When learning mathematical analysis...I would not recommend Baby Rudin as your first book on the subject. It will test your wits and give you solid nightmares. Trust me.
@@mathslover98 that book is legendary but if you have never studied any set theory or logic or proof writing, it will prove very tough going for a beginner.
Interestingly enough, I never had to take a Trig course, neither in high school nor university. All of my experience with Trig came from what was taught in the Pre-Calc and high school geometry. That being said, I think the trig from precalc helped me understand justttt enough trig to make it through and do ok Physics I and Calc II (trig comes up a lot it seems). Curious if anyone else has a similar experience.
Exact same experience for me, the way my school did it was pre-calculus was an accelerated class with half of it reviewing Algebra and the other half reviewing trigonometry, so essentially you were taking two classes in one semester. I loved that structure personally and I think it set me up well in half the time.
Great vid as always! I took Discrete Math in college and we used Susanna Epp textbook....in my option the best introduction to Logic and Set Theory I have seen (beginning chapters). Highly recommended.
"Discrete Mathematics and Its Applications" (4th ed) by Kenneth Rosen is a gem. (My copy is a Recycle Bin Rescue.) In addition to presenting Discrete Mathematics (aka, "Counting Gone Wild"), it details the tools needed to write proofs and think Mathematics like a Mathematician. There are also lots of examples and applications, so you can have good answers when your friends ask, "Why do you study this crazy Math Stuff?"
That Mathematical Statistics book by Wackerly is THE ONE. I have about a dozen books on the topic, and this is the one that got me through math stat. I found that if I mostly tuned out the instructor and instead read the class text plus Wackerly, and worked problems in Wackerly, I understood the course much better. The solutions manual is a must, too.
In other universities outside the US, advanced calculus 1 and advanced calculus 2 are the prerequisites before you could take Real analysis and Complex analysis. Adv Calc 1-2 (Intro to Analysis) are like the stepping stones to further analysis courses.
Would it be alright if I made a request? I'm trying to get some of the deeper theory behind AI in a nutshell, and I think I've identified some milestone topics I need to traverse: - Phase transitions in statistical mechanics (only classical though), and the renormalisation group - Bayesian Stats (up to the point where I'm comfortable with the theta parameter) - Theoretical CS (akin to Sipser's book/MITOCW course) - I guess I'll know that I'm good when I fully understand all the reasons the Gaussian is critical to statistical theory (at the moment I know about CLT, identity under FT, the relationship it has with the Frobenius norm) - Information and Control Theory, particularly their combination. I understand there are four big asymptotic limiting theorems here, and I need to understand all of them. So far I have the Shannon-Nyquist limit down. - Less important, Topois theory for the generalisation of classical logic, and Category theory for the foundations of mathematics (plus I like the commutative diagrams). I was wondering if you have any recommended prerequisite material, maybe even a course structure like this video, where one could end up with the above points satisfied by the end of it all? Cheers.
That ODE by Ross is excellent. I used it in the mid 1970s for a summer class. It almost made the class fun. Almost. Had occasion to use it again within the last 10 years.
I think a mistake i made when taking mathematical stats is not understanding probability theory enough beforehand. it would be a good idea to read a book like a first course in probability before taking matematical statistics
I took Calculus and got a D in it. But was somehow able to take Physics. Physics was tough but it was much more interesting so the math seemed easier to tackle. I loved how Physics put a lot of Calculus into real world situations making it easier for me to grasp. I am currently retaking Calculus 1 and taking Statics! I'm studying 5+ hour everyday nowadays
My personal journey in lower level math at Riverside City College: 1) Intermediate algebra, 2) college algebra, 3) trigonometry, 4) precalculus, 5) calculus 1 and physics mechanics, 6) calculus 2 and physics electricity and magnetism, 7) calculus 3 and physics light heat and waves, 8) ordinary differential equations and statistics, 9) linear algebra My thoughts: First become proficient in algebra with (1) and (2). Then become proficient in trigonometry with (3). Precalculus (4) is half algebra and half trigonometry, so it was a thorough review and a way to gain advanced proficiency. Calculus 1 and the first physics course just complement each other well; there's a section in calculus 1 with acceleration, velocity, position functions which are related with derivatives. These formulas are used in the physics of trajectories or 2D motion aka projectile motion of any object under only the influence of gravity aka when a mass is in a state of freefall. The (6) combo was very difficult because the tests in calculus 2 could be really hard with integration techniques unless you get nice test problems and physics of E&M (electricity and magnetism) is very difficult when compared to thermodynamics physics and modern physics that come after. If you can do well with both calc 2 and physics E&M then you are awesome, hit me up, I need smarter friends. Oddly, now for (7) calculus 3 is multivariable calc and repeats topics in Calc 1 and 2 and should be pretty easy but some topics will be more confusing in 3D or R3 which is the x,y,z-coordinate system. It was super interesting to make it up to differential equations while also taking Intro Stats. I knew more complex math and could also take a step back to the beginning where I'm mainly evaluating basic formulas or learning to use charts to find probabilities or areas under a curve. Differential equations is by far my favorite subject with instructions that all remain very similar throughout the entire course (they all ask you to solve the ODE, ordinary differential equation, just using different methods depending on the difficulty or form of the ODE). Lastly, linear algebra is about knowing math in a more generalized way. It starts by using matrices which is at first a way to organize a system of equations but because the solutions of a system of equations can be a set (as with infinitely many solutions), it opens up the subject to the properties of vector spaces (a vector space is a set of "vectors" in which the vectors can be matrices, ordered pairs, ordered triples, ordered tuples or any math object where all the values in each vector obey a list of rules or conditions). Matrix operations and notations should be learned in precalculus and learning more about them in linear algebra is a way to expand what you know and think more about sets (sets become more important in upper level math classes). Last thing is that in a differential equations course you may use matrices to solve a system of differential equations by using eigenvalues and eigenvectors; this is also done in linear algebra, so there is some important overlap.
