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Matthew Salomone
United States
Приєднався 19 жов 2009
596.C4.2 Derivative of a Complex Function: The Amplitwist
Geogebra tool for derivatives of complex functions:
geogebra.org/classic/p2txrrfu
0:00 Introduction
1:11 Derivative operates on tangent vectors ("directions") not points
4:50 "The derivative is a matrix"
7:22 Derivative of a complex function
9:08 Derivative is a ratio of tangent vectors
10:20 Visualizing the derivative of f(z) = z²
13:44 Multiplying both amplifies and twists
15:57 Summary of the "amplitwist" effect
18:29 Wrapup
geogebra.org/classic/p2txrrfu
0:00 Introduction
1:11 Derivative operates on tangent vectors ("directions") not points
4:50 "The derivative is a matrix"
7:22 Derivative of a complex function
9:08 Derivative is a ratio of tangent vectors
10:20 Visualizing the derivative of f(z) = z²
13:44 Multiplying both amplifies and twists
15:57 Summary of the "amplitwist" effect
18:29 Wrapup
Переглядів: 187
Відео
596C.C4.1 Rethinking the Meaning of Derivative: Real Case
Переглядів 282Місяць тому
Geogebra tool for derivative of real functions: www.geogebra.org/classic/my2ffwjy 0:00 Introduction 1:15 A Grad School Story 2:38 Sophistication 1: Real Tangent Vectors 5:44 LOCATIONS are not DIRECTIONS 7:07 Limit Definition of Derivative 8:05 Slope of a Tangent Line 8:43 Reframing "Slope" as "Ratio" 9:38 Slope is a Relationship of Tangent Vectors 10:55 Visualizing the Derivative as Ratio of Ta...
596C.C3.A Applying Complex Exponential and Log to M.C. Escher
Переглядів 2062 місяці тому
Burgiel, H., & Salomone, M. (2012). Logarithmic Spirals and Projective Geometry in MC Escher's" Path of Life III". Journal of Humanistic Mathematics, 2(1), 22-35. scholarship.claremont.edu/jhm/vol2/iss1/4/ Marcotte, J., & Salomone, M. (2014). Loxodromic spirals in MC Escher's sphere surface. Journal of Humanistic Mathematics, 4(2), 25-46. scholarship.claremont.edu/jhm/vol4/iss2/4/
596.C3.9 Complex Exponential and Logarithm - Mapping Properties
Переглядів 1152 місяці тому
Desmos tool: www.desmos.com/calculator/usych6intq
596C.C3.8 The Complex Exponential Function - Algebra
Переглядів 892 місяці тому
596C.C3.8 The Complex Exponential Function - Algebra
596C.C3.7 The Truth About Polar Form: Euler's Identity
Переглядів 2112 місяці тому
0:00 Introduction 0:32 Recap of our Polar Form Discoveries 1:34 Goal: What is b? 2:58 Using Maclaurin Series 6:28 How to Combine the Two Series? 8:34 Bringing In Powers of i 10:49 Summing the Two Series 11:32 Recognizing the Series... 12:02 "E" is for "Eureka" 13:26 Euler's Identity 15:34 Wrapup
596C.C3.6 When Complex Numbers Multiply: A Clue About Polar Form
Переглядів 652 місяці тому
0:00 Introduction 0:46 Polar Form of Complex Numbers 1:34 What Happens When We Multiply? 3:24 Trigonometric Addition Identities to the Rescue 4:03 Moduli Multiply, Arguments Add 5:15 What Else Adds When Multiplied? 6:20 Exponents: A Motivated Guess 7:56 But Who Is Base?
