This is a classic lecture with the introduction of Euler's equation, Complex Numbers and Complex Exponentials. Professor Mattuck thank you for another fantastic lecture.
"t is always a real variable, i dont think we have complex time yet although im sure there will be some day, when the next einstein appears." that made my day
well, imaginary time is nothing but real time*i it's just to remove the minus sign that is in front of t^2 in the minkowski spacetime. I think he means it as a concept not as a mathematical tactic.
Oddly enough Stephen Hawking pointed out in an interview with John Oliver that the one thing in his theories that haven't made it into sci-fi is complex time because people don't understand it...
The man is a Math Professor for MIT. I am 100% certain that he knows the derivation by Taylor-Series expansion... but the use of complex coordinate system is the basis by which Euler first hypothesized and demonstrated the formula. You are in fact the ignorant one is you simply want to jump to taylor's theorem without a symbolic explanation.
"The way to do it if you're on a desert island..." Yeah, because I'm sure that if I'm struggling to survive off the land that my top priority would be solving integrals. :D Anyway, Great Lecture!
Well, you never know. If you have a plant who's output of fruit is equal to some function of how much water it's given, how many plants you want and how much water you want to give it probably becomes a differential equation. :P
He seems to have interchanged Re() and integral operators without justification. In other words, he assumes that Integrate( Real( f(x) ) ) = Real( Integrate( f(x) ) ), which may be trivial to show. For instance, see here: math.stackexchange.com/questions/2139671/why-does-integral-and-the-imaginary-part-commute
Geometrically, you can think of this. When you're integrating a complex number, you're integrating a vector which is the rate of change of some other vector. This vector (the one being integrated) will cause both a rotation and a scaling to that other vector. However, the real part of the vector being integrated is the rate of change of the other vector's real part, and the imaginary part of the vector being integrated is the rate of change of the other vector's imaginary part. Therefor, you may as well just integrate each of the parts separately.
Hmm, so I distributed (-i-1)(cosx+isinx) = -icosx-i^2sinx-cosx-isinx. When simplified I get sinx - cosx -i(cosx+sinx). Can someone tell where I'm going wrong?
Because cos(x) is Re{e^(ix)}, we only want the real part of e^(ix) as our answer it's ok to integrate imaginary part, but it's not what the question demands.
@jahs389 well said. there are many fools on youtube, the guy just learned about taylor and want to manifest his knowledge all over the place. so lame !
The 18.03SC playlist matches the instructor's reordering of the material. It's probably best to view 18.03SC through the course on MIT OpenCourseWare at: ocw.mit.edu/18-03SCF11. The website includes lecture notes, assignments, inline quizzes. To see the original order see the Spring 2006 playlist: ruclips.net/p/PLEC88901EBADDD980. Best wishes on your studies!
Think, man, think. Sin and cos were DEFINED by their infinite series in the 1400s by the Indian mathematician Madhava. They were again discovered in the early 1600s by Isaac Newton and Wilhelm Leibniz. Euler's definition of the complex exponential was published in 1748. Unless Rudin also invented time travel, your statement doesn't hold water. One could go back and imagine what would have happened if what you said was true, and perhaps you could make an argument that the development of analysis would have been cleaner. What you cannot do, however, is revise history.
his explanation for euler's formula being a definition not a formula sounds a little fishy to me. either he's pandering to the high-school students - or he genuinely hasn't heard of the derivation of this from taylor's theorem.
Who's the geniuses here that can't realize sarcasm? I suppose I should explain that the use of genius in this comment is meant with a sarcastic tone....
What the HELL are you talking about? First of all, I already know the proof. It comes from the Taylor series of the exponential and trigonometric functions and the fact that odd powers of the imaginary unit are imaginary and even powers are real. Second, your comment was completely unrelated to mine. Third, what in the name of Kelvin and Stokes does 0.375 of self respect look like? What does ONE self respect look like? These students are fresh out of high school....
...Your attention to rigour may be mathematically helpful, but it's woefully misplaced in a pedagogical setting. The students will be proving it later in a complex analysis class. Quit your whining. God, I hope you're not a teacher; being in your class would make me wonder if I was being punished. I'm sure I wouldn't learn a thing.
