Reply to Dom: Hi Dom. Your message is very interesting to me. When we decided to have a part 3 to the calculus series we wanted something that would make a good sequel to parts 1 and 2. And that’s why we decided to link the three mini-courses, differential equations, linear algebra and complex variables. What amazes me is how much help these mini-courses have given to students who are currently taking these courses. Quite frankly I had no idea how the modern viewer of today would react to my “ancient” black-and-white videos.
Thanks for the comment. The person who deserves the giant share of the thanks is Madge Goldman, President of the Gabriella and Paul Rosenbaum Foundation. She is the person who, after seeing the original videos, made the funds available so that Open Course Ware could restore, digitize and upload the original videos which were beginning to fall into a state of disrepair.
Herb Gross You are a great teacher. I find it interesting how I can learn more and be more engaged on a 15~30 min video than on my 4 hour class in college.
Oh man, I came here and got disappointed at first because of the age of the video, and now after watching it I just can't think of any other video that is nearly as good as this one on this subject, the professor is so passionate to teach that everything comes easily to your mind! Thank you professor Herbert for the amazing lecture!
Good science is never outdated, good teaching isn't either! I'm deeply impressed by watching this clear and concise presentation filmed nearly 50 years ago, which is still absolutely valuable today. Notice how Prof Gross was able to run a presentation of more than 30 minutes without a jump cut!
He makes me smile alot. I seem to always enjoy this professor's lectures and presentations. I wished I would of meet him. It just goes to show, there are good professors that can make a topic enjoyable and exciting. Thank you MIT for showing a quality presentation. I enjoyed the black and white color, brings back memories.
When I took differential equations I originally had a hard time understanding how they altered the value of n. I think most people regarded it as trivial because no one really tried to explain it. I have to thank you for finally clearing this up for me.
I would just like to thank Mr. (Dr.?) Herbert Gross for these exceptional video lectures on mathematics. As a current mechanical engineering undergraduate student, seeing such videos makes me wish I could go back in time and experience such quality edification first-hand. Since such time travel is currently impossible, I suppose these video will have to suffice. Again, thank you!
Thanks David. I always enjoy hearing from viewers, especially when they refer to my lectures as "life savers":). If you would like to contact me privately I can be reached at hgross3@comcast.net. In the meantime, I send you my best wishes for success in all of your endeavors. With warmest regards, Herb
What a wonderful series. I wish I had found these before! It really emphasizes how amazing the use of power series are in this instance, and tying it all together before hand helped, because my book generally just spits out stuff without saying what any of it has to do with anything else. Very jarring.
Thanks Jennifer. It hs always been my belief that when students understand the "why", the "how" becomes even easier for them to internalize. Not only that but keep in mind that thanks to the Internet there are plenty of resources that explain the "how", but very few that delve into the "why".
Thanks 1 Iota! And too think how worried I was that the black-and-white-talking-head format would turn viewers off (and I imagine that some surfers skipped over the videos because they were in that format!). It is reassuring to know that there is always a chance that content will trump production values.
I love having my mind blown. This makes explicit the vague connection between recurrence relations and differential equations my instructors had alluded to. It also introduces a new vague (vague for me) connection between differential equations and orthogonality since Legendre polynomials pop up with respect to a certain inner product.
Thank you for the kind words, Daniel. In truth, I had as much time as I wanted to “perfect” a lecture. It took me several days to prepare each lecture. However, once that was done, I had a sort of timeless lecture which any viewer, in any place and at any time would be seeing exactly the same thing. I hope that as time goes on more and more well-thought-out lectures will be available on the internet for anyone to view free of charge!!!
One of the most informative and entertaining lectures I have seen to date; I just love the passion shown for the material! The world (myself included!) thanks you, Professor Gross!
Your very kind words make me feel quite blessed. I am pleased that my lectures will be my legacy to future generations and will result in viewers still being taught by me even when I am no longer here!