Honestly I don't know why most people recommend linear algebra so late. My uni had me take proof-based linear algebra at the same time as calc 2, and honestly as long as you have a little experience with proof writing you should be ok.
I saw a patient with a PhD is math. I said oh my grandfather also had a PhD in Mathematics. She asked what subject, so I answered, "Algebraic Topology." Her eyes got so big
15:44 🤣❤️ I love it when math books show a sense of humor. It actually seems to make a real difference for me, like I'm in a relaxed environment. I just started a Differential Equations book by Blanchard, Daveney and Hall, and its humorous tone in it helps me a lot more than the other DiffEQ books I've seen so far.
I keep tabs on what a dozen or so larger universities require for numerous fields including math, and I would note two things; the first is that they typically want you to take the proof writing course (Intro to Abstract Mathematics, Intro to Higher Mathematics etc...) concurrently with Calculus III. The second is that at larger universities (not small liberal arts colleges) they prefer for science and engineering students to take math in the applied math department where they take the differential equations class immediately after calculus with some linear algebra mixed in; while for pure math students they do make linear algebra a prerequisite for ordinary differential equations.
Thank you so much for the videos! It sounds like your videos are very accessible for all. I have always been inspired by channels like yours that bring the joy of math to people. I am actually completely blind myself, so it has always been hard to understand concepts without being able to visualize. I always wondered if given the appropriate isomorphism, if I can take anything I hear someone talk about geometrically or graphically, and just reframe the exact thing I am hearing, to different branches of mathematics like number theory, combinatorics, or linear algebra -- where it is not immediately necessary to draw a picture or visualize. My mission is to make math and science accessible to everyone; even people with disabilities. I am continuing to feel inspired every day by the work of mathematicians like yourself; keep up the excellent work !
Interesting. When I was studying Dr.Epp always taught the statistics courses. Although then I don’t remember that Discrete Mathematics was even a separate course.
I started college in '67 I went to the University of Rhode Island. I was not allowed to take anything but calculus 1. Entry level depended on SAT scores. I did have all the courses you mentioned in HS. Even in language I had to enter at Explication de text not French 1,2,3,4. It was ok.
I've always been kind of curious why US schools seem to be a couple of years behind Canada (it's almost like the first few years of US-equivalent grad school is done in our last few years of undergrad). I used Hoffman and Kunze in my second semester linear algebra course and differential geometry and topology were mandatory as a 2nd/3rd year courses. Most of the people I know also took smooth manifolds, algebraic geometry (essentially the first chapter of Hartshorne), representation theory, algebraic number theory, analytic number theory, model theory, functional analysis, + various grad courses on category theory, operator theory, harmonic analysis, lie algebras, more algebraic geometry (rest of Hartshorne), etc.
I can think of two reasons. First, in the US, literally anyone can major in math. You can start at College Algebra (essentially American 10th grade math), go through Pre-Calculus, into Calculus computation, then into Linear Algebra, Proof Writing, and ODEs. Then you go into the all-proofs courses. Take 6 years instead of 4? Fine. As I understand it, in most of the world, universities require calculus and proof writing in high school before they will let you enter the math major at all. Second, American universities require a large number of courses outside the math major, often in writing, arts, and history. As I understand it, in most of the world, math majors take almost nothing but math classes.
Also consider taking a course on numerical methods, often called numerical computation or numerical analysis. Covers how you calculate good numerical approximations to integrals, differential equation solutions, linear equation system solutions, eigenvalues/vectors of matrices, etc.
Nice video! I have to admit that from my (european) perspective it seems like there is quite the emphasis on analysis/diff equations and not so much algebra. I noticed the same in a similar video of another (american) creator, so I was wondering why this is the case. Are there generally no further algebra courses offered than the introductory ones or is this just a choice made in this video since it would be too long otherwise?
This was exactly what I was looking for. Hey, I watch your videos often dreaming of the connection between symbology and math. Thank you for the experience!
I've mentioned my mathematical journey in comments of some of your other videos covering subjects. This sequence you lay out here for Math majors is very good and mine was very similar with a few minor differences in ordering, involving elective math courses and upper level required courses. (One or two my courses I list may seem oddball compared to the year I took them, but I got special permission to take them when I did.) First off, in middle school and high school, I had a full year of Algebra 1 (8th grade,) a full year of Geometry (Freshman year,) a full year of Algebra 2 (Sophomore year,) a full year of Pre-calculus (Junior Year, half being Trig and other half elementary Calculus,) and a full year of Calculus (Senior Year). Going into college, I could've paid extra money to test out of both Analytic Geometry and Calculus, but I decided to take Calculus in College too since I assumed it would be a lot harder than High School Calculus. Freshman Year: Analytic Geometry - First Semester A Primer of Modern Algebra - First Semester - A simplistic look at some of the properties covered in the introductory topics of Abstract Algebra, such as sets, relations, associative laws, commutative laws, distributive laws, etc. Also, very fundamental introductions to groups, cyclic groups, rings, and fields. Very, very light on theory in this course. Mostly just examples and applications. Calculus I - Second Semester - Breezed through it (In hindsight, I should've paid to test out of it since I already had a full year in High School.) Axiomatic Geometry - Second Semester - Elective Course - Study of Non-Euclidean Geometries including Neutral Geometries, Spherical Geometry and Hyperbolic Geometry. Note: This course usually has prerequisites and couldn't be taken until at least Junior year in college, but I got special permission to take it my second semester of my Freshman year because I was already way ahead of my class at the time with my strong background and good grades. Besides, this course was a course that was not offered every year. My university only offered it once every 2 years Sophomore Year: Calculus II - First Semester - Breezed through it (Like Calculus I, I should've paid to test out of Calculus II.) Calculus III (Multi-variable) - Second Semester Linear Algebra - Second Semester Differential Equations - Second Semester (Most people don't take Calc III, Linear Algebra, and Differential Equations at the same time, but I was allowed to and, as a result, I got way ahead of the rest of my class.) Junior Year: Abstract Algebra I - First Semester - This was my first intense theory course and was the only math course in Undergrad I ended up getting a grade that was not an 'A'. I was in a class with mostly Seniors. This was the course where I had to learn how to write actual proofs (My university, at the time, didn't offer a course in proof writing. So it was up to the professors who taught the upper level Math courses to teach the various proof techniques along