv2gx3p 22 дні тому

    Love your videos. All are well done and clearly presented. Wish you are well; as I saw that the last video was a year ago.

  • @ebtisamhussien7389
    @ebtisamhussien7389 23 дні тому

    Thank you

  • @dinapapadopoulou224
    @dinapapadopoulou224 28 днів тому

    Great work ! Ty so mutch . Love from a future chemeng

  • @jeromemalenfant6622
    @jeromemalenfant6622 29 днів тому

    It's easy to show the square root of any prime number is irrational. Any number can be expressed as the product of prime numbers raised to some non-zero power. If, for some number n, all those exponents are even integers then n is a perfect prime. If one or more of the exponents is odd, n is not a perfect square. Then, when you take the square root of n you're left with at least one square root of a prime number in the product; i.e. an irrational number. So the square root of n is irrational.

  • @rajashriasangi8988
    @rajashriasangi8988 29 днів тому

    Nice:) Thank you so much!!

  • @everythingisalllies2141
    @everythingisalllies2141 Місяць тому

    Nonsense. Now redo the math but don't make the light have a 1:1 ratio on your graph. I like to draw light velocity practically along the x axis, because nothing can go faster. This gives me the full plotting area to enter real data. Not only half of it. The weird way you choose to have light at 45 degrees is the only reason you math works the way it does. You have (minkowski has) created a "special case". Only if light is drawn at 45 can you get the result you want to see. The whole theory collapses if you change that one thing.

  • @elreturner1227
    @elreturner1227 Місяць тому

    Are you writing this on your wall?

    • @MuPrimeMath
      @MuPrimeMath Місяць тому

      I'm using a stick-on whiteboard from writeyboards.com

    • @elreturner1227
      @elreturner1227 Місяць тому

      @@MuPrimeMath oh alr thanks

  • @theboombody
    @theboombody Місяць тому

    I will need to look this over a few times and digest it a bit, but it sure seems to be helpful for understanding the function and purpose of normal subgroups, which I darn sure have been stuck on for a long time. Thanks for the upload.

  • @diyanair214
    @diyanair214 Місяць тому

    Amazing

  • @user-hm2gb6pm6b
    @user-hm2gb6pm6b Місяць тому

    Not visible

  • @lucasgroves137
    @lucasgroves137 Місяць тому

    Lmao.

  • @rodioniskhakov905
    @rodioniskhakov905 Місяць тому

    3:48 I don't understand why do you say that x' = e^(rt) but not x' = e^(At). Why r = A? Or rather, why e*I = A?

  • @punikachi
    @punikachi Місяць тому

    Fantastic explanation thank you so much

  • @angelinajuly8637
    @angelinajuly8637 Місяць тому

    Excellent excellent excellent, I loved how you explained EVERYTHINGGGGGGGGGGGGGG!!!!!!!!!!! Thank you.

  • @CynthiaGondwe-de9rp
    @CynthiaGondwe-de9rp Місяць тому

    Thanks alot

  • @fahrenheit2101
    @fahrenheit2101 Місяць тому

    Nice. Setting aside the pretty poorly drawn parallelograms, this is a slick proof.

  • @atonumukherjee4398
    @atonumukherjee4398 Місяць тому

    too many proofs of v tensor 0 is 0 😂

  • @md.hamidulhaque5816
    @md.hamidulhaque5816 Місяць тому

    This channel should get more views. All of the videos are too intuitive and amazing. You explained the main things and confusions within few minutes. That is incredible.

  • @ignantxxxninja
    @ignantxxxninja Місяць тому

    For the most part, trig functions will diverge unless you multiply it with something that converges

  • @mimi30156
    @mimi30156 Місяць тому

    You're so amazing i love you

  • @surenazand
    @surenazand Місяць тому

    Thx to this proof.in short if p is divisible by q then n will be perfect square but if p is not divisible by q we have a fraction which can be simplified to other or not.in both case n will be a fraction not integer so contradiction and then wron assumption

  • @mosarof_cp
    @mosarof_cp Місяць тому

    It's absolutely Rong!