I never noticed this until I watched the videos. I think it was a consequence of having to concentrate on thinking about what I had to say next and knowing that if I made a mistake we would have to reshoot the entire video. In those days, editing was both tedious and expensive! My mandate was to never make the same mistake ONCE!
this guy doesnt blink, but he is brilliant, very good, i wish we had professors as good as him at ucsd, learning math there is so hard because all of the professors are awful teachers
This is so helpful >< thank you so much! It's clear, concise, and easy to understand this way. Since I'm taking elem. Difeq in highschool, its a bit confusing with the teacher&all.
Thank you Kidda. It excites me to know that the videos a re viewed by people all over the world and that this (hopefully) will contnue to happen even when I am no longer here
Thank you for your kind comment, Isabelle. Unfortunately I do not have such a lecture. In fact most of my lectures are at the Precalculus level; mainly on arithmetic and algebra that are designed to help mathematically at-risk adults overcome their “fear” of mathematics. My own website is at www.mathasasecndlanguage.com. I send you my best wishes for success in all of your endeavors. Thanks again for taking the time to comment.
What colour chalk appears black like that? It makes a contrast with the white chalk that is actually really helpful. I love how organized and well put together these videos are. Many thanks.
Can Example 1 be solved using differential operator D and ; I think you have showed this method to illustrate power series method for const coefficients. Fantastic job sir.😊
Thank you so much Mr. I am currently working on a big equation that I have struggled with for days. This made things clearer. Is it possible to write the second derivative with a summation limit that starts at 0 without altering anything else? (we would of course get two zero terms from it) but this would make it easier to operate with the other summations.
I have a tricky question. Wherever i search on the web i find that in the Legendre Differential Equation at the point where we have y,y',y'' the index of sumation stays at n=0 and does not increase. Why exactly is that? From what i see here it should increase.
Aggelos Karaglanis Hi Aggelos. It is now more than 40 years since I’ve worked with differential equations and some topics are no longer on the tip of my tongue. However my guess is that when you look at the series term by term the first and/or second term will equal 0. I may be wrong and perhaps others can answer your question better than I did.
I began teaching this course live in the summer of 1968; and my the middle of 1969 it was decided to produce the lectures on video tape. And by the time I wrote all of the supporting material it was late 1972. Shortly thereafter I left MIT on a full time basis (but I remained as a visiting senior lecturer during the summers until 1988) to become the founding math department chairperson at Bunker Hill Community College in Boston, where I remained until my retirement in 2003. Since then I have devoted my time to helping elementary school teachers and developing my own website (www.mathasasecondlanguage) where I have uploaded all of my materials in arithmetic and algebra for anyone to use free of charge.
Herb Gross Your immortality will be complete when they send these recordings into deep space to be picked up by an advancing space-faring civilization. Your work epitomizes all the good that humanity has to offer.
+Gray Wagner Thanks for your question, Gray. The original purpose of this course (as its title implies) was as a refresher course for persons who had already taken calculus courses. The result is that at times I may have been a bit sloppy in my presentation. The point is that when I said “interval of convergence”, the correct wording is “the interval of absolute convergence”; and that is what allows me to rearrange terms etc. I’m hoping that I was a bit more explicit when I introduced this topic in Part 1 of this trilogy of courses. As I reviewed the video prior to sending this message, I noticed that I was also very sloppy in my referring to (- a) as “minus a”. The word “minus” should be reserved for the operation of subtraction; and to be correct (-a) should be read as “The opposite (or additive inverse) of a”. Too many students jump to the conclusion that (-a) is a negative number. However if a = (-3) then (-a) = (+3). Notice the three different meanings of the minus sign in the expression below: (-3) - (-a) To be mathematically precise the above expression should be read as negative 3 minus the opposite of a Hopefully after some 45 years I hope that the statute of limitations prevents me from being tried on the charge of sloppiness!!!