with teaching the subject matter.) Probability & Statistics I - First Semester - This course was for Math Majors, Engineering Majors, and other technical/science majors. More intensive than your usual statistic courses that gloss over the math. Design & Analysis of Experiments - First Semester - Elective Course - Basically an applied statistic course that was light on the mathematics and heavy on application using SAS programming to generate results to analyze and interpret, given an input data set. Classical Applied Analysis (aka Mathematical Physics) - First Semester - Elective course for Math majors, but required for Physics majors. This was taught out of the Math Department but the class was mostly physics and engineering majors with a few mathematics majors. It covered your usual topics that you see in standard Math Physics courses, such as even & odd functions, Fourier series (discrete and continuous) and Fourier transforms, Laplace Transforms, applied partial differential equations, heat equations, wave equations, etc. Abstract Algebra II - Second Semester - Much smoother and easier for me than Abstract Algebra I, after getting the hang of theoretical mathematics. Probability & Statistics II - Second Semester - Just the continuation of Prob & Stats I. Complex Variables - Second Semester - We used the same Fisher Book that is presented in this video. I took the course in the Spring of 1987. So, it seems this book has been around for awhile. I still have this book, along with all my other math books from undergrad and grad school. Senior Year: Advanced Calculus I (Real Variables I) - First Semester Number Theory - First Semester - Elective Course Numerical Analysis - First Semester - Elective Course Advanced Calculus II (Real Variables II) - Second Semester Non Parametric Statistics - Second Semester - An Elective Special Topics Course that wasn't part of the normal curriculum - Only a few people, including myself, signed up for the course. A very small class which resulted in a personal experience with the Professor. It was a very interesting course covering ways to analyze data without the use of the usual parameters covered in classical statistics. Discrete Mathematics - Second Semester - At the time, this was a new course which was a special topics elective course. When I took it, my class was the guinea pigs class. First time ever taught at my University back in Spring of '88. Since then, it has become part of the curriculum that is still an elective for Math Majors (but strongly recommended) and required for all Computer Science Majors. Applied Mathematical Modeling - Second Semester - An elective course that was taught every 2 years - A very fun class on learning about various everyday things you see in applied mathematics, natural sciences, engineering, computer science, and social sciences and how to develop mathematical models for them. Covered many types of models (Linear vs. Non-linear, Static vs. Dynamic, Discrete vs. Continuous, Stochastic vs. Deterministic, Implicit vs. Explicit, etc.) I don't remember some of the real-world examples we covered in the course, but I do remember covering various Growth and Decay models, regarding various populations and various interests (compound, continuous, etc.)
The above was my Undergrad Mathematical Journey. For Grad school, since I already had a full year of Advanced Calculus and and a full year of Abstract Algebra in undergrad, I didn't have to take them again in Grad School. Besides myself, I think there was only one other incoming American grad student who didn't have to take either, since they had them both in undergrad too. Most, not all, incoming foreign grad students didn't have to take them either since many were already ahead of the curve compared to the incoming American students. Since I was free to take what I wanted in my first year of grad school, I jumped right into the 600-level (2nd year grad) courses, courses along with some 500-level courses that weren't offered at my undergrad university. Mathematical Statistics I, II & III (My grad school at the time was on the quarter system. Since I graduated, they eventually moved to the Semester system like my undergrad was.) - This 500-level course was the study of the mathematics behind all those formulas you used in an undergrad Prob & Stats course. Complex Analysis I, II & III - This 600-level course was the course to prepare people for one of the Qualifier exams if they wished to pursue a PhD. After passing the qualifiers, if they wished to pursue Complex Analysis as their field of study, they'd have to take even a more advanced grad course in Complex Analysis, followed by taking and passing another Qualifier Exam before beginning their research and writing/defending their dissertation. Mathematical Programming I, II & III (covered Linear & Nonlinear) - A 500-level elective course - This course was mostly about using mathematical models to solve problems. Topics included Game Theory, Systems of equations, Decision Problems, etc.) Group Representation Theory - This was just a 1-term 600-level elective course that many people didn't sign up for. Prerequisite was Abstract Algebra. Theory Of Ordinary Differential Equations I, II & III - This course was the course to prepare people for one of the Qualifier exams if they wished to pursue a PhD. After passing the qualifiers, if they wished to pursue Ordinary Differential Equations as their field of study, they'd have to take even a more advanced grad course in Ordinary Differential Equations, followed by taking and passing another Qualifier Exam before beginning their research and writing/defending their dissertation. Numerical Analysis I, II & III - This was just a 500-level course covering the usual topics in Numerical Analysis. More intensive 600-level and 700-level courses in Numerical Analysis were offered also for those who wished to pursue a PhD and specialize in the field. Time series - This was just a 1-term 500-level course about Time series data analysis. Linear Models & Multivariable Analysis I - This was a 600-level mathematical statistics course that leaned heavily on data analysis and the use of SAS programming. After the first term, I decided that I wasn't interested in this type of course and opted out of taking the 2nd and 3 terms since I wasn't going to pursue it any further. Partial Differential Equations & Fourier Analysis I & II - This was a 2-term 500 level applied course in partial differential equations and Fourier Analysis. There was also a 600-level theory course in Partial Differential Equations for people who wanted to pursue that. Since I was taking Theory of Ordinary Differential Equations at the same time, I decided to just take the applied course instead of the theory course for partial differential equations. After I received my Masters degree, I continued taking graduate mathematics courses while working on my Computer Science Masters Degree. During that time, I took courses I didn't take while working on my Math degree. I took courses in Combinatorics, Applied Linear Algebra, Applied Regression Analysis, Applied Probability, Numerical Methods & Applied Math, and Graph Theory via the Math Department. Also, took Computability Theory, Computational Complexity Theory, Automata Theory and Formal Languages via the Computer Science Department and a course in Stochastic Processes via the Electrical & Computer Engineering Department. For Masters Degrees, I exceeded both the breadth and depth requirements for Math Study. Yet, I still feel I needed/wanted more. For example, I would've liked to have taken courses in Functional Analysis and Theory of Partial Differential Equations, but just couldn't find the time. The only things I do not regret not taking were Topology (never interested me) and Advanced Graduate courses in Abstract Algebra (my least favorite math subject) and Advanced Real Analysis (Advanced Calculus as an undergrad was enough and sufficient for what I wanted out of Grad School.)