596C.C3.5 The Principal Square Root of a Complex Number
Переглядів 1522 місяці тому
Principal square root function in Desmos: www.desmos.com/calculator/do61nnowv6 0:00 Introduction 1:00 "Branch" of a Complex Function 2:00 Example with f(z) = z² 2:58 Domains of One-to-One-ness for f(z) = z² 4:50 Branches of f(z) = z² 9:28 The Principal Square Root 11:07 Examples 14:09 Domain of Principal Square Root 15:29 Discontinuity of Principal Square Root 17:15 Wrapup
596C.C3.4 Recap: There's More Than One "Square Root Function"
Переглядів 762 місяці тому
0:00 Introduction 0:49 f(x) = x² Is Not One-to-One Unless We Restrict Domain 1:36 The "Typical" Domain Restriction and Inverse 3:20 An "Alternative" Choice and Different Inverse 6:18 Summary
596C.C3.3 Left and Right Inverses and the Real Square Root
Переглядів 712 місяці тому
0:00 Introduction 1:54 Left and Right Inverses 4:59 Which Functions Have Left/Right Inverses? 8:31 Which Functions Have Inverse Functions? 8:57 The Real Function f(x) = x² and Its Inverses (Plural!) 12:50 Restricting Domain of f(x) = x² 13:29 The "Usual" Choice of Inverse of f(x) = x² 14:30 A "Different" Inverse for f(x) = x²
596C.C3.2 One-to-One and Onto Functions
Переглядів 1092 місяці тому
0:00 Introduction 1:41 One-to-One Functions 6:19 Onto Functions 8:04 Bijective and Invertible Functions
596C.C3.1 Some Square-Root Conundrums
Переглядів 1092 місяці тому
0:00 Introduction 0:55 A Teaching Conundrum about Square Roots 7:16 A Mathematical Conundrum: Does 1 = -1 ? 8:54 Goals of This Week
596C.C2.4 Complex Functions as Transformations, Part 2
Переглядів 1502 місяці тому
Linear algebra explorer (Geogebra) www.geogebra.org/classic/qhasvqcy Complex multiplication explorer www.geogebra.org/m/c5D8YUwp Möbius transformation explorer www.desmos.com/calculator/odyg1hjtpf Julia maps explorer: www.desmos.com/calculator/nhr05xvu5e Complex function point explorer : www.geogebra.org/classic/bjknjpdw 0:00 Introduction 0:48 Analogy to Linear Algebra 4:36 Example: Complex Add...
596C.C2.3 Complex Functions as Transformations, Part 1
Переглядів 1122 місяці тому
Complex function point explorer : www.geogebra.org/classic/bjknjpdw By changing the expression for W, you can use this tool to explore a complex function one point at a time. (Note, be sure to use capitalized Z for the variable.) 0:00 Introduction 0:50 Defining How Transformation Happens 1:47 Evaluation Via Complex and Real Viewpoints 3:32 Why Can't We Just "Graph" w = f(z) ? 5:55 A Better Idea...
596C.C2.2 Real and Imaginary Parts of a Complex Function
Переглядів 1022 місяці тому
0:00 Introduction 1:12 The Real and Imaginary Part of w = f(z) 2:44 Example of Determining Re(f) and Im(f) 4:34 (When) Do Two Real Functions Make a Complex? 5:53 These Two Real Functions Do "Work" 9:04 Conclusion & Preparing for Visualization
596C.C2.1 Introducing Functions of a Complex Variable
Переглядів 1192 місяці тому
596C.C2.1 Introducing Functions of a Complex Variable
Using Adobe Scan to Capture Your Work
Переглядів 2092 місяці тому
Using Adobe Scan to Capture Your Work
596C.T2.1 Your Portfolio and My "Ungrading" Philosophy
Переглядів 792 місяці тому
596C.T2.1 Your Portfolio and My "Ungrading" Philosophy
596C.T1.1 Reading Klein's "Plan A/Plan B" History
Переглядів 2232 місяці тому
596C.T1.1 Reading Klein's "Plan A/Plan B" History
596C.C1.3 Investigations of Complex Number Arithmetic and Geometry
Переглядів 1322 місяці тому
596C.C1.3 Investigations of Complex Number Arithmetic and Geometry
596C.C1.2 Rectangular and Polar Coordinates in the Complex Plane
Переглядів 1182 місяці тому
596C.C1.2 Rectangular and Polar Coordinates in the Complex Plane
596C.C1.1 Introducing Complex Numbers and the Complex Plane
Переглядів 1522 місяці тому
596C.C1.1 Introducing Complex Numbers and the Complex Plane
202.V4.3 Intersection of Vector Subspaces
Переглядів 6012 роки тому
202.V4.3 Intersection of Vector Subspaces
202.V5.2 Adding a Vector to a Linearly Independent Set
Переглядів 4832 роки тому
202.V5.2 Adding a Vector to a Linearly Independent Set
202.G4 Finding the Eigenspace Associated with an Eigenvalue
Переглядів 2182 роки тому
202.G4 Finding the Eigenspace Associated with an Eigenvalue
202.G3 Characteristic Polynomials and Geometry of Eigenvalues
Переглядів 2702 роки тому
202.G3 Characteristic Polynomials and Geometry of Eigenvalues
202.G2 Using Both Row Operations and Dimension Reduction Together for Determinants
Переглядів 752 роки тому
202.G2 Using Both Row Operations and Dimension Reduction Together for Determinants
202.G2 Dimension Reduction in Finding Determinants
Переглядів 932 роки тому
202.G2 Dimension Reduction in Finding Determinants
These geometric interpretations sometimes mess with my brain but are really cool to think about. I'd also never heard the term "amplitwist" before!
Wait.. does Abstract Algebra abstract away numbers?? What an amazing intro
@@boutiquemaths You better believe it! L
This lecture was irreducibiliterrific! 👍
8:11 Did you mean to say "If the coefficient is not *1*?" Either way, thanks for another amazing lecture! 🙏
Thanks for helping to keep things real 👍
I feel like if more people were able and willing to learn this information, memes would start getting made from Matt's videos and his channel would blow up!