+Herb Gross Hi Herb, Thanks for clearing that up. I am currently in my third year of maths at the University of Queensland. In one of my Computational-Math subjects we are approximating ODE's using Taylor series. I was able to correct my lecturer on this exact problem. He said that the Taylor series could approximate a function at any point. I used 1/1-x at x=2 as a counter example, like you used in Single variable Calculus. It impressed everyone in the class. (That's what I like to think anyway) On a personal note, I am a big fan! I have watched all Calculus revisited, read the supplementary notes and even watched a lot of the Maths as a Second Language videos. You would be surprised to find out how little mathematics a third year maths student actually knows.
SWiSHRoyal Until I saw your comment i had never noticed it. I wonder if others have noticed the same thing. I looked at the videos again but didn't notice anything that was distracting but that could just be because my eyes are 86 years old:)
+Mustafa M Hello Mustafa. I just now noticed the date of your message and I apologize for having missed seeing it before now. To answer your question I was born in Boston on April 2, 1929. Although I am quite old now, I rejoice in the fact that the young Herb Gross in the calculus videos will always remain a young and enthusiastic instructor! I have been very blessed.
Mustafa M We used a room at MIT to make the videos and we were not able to find a way to set up the lighting in a way that eliminated all shadows. But the black boards (except that they were green) were the usual type you are used to seeing. We did try to make sure that the shadows did not obstruct the viewing of the boards
They weren't ancient when I made them but that was way back in 1970 - 71:) I am very pleasantly surprised by how well they seem to have withstood the test of time. Thanks for asking and I send you my best wishes.
Fantastic lecture! 40+ years later it stands as one of the clearest explanations on power series solutions on the internet. Thank you!
Reply to Dom:
Hi Dom.
Your message is very interesting to me. When we decided to have a part 3 to the calculus series we wanted something that would make a good sequel to parts 1 and 2. And that’s why we decided to link the three mini-courses, differential equations, linear algebra and complex variables. What amazes me is how much help these mini-courses have given to students who are currently taking these courses. Quite frankly I had no idea how the modern viewer of today would react to my “ancient” black-and-white videos.
thank you for your black and white videos , the board you used in the video is a mind blown
Prof Gross I m self teaching differential Equations with these videos of yours.
Very grateful to you.
Wish you❤️ happiness and good health from 🇮🇳.
Thanks for the comment. The person who deserves the giant share of the thanks is Madge Goldman, President of the Gabriella and Paul Rosenbaum Foundation. She is the person who, after seeing the original videos, made the funds available so that Open Course Ware could restore, digitize and upload the original videos which were beginning to fall into a state of disrepair.
The way he expresses the ideas and his enthusiasm is remarkable and inspirational. Rest in peace.
This is the clearest lecture I have seen on this topic
Thanks Prof. Gross and MIT open courseware!!
Herb Gross You are a great teacher. I find it interesting how I can learn more and be more engaged on a 15~30 min video than on my 4 hour class in college.
Oh man, I came here and got disappointed at first because of the age of the video, and now after watching it I just can't think of any other video that is nearly as good as this one on this subject, the professor is so passionate to teach that everything comes easily to your mind! Thank you professor Herbert for the amazing lecture!
Good science is never outdated, good teaching isn't either!
I'm deeply impressed by watching this clear and concise presentation filmed nearly 50 years ago, which is still absolutely valuable today.
Notice how Prof Gross was able to run a presentation of more than 30 minutes without a jump cut!
Now lookit: even though this video is 40+ years old, good pedagogy is timeless. Great video!
Just stumbled upon this video. Professor Gross is a hidden gem!
I've been blessed by this video. Didn't think I would watch it for half an hour but ended up wanting to watch for an even longer time.
He makes me smile alot. I seem to always enjoy this professor's lectures and presentations. I wished I would of meet him. It just goes to show, there are good professors that can make a topic enjoyable and exciting. Thank you MIT for showing a quality presentation. I enjoyed the black and white color, brings back memories.