Hope, you have the same for Engineering. I have skimmed through your videos and almost anything related to Engineering is - advice, hardness etc. I would be nice, if there is a video that specifically says something like - "Exact pathway (course by course && book by book) for Scientists and Engineers". The exact instruction following which, someone becoming non-Math-majon scientist or engineer, never ever in his life would have any kind of Math-related problems ever! He just does his, let us say, Quantum Physics pathway, or Theoretical Physics, or Mechanical Engineering, or Electrical Engineering, Aerospace anything anything anything and what he does - he just picks any book on his speciality and just reads it like a novel - so much proficiency he's got in Math. So, what might be the pathway for that level? Or, does it differ drastically for Science and Engineering? So what is it for Engineering, in that case? By the way, I have skimmed through your website and it does not include any pathway data. Would be nice to add something on that. Like, what do you need math for? Different pathways, with books, workbooks, your tutorials and avice videos from youtube for Math majors, Engineering Majors and Physics majors and so on.
Hey math sorcerer! Any thoughts about if a Functional Analysis course should be in the essentials for the math major? To most places even offer a course like this in undergrad? Great video
You should also check out the textbook “Aerodynamics for Naval Aviators”. You don’t necessarily have to be a pilot, as the book focuses on aerospace engineering concepts rather than flying techniques. Has everything from advanced weather, aerodynamics, engine characteristics & performance, etc. I’d definitely recommend you pass College Physics at a minimum though before even touching that book!
Man, you should do a review of I.E Irodov's Problems in General Physics... it is a book that is notorious among JEE aspirants in India... it's also a classic, so I hope you' re gonna like it! Also, you make absolutely genius and relatable content! Love from India.
Can you also make video on physics books. You gave a book on Physics 1. Can you also share books Physics 2 and 3 if you have and some book on other physics related courses math grads take.
Do you have any thoughts on using AI for math learning? Should I be asking GPT its opinions on calculus or just stick with the books? Will AI videos like on the Onlocklearning channel for example will be the way new students learn it? ruclips.net/user/shortsPGK-RXA8Ye8
I would like help. People keep telling me to go back to the fundamentals to strengthen my math. But i don’t know what the fundemntals are. Just a little bit of guidance please 😊
Hey dude i bought your calc II course and not even a day went buy and I see that the course is now 12.99 instead of 84.99. I am a college student so that really hurts. I was wondering if you could help me out?
I have a lot to critique about math majors programs. These needs more than 4 years and a betters sequence of classes. The same applies to physics. Mathematics requires exhaustive study and every theorem proven in detail without skipping steps. Calculus needs four courses. Ordinary differential equations needs 2 courses. Tensor calculus needs three courses. Combinatorial techniques needs two courses. Discrete math needs 3 courses. Proof techniques needs 3 courses. Abstract algebra needs 4 courses. Special functions needs 3 courses. Differential geometry needs 4 courses where geometry is included. Probability and Statistics needs 4 courses. Advanced calculus needs 3 courses. Logic 2 courses. Topology 2 courses. This is the very basics to do something with math such as physics, programming and simulation, actuarial science, rocketry and aerospace. People may carry on studying math much deeper if they want to exploring topics in metamathematics. This video gives a gentle introduction. But this needs more detail and depth. Serious practitioners dedicates 8 to 10 years forming this skill.
Depends how deep you go into each subject, but in general textbooks are planned around a 1 - 3 semester course. So anywhere from 4 - 12 months per book if you follow that format. Most authors discuss course planning in the intro. Depends how quickly you learn as well, obviously.
I have autism and I wanted to know if I can succeed by getting a PhD in physics at the university of Cambridge. But I’m having doubts if my IQ is high enough to master the concepts because I know that IQ is important when getting into these sort of fields
When learning mathematical analysis...I would not recommend Baby Rudin as your first book on the subject. It will test your wits and give you solid nightmares. Trust me.
Amann and Escher (I-III) is good
It's very good book I think, even for beginners. When you pass the first chapter, everything follows imo.
@@mathslover98 that book is legendary but if you have never studied any set theory or logic or proof writing, it will prove very tough going for a beginner.
@@mathslover98Rudin is absolutely not beginner friendly. Understanding Analysis by Abbot is much better for beginners.
Yes, baby Rudin is very hard to understand but elementary classical analysis by Marsden is more friendly and complete for beginners
Interestingly enough, I never had to take a Trig course, neither in high school nor university. All of my experience with Trig came from what was taught in the Pre-Calc and high school geometry. That being said, I think the trig from precalc helped me understand justttt enough trig to make it through and do ok Physics I and Calc II (trig comes up a lot it seems). Curious if anyone else has a similar experience.
Interesting !!