I don't quite understand everything, and I probably need to rewatch about 15 of these videos, but I just want to say that it's been a really fun ride. At this point it's cliché, but I really would like to thank you for making all of these wonderful videos and sharing them on UA-cam!
FTA? More like "Fantastic videos; hip hip hooray!" 👍
14:36 Peak math education 👌
At this point, I would have expected nothing else besides the Mission Impossible music! 🤣
Ah man this is melting my brain.
Oh man, a notecard *attached to a pencil*!? Things are just crazy now 😃
When Matt looks at the camera and tells you something is important, you better buckle up!
1:28 I don't see anything to click on screen 😥 But in any case, thanks for another outstanding video!
Automorphisms? More like "Amazing lectures that can lead to wisdom!" 👍
Thanks for helping us get to the root of the issues!
Normal extensions? More like "Nice examples, and thanks for the instruction!" 🙏
Galois group examples? More like "Great videos, with information that's ample!" 👍
3:14 Ah, so that's what Galois groups are!
I thought it was so cool when I found out that every possible group is just a subgroup of some symmetric group. I still need some time to get comfortable with material beyond the basics, like group actions and automorphism groups, but these concrete and colorful examples definitely help!
I was curious what character you drew with the text "I generate the basis!" at 9:32, then later realized it was a rabbit. Now it makes sense!
Cyclotomic fields? More like "Cool videos, and hilarious puns that bring the feels!" 🤣👍
Just like the Bridgewater logo, that tower is nice! 👍
0:30 0:46 What, how did tearing the paper automatically factor the polynomial like that!? 😮😆 4:54 It's been awhile since I used Mathematica, and I never knew you could try factoring polynomials over different fields like that!
Thanks for helping us get to the roots of the concepts!
Existence and uniqueness? More like "An amazing playlist this is!" 👍
"Cinder-alpha" 😆 Man, you're killin' it with these puns!
The flashcards in the beginning really sold it!
Having just watched these several videos on constructions, I do kind of miss the days in high school geometry where we used compasses and straightedges. Thanks for the fond memories Mrs. K...
9:18 It's cool to know there's more algebra beyond this, and I'm sure it's irrationally great!
Oh man, this video was triangularwesome! 👍📐
The only video that helped me to understand the triangle inequality🥹❤🙏 Thank you!
Adjoin? More like "Amazing lectures, and in on this abstract algebra journey, I'm glad I could join!" 👍
I knew the rationals were countable, but didn't even consider the countability of the algebraic numbers before. Very interesting stuff!
Adobe Scan? More like "Awesome lectures; thanks for helping us understand!" 👍
I was following along through the MATH 302 Abstract Algebra II playlist and got a bit worried it ended, but I'm glad to see more videos in this Abstract Algebra II playlist. Thanks again for making such high quality educational material and sharing it for free!
What's "irreducible" is the outstanding quality of your videos!
Hey there! Is this video supposed to pick up where your Algebra II playlist left off? ua-cam.com/play/PLL0ATV5XYF8CP3A00vb4qjTt6jWmXL2K_.html&feature=shared
Constructing finite fields? More like "Cool videos, and much knowledge they yield!" 👍 Also, did you mean to keep some of the videos in this playlist hidden? Out of the 36 videos, 10 are hidden, and it seems this wasn't meant to be the last video in the course. If it's not too much trouble, would you mind unhiding the other videos, or at least, pointing us to the next set of videos? Thanks in advance!
I'd say these videos are definitely both prime and maximal!
Thanks for another "twisted" video! 🤣
2:19 Whoa, negative norms!? Very cool 👍 Also, thanks for clearing up one of my misconceptions at 6:09!
I'm absolutely loving all these ring puns and classic media! 🤣💍👍
Absorbing multiplication? More like "Amazing lectures; thanks for all the great information!" I do want to call out that issue with the polynomial ideals in the example at 11:55 (as other commenters have), but other than that, thanks for another great video 👍
Quaternions? More like "Quite interesting lectures to think upon!" 👍
0:33 🤣This channel is solid gold!💍
I never thought about the idea of trying to "trick" polynomials before; this is all really neat stuff!
The genuine in the intro was one of the most wholesome things I've seen in awhile! And also, just imagine how many people pay to go to lectures to learn this stuff, while all of Matt's playlists are freely available on UA-cam. I feel sorry for people who paid money and didn't know these were here...
I absolutely appreciated the humor in this video! 🤣 On a side note, and while continuing to acknowledge how good these videos are, I'd highly suggest double-checking the examples. Unfortunately I think the errors can get to the point where they create genuine confusion.
Even though the example wasn't quite right, this lecture was polynomialwesome!👍