When I took differential equations I originally had a hard time understanding how they altered the value of n. I think most people regarded it as trivial because no one really tried to explain it. I have to thank you for finally clearing this up for me.
Thank you for sharing this with you. It reminds me of the saying that the job of the teacher is not to cover ground, but rather to uncover it.
I would just like to thank Mr. (Dr.?) Herbert Gross for these exceptional video lectures on mathematics. As a current mechanical engineering undergraduate student, seeing such videos makes me wish I could go back in time and experience such quality edification first-hand. Since such time travel is currently impossible, I suppose these video will have to suffice. Again, thank you!
First Herbert Gross video I've watched and I can honestly say that he is an extremely gifted teacher!
Thanks Dr. Gross. I'm an older student taking a DE course and these videos are a life saver.
Thanks David. I always enjoy hearing from viewers, especially when they refer to my lectures as "life savers":). If you would like to contact me privately I can be reached at hgross3@comcast.net. In the meantime, I send you my best wishes for success in all of your endeavors.
With warmest regards,
Herb
What a wonderful series. I wish I had found these before! It really emphasizes how amazing the use of power series are in this instance, and tying it all together before hand helped, because my book generally just spits out stuff without saying what any of it has to do with anything else. Very jarring.
Thanks Jennifer. It hs always been my belief that when students understand the "why", the "how" becomes even easier for them to internalize. Not only that but keep in mind that thanks to the Internet there are plenty of resources that explain the "how", but very few that delve into the "why".
One of the best lecture on power series.Just loved the way you taught. None of the lectures are comparable to this one.
I love these lectures! Thanks to Doc Gross, M.I.T., and the donors/sponsors for posting them.
Wow such clear and well-organized explanations! And the black & white effect is totally jamming for this midnight study session. Vintage. I dig it.
Thanks 1 Iota! And too think how worried I was that the black-and-white-talking-head format would turn viewers off (and I imagine that some surfers skipped over the videos because they were in that format!). It is reassuring to know that there is always a chance that content will trump production values.
I love having my mind blown. This makes explicit the vague connection between recurrence relations and differential equations my instructors had alluded to. It also introduces a new vague (vague for me) connection between differential equations and orthogonality since Legendre polynomials pop up with respect to a certain inner product.
That was amazing! 6 years ive been needing a simple/complete explanation and all i had to do was go back to the 70's who knew?
Thank you for the kind words, Daniel. In truth, I had as much time as I wanted to “perfect” a lecture. It took me several days to prepare each lecture. However, once that was done, I had a sort of timeless lecture which any viewer, in any place and at any time would be seeing exactly the same thing. I hope that as time goes on more and more well-thought-out lectures will be available on the internet for anyone to view free of charge!!!
Your smile is as pleasant as the way you teach. Thank you
One of the most informative and entertaining lectures I have seen to date; I just love the passion shown for the material! The world (myself included!) thanks you, Professor Gross!
Your very kind words make me feel quite blessed. I am pleased that my lectures will be my legacy to future generations and will result in viewers still being taught by me even when I am no longer here!
This is how all good math lectures are-organized and methodical.
I was a little bit disappointed by the age of the video, but it turned out to be the best explanation of the power series solution I have seen.
Amazing lecture. An explanation from first principles well worth your time to understand power series solutions.
God bless for this professor, amazing ! 44 years ago . have added my knowledge
Great explanations! Clear, intuitive, and inspiring.
I never noticed this until I watched the videos. I think it was a consequence of having to concentrate on thinking about what I had to say next and knowing that if I made a mistake we would have to reshoot the entire video. In those days, editing was both tedious and expensive! My mandate was to never make the same mistake ONCE!
Wow, thank you so much for making these videos Professor Gross. You always explain things so well!