Exact same experience for me, the way my school did it was pre-calculus was an accelerated class with half of it reviewing Algebra and the other half reviewing trigonometry, so essentially you were taking two classes in one semester. I loved that structure personally and I think it set me up well in half the time.
Great vid as always! I took Discrete Math in college and we used Susanna Epp textbook....in my option the best introduction to Logic and Set Theory I have seen (beginning chapters). Highly recommended.
I took PDEs my senior year. Took differential equations for the first time in grad school, no problem.
It's also important to have a solid introduction to probability and stochastic processes
"Discrete Mathematics and Its Applications" (4th ed) by Kenneth Rosen is a gem. (My copy is a Recycle Bin Rescue.) In addition to presenting Discrete Mathematics (aka, "Counting Gone Wild"), it details the tools needed to write proofs and think Mathematics like a Mathematician. There are also lots of examples and applications, so you can have good answers when your friends ask, "Why do you study this crazy Math Stuff?"
"What’s all this *Discrete Math* stuff?", as Bob Pease might have said.
(He was an Analog Electronics Master.)
literally, I thank you from the deepest point ever discovered in my heart for your existence.
That Mathematical Statistics book by Wackerly is THE ONE. I have about a dozen books on the topic, and this is the one that got me through math stat. I found that if I mostly tuned out the instructor and instead read the class text plus Wackerly, and worked problems in Wackerly, I understood the course much better. The solutions manual is a must, too.
10:25 That linear algebra book came from my alma mater, Grossmont College (AS, 1999)!!!!! Thank you lots for featuring THAT copy!!!!! ♥♥♥♥♥ 🙂🙂🙂🙂🙂
In other universities outside the US, advanced calculus 1 and advanced calculus 2 are the prerequisites before you could take Real analysis and Complex analysis. Adv Calc 1-2 (Intro to Analysis) are like the stepping stones to further analysis courses.
Would it be alright if I made a request?
I'm trying to get some of the deeper theory behind AI in a nutshell, and I think I've identified some milestone topics I need to traverse:
- Phase transitions in statistical mechanics (only classical though), and the renormalisation group
- Bayesian Stats (up to the point where I'm comfortable with the theta parameter)
- Theoretical CS (akin to Sipser's book/MITOCW course)
- I guess I'll know that I'm good when I fully understand all the reasons the Gaussian is critical to statistical theory (at the moment I know about CLT, identity under FT, the relationship it has with the Frobenius norm)
- Information and Control Theory, particularly their combination. I understand there are four big asymptotic limiting theorems here, and I need to understand all of them. So far I have the Shannon-Nyquist limit down.
- Less important, Topois theory for the generalisation of classical logic, and Category theory for the foundations of mathematics (plus I like the commutative diagrams).
I was wondering if you have any recommended prerequisite material, maybe even a course structure like this video, where one could end up with the above points satisfied by the end of it all? Cheers.
This is the best course sequence for everyone, not only math majors!
That's what i looking for! Thnank you very much!
That ODE by Ross is excellent. I used it in the mid 1970s for a summer class. It almost made the class fun. Almost. Had occasion to use it again within the last 10 years.
I think a mistake i made when taking mathematical stats is not understanding probability theory enough beforehand. it would be a good idea to read a book like a first course in probability before taking matematical statistics
Great list! I’m a current math undergrad, and I love collecting older textbooks, so I’ll be on the look for these!
I took Calculus and got a D in it. But was somehow able to take Physics.
Physics was tough but it was much more interesting so the math seemed easier to tackle. I loved how Physics put a lot of Calculus into real world situations making it easier for me to grasp.
I am currently retaking Calculus 1 and taking Statics!
I'm studying 5+ hour everyday nowadays
My personal journey in lower level math at Riverside City College: 1) Intermediate algebra, 2) college algebra, 3) trigonometry, 4) precalculus, 5) calculus 1 and physics mechanics, 6) calculus 2 and physics electricity and magnetism, 7) calculus 3 and physics light heat and waves, 8) ordinary differential equations and statistics, 9) linear algebra
My thoughts: First become proficient in algebra with (1) and (2). Then become proficient in trigonometry with (3). Precalculus (4) is half algebra and half trigonometry, so it was a thorough review and a way to gain advanced proficiency. Calculus 1 and the first physics course just complement each other well; there's a section in calculus 1 with acceleration, velocity, position functions which are related with derivatives. These formulas are used in the physics of trajectories or 2D motion aka projectile motion of any object under only the influence of gravity aka when a mass is in a state of freefall. The (6) combo was very difficult because the tests in calculus 2 could be really hard with integration techniques unless you get nice test problems and physics of E&M (electricity and magnetism) is very difficult when compared to thermodynamics physics and modern physics that come after. If you can do well with both calc 2 and physics E&M then you are awesome, hit me up, I need smarter friends. Oddly, now for (7) calculus 3 is multivariable calc and repeats topics in Calc 1 and 2 and should be pretty easy but some topics will be more confusing in 3D or R3 which is the x,y,z-coordinate system. It was super interesting to make it up to differential equations while also taking Intro Stats. I knew more complex math and could also take a step back to the beginning where I'm mainly evaluating basic formulas or learning to use charts to find probabilities or areas under a curve. Differential equations is by far my favorite subject with instructions that all remain very similar throughout the entire course (they all ask you to solve the ODE, ordinary differential equation, just using different methods depending on the difficulty or form of the ODE). Lastly, linear algebra is about knowing math in a more generalized way. It starts by using matrices which is at first a way to organize a system of equations but because the solutions of a system of equations can be a set (as with infinitely many solutions), it opens up the subject to the properties of vector spaces (a vector space is a set of "vectors" in which the vectors can be matrices, ordered pairs, ordered triples, ordered tuples or any math object where all the values in each vector obey a list of rules or conditions). Matrix operations and notations should be learned in precalculus and learning more about them in linear algebra is a way to expand what you know and think more about sets (sets become more important in upper level math classes). Last thing is that in a differential equations course you may use matrices to solve a system of differential equations by using eigenvalues and eigenvectors; this is also done in linear algebra, so there is some important overlap.