![SahBabii - Viking [Official Music Video]](http://i.ytimg.com/vi/D37cL92JX0I/mqdefault.jpg)
rest in piece to such a brilliant man. MIT uploading this to the general public has immortalized him
I never understood this sentence (general public)*
This is a classic lecture with the introduction of Euler's equation, Complex Numbers and Complex Exponentials. Professor Mattuck thank you for another fantastic lecture.
This is, THE, best lecture on this topic EVER because, Professor Mattuck follows directions! Don't turn writing on the wall into a complex problem.
hehehehehe that pun
I understood and learnt everything he taught in this video except how to draw a zeta...
Watching this at 1.5x speed makes for a good quick review. It is still clear and easy to follow.
33:15 keeping the chalk going like a boss.
"t is always a real variable, i dont think we have complex time yet although im sure there will be some day, when the next einstein appears."
that made my day
@@friedrichnietzsche5664 a complex number isn't always imaginary.
well, imaginary time is nothing but real time*i it's just to remove the minus sign that is in front of t^2 in the minkowski spacetime. I think he means it as a concept not as a mathematical tactic.
@@friedrichnietzsche5664 but not every complex number is imaginary.
Oddly enough Stephen Hawking pointed out in an interview with John Oliver that the one thing in his theories that haven't made it into sci-fi is complex time because people don't understand it...
The man is a Math Professor for MIT. I am 100% certain that he knows the derivation by Taylor-Series expansion... but the use of complex coordinate system is the basis by which Euler first hypothesized and demonstrated the formula. You are in fact the ignorant one is you simply want to jump to taylor's theorem without a symbolic explanation.
Lecture 2: "High Euler..."
Lecture 6: "Euler sells himself..."
Euler sure was wild
"the master of us all."
"So, I'm going to do the fun things and assume they are true, because they are."
"The way to do it if you're on a desert island..." Yeah, because I'm sure that if I'm struggling to survive off the land that my top priority would be solving integrals. :D
Anyway, Great Lecture!
Well, you never know. If you have a plant who's output of fruit is equal to some function of how much water it's given, how many plants you want and how much water you want to give it probably becomes a differential equation. :P
Awesome!!!! This professor is so good
12:08 this man is woke to another level
good way to remember the formulae for cos(t1+t2) and sin(t1+t2) %)
Thank you for the clarity
lol it was fun calculating the integral of e^x cos x using complex numbers :)
By the Way,
Hawking already defined imaginary Time.
en.wikipedia.org/wiki/Imaginary_time
Great teacher 😊😃😃😃
dude, he is professor in M fucking I fucking T, u get it? MIT? he knows like everything.
This is the way math should be taught everywhere.
This is a very small part of math.
Excellent!
The interpretation in the complex plane came after Euler, not before.
11:50 It's funny that I'm listening to light music as he said all that.
Then he might have been hearing the music you were listening to at that moment. Fear our Lord.
He seems to have interchanged Re() and integral operators without justification.
In other words, he assumes that Integrate( Real( f(x) ) ) = Real( Integrate( f(x) ) ), which may be trivial to show.
For instance, see here:
math.stackexchange.com/questions/2139671/why-does-integral-and-the-imaginary-part-commute
Geometrically, you can think of this. When you're integrating a complex number, you're integrating a vector which is the rate of change of some other vector. This vector (the one being integrated) will cause both a rotation and a scaling to that other vector. However, the real part of the vector being integrated is the rate of change of the other vector's real part, and the imaginary part of the vector being integrated is the rate of change of the other vector's imaginary part. Therefor, you may as well just integrate each of the parts separately.
I Really Like The Video From Your Complex Numbers and Complex Exponentials.
believe me there were better days in Euler's life
Also, I wonder if you can do the integration easier if you just express cos x = (e^(ix) + e^(-ix) ) / 2
actually thats quite smart , as u dont have to remember to take the real part of the integral and so on.I dont see any reason it shouldnt work
@soom66 you derivated euler's formula in 8th grade?! yeah sounds perfectly realistic to me
Exponential law. Fail. Rules for Exponents. Win.
@soom66 wait a second, I assumed he was going to derive it but he didnt... unbelievable... my bad
the argument beats the angle :p
@waterskippers hilariously creative
29:14 i feel like im being photographed too so i wrote arguments as well
Amazing
well that's easily fixed... 16:03
Yeah, I'm sure he's ignorant to one of the most frequently used derivation in mathematics.....