Way better than my prof. :) I enjoyed your lecture
Is there any other Power Series solution lecture as good as this one on RUclips? Learnt more in a 34min video than a 6month course
He's so happy!
this guy doesnt blink, but he is brilliant, very good, i wish we had professors as good as him at ucsd, learning math there is so hard because all of the professors are awful teachers
Professor Herbert excellent explanation thank you sir.superb
Explictly explained concept ...Thank you
Herb Gross is the best ever.
This is so helpful >< thank you so much! It's clear, concise, and easy to understand this way. Since I'm taking elem. Difeq in highschool, its a bit confusing with the teacher&all.
Best lecturer I've found, besides Kahn. Very cool.
I am from Saudi Arabia, I do study abroad and god bless this man .
Thank you Kidda. It excites me to know that the videos a re viewed by people all over the world and that this (hopefully) will contnue to happen even when I am no longer here
I'm so happy to hear from you Sir , we do respect and value all your efforts for humanity , I Wish you always the good health and long live .
Herb Gross thank you very much for sharing your knowledge. Long live!!!
It's my pleasure! Enjoy and keep learning!
I will
i could never thank you enough
Basma Abumahfouz You just did!!!
legend, the pioneer of distance learning
Thank you for your kind comment, Isabelle.
Unfortunately I do not have such a lecture. In fact most of my lectures are at the Precalculus level; mainly on arithmetic and algebra that are designed to help mathematically at-risk adults overcome their “fear” of mathematics. My own website is at www.mathasasecndlanguage.com.
I send you my best wishes for success in all of your endeavors.
Thanks again for taking the time to comment.
this man is good years later this the best
If only every prof could explain like this...
Amazing teacher! This is incredibly helpful, thank you very much!
Rest in peace, you'll be immortalized doing what you loved, in your lecturers!
This video was so incredibly helpful. Thank you!!!
This guy is great
I absolutely love this!
This is such a good lecture!!!
Thank you I think I finally get it.
I love this video and this guy!
Rest in peace, great man!
What colour chalk appears black like that? It makes a contrast with the white chalk that is actually really helpful. I love how organized and well put together these videos are. Many thanks.
Herb Gross This was an awesome lecture! So helpful! Do you have one on regular singular points?
perfect professor
This lecture is sooooooooo juicy. Amazing.
Can Example 1 be solved using differential operator D and ; I think you have showed this method to illustrate power series method for const coefficients. Fantastic job sir.😊
Thank you :)
God bless you :)
Thank you so much Mr. I am currently working on a big equation that I have struggled with for days. This made things clearer. Is it possible to write the second derivative with a summation limit that starts at 0 without altering anything else? (we would of course get two zero terms from it) but this would make it easier to operate with the other summations.
math with such smiling face :) , it's cool to learn :D
WOW THAT WAS GREAT
Awesome
I feel so high to see his smiling face hahaha. Nice video.
Math.......Math never changes........
It would be great if I could watch all sections of power series in diff equation.
Rest in peace legend
"Lookit"
Great vedio
When was this Video make?
The series was first released in 1972, but equally valuable today for students who are learning these topics for the first time.
@@mitocw I’m a student and this is EXTREMELY helpful!
Much much better than anything On the webs
I have a tricky question. Wherever i search on the web i find that in the Legendre Differential Equation at the point where we have y,y',y''
the index of sumation stays at n=0 and does not increase.
Why exactly is that? From what i see here it should increase.
Aggelos Karaglanis
Hi Aggelos. It is now more than 40 years since I’ve worked with differential equations and some topics are no longer on the tip of my tongue. However my guess is that when you look at the series term by term the first and/or second term will equal 0. I may be wrong and perhaps others can answer your question better than I did.
BEST fucking review ever !
I might have used a different choic eof words but thank you for the kind comment!!!
My professor couldn't explain us in 240 minutes what he did in 34 min... and better.
When exactly was this filmed?