Honestly I don't know why most people recommend linear algebra so late. My uni had me take proof-based linear algebra at the same time as calc 2, and honestly as long as you have a little experience with proof writing you should be ok.
I saw a patient with a PhD is math. I said oh my grandfather also had a PhD in Mathematics. She asked what subject, so I answered, "Algebraic Topology." Her eyes got so big
Great recommendations! Saludos desde Monterrey, México.
Saludos!!
15:44 🤣❤️ I love it when math books show a sense of humor. It actually seems to make a real difference for me, like I'm in a relaxed environment. I just started a Differential Equations book by Blanchard, Daveney and Hall, and its humorous tone in it helps me a lot more than the other DiffEQ books I've seen so far.
Same!!
I keep tabs on what a dozen or so larger universities require for numerous fields including math, and I would note two things; the first is that they typically want you to take the proof writing course (Intro to Abstract Mathematics, Intro to Higher Mathematics etc...) concurrently with Calculus III. The second is that at larger universities (not small liberal arts colleges) they prefer for science and engineering students to take math in the applied math department where they take the differential equations class immediately after calculus with some linear algebra mixed in; while for pure math students they do make linear algebra a prerequisite for ordinary differential equations.
My real analysis teacher in undergrad had us do *all* the proofs in front of the class. We had no book, the teacher led us through the concepts.
Thank you so much for the videos! It sounds like your videos are very accessible for all. I have always been inspired by channels like yours that bring the joy of math to people. I am actually completely blind myself, so it has always been hard to understand concepts without being able to visualize. I always wondered if given the appropriate isomorphism, if I can take anything I hear someone talk about geometrically or graphically, and just reframe the exact thing I am hearing, to different branches of mathematics like number theory, combinatorics, or linear algebra -- where it is not immediately necessary to draw a picture or visualize. My mission is to make math and science accessible to everyone; even people with disabilities. I am continuing to feel inspired every day by the work of mathematicians like yourself; keep up the excellent work
!
Theoretical statistics and probability theory .
Interesting. When I was studying Dr.Epp always taught the statistics courses. Although then I don’t remember that Discrete Mathematics was even a separate course.
12:44 Mathematical Statistics with Application by WMS is currently the bane of my existence 😤
This is going to be very helpful to me.
It's good that you included Physics, because why else would one study Math?
Hehehehehe
@@TheMathSorcerer Raise your hand if you took Calculus-based Physics with Calculus being a co-requisite!
YEEE!! HAAW!!
I started college in '67 I went to the University of Rhode Island. I was not allowed to take anything but calculus 1. Entry level depended on SAT scores. I did have all the courses you mentioned in HS. Even in language I had to enter at Explication de text not French 1,2,3,4. It was ok.
That Erdos cameo was enthralling
Great talk on the best sequence.
I've always been kind of curious why US schools seem to be a couple of years behind Canada (it's almost like the first few years of US-equivalent grad school is done in our last few years of undergrad). I used Hoffman and Kunze in my second semester linear algebra course and differential geometry and topology were mandatory as a 2nd/3rd year courses. Most of the people I know also took smooth manifolds, algebraic geometry (essentially the first chapter of Hartshorne), representation theory, algebraic number theory, analytic number theory, model theory, functional analysis, + various grad courses on category theory, operator theory, harmonic analysis, lie algebras, more algebraic geometry (rest of Hartshorne), etc.
I can think of two reasons. First, in the US, literally anyone can major in math. You can start at College Algebra (essentially American 10th grade math), go through Pre-Calculus, into Calculus computation, then into Linear Algebra, Proof Writing, and ODEs. Then you go into the all-proofs courses. Take 6 years instead of 4? Fine. As I understand it, in most of the world, universities require calculus and proof writing in high school before they will let you enter the math major at all. Second, American universities require a large number of courses outside the math major, often in writing, arts, and history. As I understand it, in most of the world, math majors take almost nothing but math classes.
So they can vote for T 😂😂😂
This video has got me wondering if I should come back to Math.
Also consider taking a course on numerical methods, often called numerical computation or numerical analysis. Covers how you calculate good numerical approximations to integrals, differential equation solutions, linear equation system solutions, eigenvalues/vectors of matrices, etc.
Nice video! I have to admit that from my (european) perspective it seems like there is quite the emphasis on analysis/diff equations and not so much algebra. I noticed the same in a similar video of another (american) creator, so I was wondering why this is the case. Are there generally no further algebra courses offered than the introductory ones or is this just a choice made in this video since it would be too long otherwise?
This was exactly what I was looking for.
Hey, I watch your videos often dreaming of the connection between symbology and math.
Thank you for the experience!
Point of mathematics books.
Never learn statistics before probability
It's Paul "Air-dish".
I've mentioned my mathematical journey in comments of some of your other videos covering subjects. This sequence you lay out here for Math majors is very good and mine was very similar with a few minor differences in ordering, involving elective math courses and upper level required courses. (One or two my courses I list may seem oddball compared to the year I took them, but I got special permission to take them when I did.)
First off, in middle school and high school, I had a full year of Algebra 1 (8th grade,) a full year of Geometry (Freshman year,) a full year of Algebra 2 (Sophomore year,) a full year of Pre-calculus (Junior Year, half being Trig and other half elementary Calculus,) and a full year of Calculus (Senior Year).
Going into college, I could've paid extra money to test out of both Analytic Geometry and Calculus, but I decided to take Calculus in College too since I assumed it would be a lot harder than High School Calculus.