Hmm, so I distributed (-i-1)(cosx+isinx) = -icosx-i^2sinx-cosx-isinx. When simplified I get sinx - cosx -i(cosx+sinx). Can someone tell where I'm going wrong?
Wait, nevermind. He says to pick out only the real factor. My question is then, why? Is it simply because the imaginary is impossible?
Because cos(x) is Re{e^(ix)}, we only want the real part of e^(ix) as our answer
it's ok to integrate imaginary part, but it's not what the question demands.
Ah okay. Thank you for clearing that up.
lec 4,5 isnt listed in the playlist
Here's the official playlist: ruclips.net/p/PLEC88901EBADDD980. Best wishes on your studies.
@@mitocw i was following the course notes for MIT 18.03 SC version but the lectures don't match up with them is there a order mix up?
21:29 C'est la vie :P
First order linear differential eqn with same initial conditions has one answer
comlexifying the integral where is c ?
"This is what separates the girls from the women." :D
well, nothing to comment but I saw too many inappropriate comments !
Oh my god this guy is hilarious lol
him: “it’s in the notes”
me in 2019: ...
you can see the notes
@jahs389 well said.
there are many fools on youtube, the guy just learned about taylor and want to manifest his knowledge all over the place. so lame !
What about lectures 4 and 5?. They aren't in order.
The 18.03SC playlist matches the instructor's reordering of the material. It's probably best to view 18.03SC through the course on MIT OpenCourseWare at: ocw.mit.edu/18-03SCF11. The website includes lecture notes, assignments, inline quizzes. To see the original order see the Spring 2006 playlist: ruclips.net/p/PLEC88901EBADDD980. Best wishes on your studies!
37:27 isnt it -2? -1+ i^2 = -1 + (-1) =-2?
No, (-1 + i)(-1 - i) --> -1*-1 + i*-I = 1+1.
This guy might have had a bit too much to drink the previous night lol. I don't mean that in a bad way though.
One person didn't hear the music.
Read the prologue of Real and Complex Analysis by Rudin. Sine and cosine are DEFINED as the imaginary and real parts of the complex exp function.
Think, man, think. Sin and cos were DEFINED by their infinite series in the 1400s by the Indian mathematician Madhava. They were again discovered in the early 1600s by Isaac Newton and Wilhelm Leibniz. Euler's definition of the complex exponential was published in 1748. Unless Rudin also invented time travel, your statement doesn't hold water. One could go back and imagine what would have happened if what you said was true, and perhaps you could make an argument that the development of analysis would have been cleaner. What you cannot do, however, is revise history.
Please process the videos with a CNN that puts it in HD
his explanation for euler's formula being a definition not a formula sounds a little fishy to me. either he's pandering to the high-school students - or he genuinely hasn't heard of the derivation of this from taylor's theorem.
It's more an identity than definition.
He clearly said like 3 times that he wasn't going to derive it with taylors theorem because it would take too long and too much chalk
nothing to do with diff eq lol
Are you sure? Have you ever solved 2nd-order equations? Wave equations, etc?
You are being lied to. It's not the letter i that is complex, it is the glorious letter j that is complex and sophisticated!
Andrew Rodgers
J is used in physics to avoid confusion as I is already used in current!
@@nirmalpadwal1266 whoosh
Who's the geniuses here that can't realize sarcasm?
I suppose I should explain that the use of genius in this comment is meant with a sarcastic tone....
What the HELL are you talking about? First of all, I already know the proof. It comes from the Taylor series of the exponential and trigonometric functions and the fact that odd powers of the imaginary unit are imaginary and even powers are real. Second, your comment was completely unrelated to mine. Third, what in the name of Kelvin and Stokes does 0.375 of self respect look like? What does ONE self respect look like? These students are fresh out of high school....
...Your attention to rigour may be mathematically helpful, but it's woefully misplaced in a pedagogical setting. The students will be proving it later in a complex analysis class. Quit your whining. God, I hope you're not a teacher; being in your class would make me wonder if I was being punished. I'm sure I wouldn't learn a thing.
Calm down bitch