I began teaching this course live in the summer of 1968; and my the middle of 1969 it was decided to produce the lectures on video tape. And by the time I wrote all of the supporting material it was late 1972. Shortly thereafter I left MIT on a full time basis (but I remained as a visiting senior lecturer during the summers until 1988) to become the founding math department chairperson at Bunker Hill Community College in Boston, where I remained until my retirement in 2003. Since then I have devoted my time to helping elementary school teachers and developing my own website (www.mathasasecondlanguage) where I have uploaded all of my materials in arithmetic and algebra for anyone to use free of charge.
Herb Gross Your immortality will be complete when they send these recordings into deep space to be picked up by an advancing space-faring civilization. Your work epitomizes all the good that humanity has to offer.
Wow when was these videos were recorded ?
The series was first released in 1972.
@@mitocw wow! And he does a beautiful job at explaining everything too
When he says the series must be convergent, does he mean absolutely convergent or can it be conditionally convergent as well?
+Gray Wagner Thanks for your question, Gray. The original purpose of this course (as its title implies) was as a refresher course for persons who had already taken calculus courses. The result is that at times I may have been a bit sloppy in my presentation. The point is that when I said “interval of convergence”, the correct wording is “the interval of absolute convergence”; and that is what allows me to rearrange terms etc. I’m hoping that I was a bit more explicit when I introduced this topic in Part 1 of this trilogy of courses.
As I reviewed the video prior to sending this message, I noticed that I was also very sloppy in my referring to (- a) as “minus a”. The word “minus” should be reserved for the operation of subtraction; and to be correct (-a) should be read as “The opposite (or additive inverse) of a”. Too many students jump to the conclusion that (-a) is a negative number. However if a = (-3) then (-a) = (+3).
Notice the three different meanings of the minus sign in the expression below:
(-3) - (-a)
To be mathematically precise the above expression should be read as
negative 3 minus the opposite of a
Hopefully after some 45 years I hope that the statute of limitations prevents me from being tried on the charge of sloppiness!!!
+Herb Gross Hi Herb,
Thanks for clearing that up.
I am currently in my third year of maths at the University of Queensland. In one of my Computational-Math subjects we are approximating ODE's using Taylor series. I was able to correct my lecturer on this exact problem. He said that the Taylor series could approximate a function at any point. I used 1/1-x at x=2 as a counter example, like you used in Single variable Calculus. It impressed everyone in the class. (That's what I like to think anyway)
On a personal note, I am a big fan! I have watched all Calculus revisited, read the supplementary notes and even watched a lot of the Maths as a Second Language videos. You would be surprised to find out how little mathematics a third year maths student actually knows.
Omg this is exactly what we are doing in 2024
He passed 2 months ago. RIP Herbert Gross
Nooooo 😢
How comes the chalk throws shadows?
SWiSHRoyal Until I saw your comment i had never noticed it. I wonder if others have noticed the same thing. I looked at the videos again but didn't notice anything that was distracting but that could just be because my eyes are 86 years old:)
Was wondering how old you were......thank you professor for your contribution to humanity
+Mustafa M
Hello Mustafa.
I just now noticed the date of your message and I apologize for having missed seeing it before now. To answer your question I was born in Boston on April 2, 1929. Although I am quite old now, I rejoice in the fact that the young Herb Gross in the calculus videos will always remain a young and enthusiastic instructor! I have been very blessed.
There is a cool shadow on the board. Makes me think maybe this is all written on glass.
Mustafa M We used a room at MIT to make the videos and we were not able to find a way to set up the lighting in a way that eliminated all shadows. But the black boards (except that they were green) were the usual type you are used to seeing. We did try to make sure that the shadows did not obstruct the viewing of the boards
wow, ancient.
They weren't ancient when I made them but that was way back in 1970 - 71:)
I am very pleasantly surprised by how well they seem to have withstood the test of time. Thanks for asking and I send you my best wishes.
Wow
RIP
oh my, this is rather old.
A interesting history document
NO COMMENT!!
I Think the viewers have f***ed up