Freshman Year:
Analytic Geometry - First Semester
A Primer of Modern Algebra - First Semester - A simplistic look at some of the properties covered in the introductory topics of Abstract Algebra, such as sets, relations, associative laws, commutative laws, distributive laws, etc. Also, very fundamental introductions to groups, cyclic groups, rings, and fields. Very, very light on theory in this course. Mostly just examples and applications.
Calculus I - Second Semester - Breezed through it (In hindsight, I should've paid to test out of it since I already had a full year in High School.)
Axiomatic Geometry - Second Semester - Elective Course - Study of Non-Euclidean Geometries including Neutral Geometries, Spherical Geometry and Hyperbolic Geometry. Note: This course usually has prerequisites and couldn't be taken until at least Junior year in college, but I got special permission to take it my second semester of my Freshman year because I was already way ahead of my class at the time with my strong background and good grades. Besides, this course was a course that was not offered every year. My university only offered it once every 2 years
Sophomore Year:
Calculus II - First Semester - Breezed through it (Like Calculus I, I should've paid to test out of Calculus II.)
Calculus III (Multi-variable) - Second Semester
Linear Algebra - Second Semester
Differential Equations - Second Semester
(Most people don't take Calc III, Linear Algebra, and Differential Equations at the same time, but I was allowed to and, as a result, I got way ahead of the rest of my class.)
Junior Year:
Abstract Algebra I - First Semester - This was my first intense theory course and was the only math course in Undergrad I ended up getting a grade that was not an 'A'. I was in a class with mostly Seniors. This was the course where I had to learn how to write actual proofs (My university, at the time, didn't offer a course in proof writing. So it was up to the professors who taught the upper level Math courses to teach the various proof techniques along with teaching the subject matter.)
Probability & Statistics I - First Semester - This course was for Math Majors, Engineering Majors, and other technical/science majors. More intensive than your usual statistic courses that gloss over the math.
Design & Analysis of Experiments - First Semester - Elective Course - Basically an applied statistic course that was light on the mathematics and heavy on application using SAS programming to generate results to analyze and interpret, given an input data set.
Classical Applied Analysis (aka Mathematical Physics) - First Semester - Elective course for Math majors, but required for Physics majors. This was taught out of the Math Department but the class was mostly physics and engineering majors with a few mathematics majors. It covered your usual topics that you see in standard Math Physics courses, such as even & odd functions, Fourier series (discrete and continuous) and Fourier transforms, Laplace Transforms, applied partial differential equations, heat equations, wave equations, etc.
Abstract Algebra II - Second Semester - Much smoother and easier for me than Abstract Algebra I, after getting the hang of theoretical mathematics.
Probability & Statistics II - Second Semester - Just the continuation of Prob & Stats I.
Complex Variables - Second Semester - We used the same Fisher Book that is presented in this video. I took the course in the Spring of 1987. So, it seems this book has been around for awhile. I still have this book, along with all my other math books from undergrad and grad school.
Senior Year:
Advanced Calculus I (Real Variables I) - First Semester
Number Theory - First Semester - Elective Course
Numerical Analysis - First Semester - Elective Course
Advanced Calculus II (Real Variables II) - Second Semester
Non Parametric Statistics - Second Semester - An Elective Special Topics Course that wasn't part of the normal curriculum - Only a few people, including myself, signed up for the course. A very small class which resulted in a personal experience with the Professor. It was a very interesting course covering ways to analyze data without the use of the usual parameters covered in classical statistics.
Discrete Mathematics - Second Semester - At the time, this was a new course which was a special topics elective course. When I took it, my class was the guinea pigs class. First time ever taught at my University back in Spring of '88. Since then, it has become part of the curriculum that is still an elective for Math Majors (but strongly recommended) and required for all Computer Science Majors.
Applied Mathematical Modeling - Second Semester - An elective course that was taught every 2 years - A very fun class on learning about various everyday things you see in applied mathematics, natural sciences, engineering, computer science, and social sciences and how to develop mathematical models for them. Covered many types of models (Linear vs. Non-linear, Static vs. Dynamic, Discrete vs. Continuous, Stochastic vs. Deterministic, Implicit vs. Explicit, etc.) I don't remember some of the real-world examples we covered in the course, but I do remember covering various Growth and Decay models, regarding various populations and various interests (compound, continuous, etc.)
The above was my Undergrad Mathematical Journey.
For Grad school, since I already had a full year of Advanced Calculus and and a full year of Abstract Algebra in undergrad, I didn't have to take them again in Grad School. Besides myself, I think there was only one other incoming American grad student who didn't have to take either, since they had them both in undergrad too. Most, not all, incoming foreign grad students didn't have to take them either since many were already ahead of the curve compared to the incoming American students.
Since I was free to take what I wanted in my first year of grad school, I jumped right into the 600-level (2nd year grad) courses, courses along with some 500-level courses that weren't offered at my undergrad university.
Mathematical Statistics I, II & III (My grad school at the time was on the quarter system. Since I graduated, they eventually moved to the Semester system like my undergrad was.) - This 500-level course was the study of the mathematics behind all those formulas you used in an undergrad Prob & Stats course.
Complex Analysis I, II & III - This 600-level course was the course to prepare people for one of the Qualifier exams if they wished to pursue a PhD. After passing the qualifiers, if they wished to pursue Complex Analysis as their field of study, they'd have to take even a more advanced grad course in Complex Analysis, followed by taking and passing another Qualifier Exam before beginning their research and writing/defending their dissertation.
Mathematical Programming I, II & III (covered Linear & Nonlinear) - A 500-level elective course - This course was mostly about using mathematical models to solve problems. Topics included Game Theory, Systems of equations, Decision Problems, etc.)
Group Representation Theory - This was just a 1-term 600-level elective course that many people didn't sign up for. Prerequisite was Abstract Algebra.
Theory Of Ordinary Differential Equations I, II & III - This course was the course to prepare people for one of the Qualifier exams if they wished to pursue a PhD. After passing the qualifiers, if they wished to pursue Ordinary Differential Equations as their field of study, they'd have to take even a more advanced grad course in Ordinary Differential Equations, followed by taking and passing another Qualifier Exam before beginning their research and writing/defending their dissertation.
Numerical Analysis I, II & III - This was just a 500-level course covering the usual topics in Numerical Analysis. More intensive 600-level and 700-level courses in Numerical Analysis were offered also for those who wished to pursue a PhD and specialize in the field.
Time series - This was just a 1-term 500-level course about Time series data analysis.
Linear Models & Multivariable Analysis I - This was a 600-level mathematical statistics course that leaned heavily on data analysis and the use of SAS programming. After the first term, I decided that I wasn't interested in this type of course and opted out of taking the 2nd and 3 terms since I wasn't going to pursue it any further.
Partial Differential Equations & Fourier Analysis I & II - This was a 2-term 500 level applied course in partial differential equations and Fourier Analysis. There was also a 600-level theory course in Partial Differential Equations for people who wanted to pursue that. Since I was taking Theory of Ordinary Differential Equations at the same time, I decided to just take the applied course instead of the theory course for partial differential equations.
After I received my Masters degree, I continued taking graduate mathematics courses while working on my Computer Science Masters Degree. During that time, I took courses I didn't take while working on my Math degree. I took courses in Combinatorics, Applied Linear Algebra, Applied Regression Analysis, Applied Probability, Numerical Methods & Applied Math, and Graph Theory via the Math Department. Also, took Computability Theory, Computational Complexity Theory, Automata Theory and Formal Languages via the Computer Science Department and a course in Stochastic Processes via the Electrical & Computer Engineering Department.
For Masters Degrees, I exceeded both the breadth and depth requirements for Math Study. Yet, I still feel I needed/wanted more. For example, I would've liked to have taken courses in Functional Analysis and Theory of Partial Differential Equations, but just couldn't find the time. The only things I do not regret not taking were Topology (never interested me) and Advanced Graduate courses in Abstract Algebra (my least favorite math subject) and Advanced Real Analysis (Advanced Calculus as an undergrad was enough and sufficient for what I wanted out of Grad School.)
Thank you for this video
Hey, Sorcerer. I'm struggling with combinatorics. Can you do a series on it, about good combinatorics books.
Hope, you have the same for Engineering. I have skimmed through your videos and almost anything related to Engineering is - advice, hardness etc. I would be nice, if there is a video that specifically says something like - "Exact pathway (course by course && book by book) for Scientists and Engineers". The exact instruction following which, someone becoming non-Math-majon scientist or engineer, never ever in his life would have any kind of Math-related problems ever! He just does his, let us say, Quantum Physics pathway, or Theoretical Physics, or Mechanical Engineering, or Electrical Engineering, Aerospace anything anything anything and what he does - he just picks any book on his speciality and just reads it like a novel - so much proficiency he's got in Math. So, what might be the pathway for that level? Or, does it differ drastically for Science and Engineering? So what is it for Engineering, in that case? By the way, I have skimmed through your website and it does not include any pathway data. Would be nice to add something on that. Like, what do you need math for? Different pathways, with books, workbooks, your tutorials and avice videos from youtube for Math majors, Engineering Majors and Physics majors and so on.
Hey math sorcerer! Any thoughts about if a Functional Analysis course should be in the essentials for the math major? To most places even offer a course like this in undergrad? Great video
First , Please do one for Aspiring Aeronautical Engineers❤
That'd be awesome!
You should also check out the textbook “Aerodynamics for Naval Aviators”. You don’t necessarily have to be a pilot, as the book focuses on aerospace engineering concepts rather than flying techniques. Has everything from advanced weather, aerodynamics, engine characteristics & performance, etc. I’d definitely recommend you pass College Physics at a minimum though before even touching that book!
@@AV4Life No Way someone mentioned this book. Im 17 now and my Grandfather who was a pilot Gave me that Book Last year!
The thing that I don't like about fitzpatrick is that it's so think on answers.
Do you have a plaintext list of all books you have recommended throughout the years?
Man, you should do a review of I.E Irodov's Problems in General Physics... it is a book that is notorious among JEE aspirants in India... it's also a classic, so I hope you' re gonna like it! Also, you make absolutely genius and relatable content! Love from India.
I will check it out! Thank you !’
Nah bhai.
Muy bueno! quería saber si recomiendas el libro "Plane & Solid Geometry Wentworth-Smith" para aprender geometría de manera formal.
I would love if you review Rogan's recent between Eric Weinstin & Terrence Howard. Humble request❤
Love from INDIA
Can you also make video on physics books. You gave a book on Physics 1. Can you also share books Physics 2 and 3 if you have and some book on other physics related courses math grads take.
Make a list of books for learning geometry and geometric approach applicable in various domains in research
Do you have to read the entire textbook? This will take forever, especially for non students. Do you have a video for going through textbooks?
Nah, you can try out the exercises straight away and refer back to the text when needed. Kinda like an iterative approach.
@@Postmodern368 ty
Thank you man! 🙏
Can you review Introduction to smooth manifolds by John m lee
Do you have any thoughts on using AI for math learning?
Should I be asking GPT its opinions on calculus or just stick with the books?
Will AI videos like on the Onlocklearning channel for example will be the way new students learn it?
ruclips.net/user/shortsPGK-RXA8Ye8
gr8
When will you talk about surreal numbers?
can you do best math courses to take for engineers?
I would like help. People keep telling me to go back to the fundamentals to strengthen my math. But i don’t know what the fundemntals are. Just a little bit of guidance please 😊
i just know bro gonns to his bookshelf
is it for general math or applied math or pure math?
Hey dude i bought your calc II course and not even a day went buy and I see that the course is now 12.99 instead of 84.99. I am a college student so that really hurts. I was wondering if you could help me out?
Any astronomy book recommendations?
I have a lot to critique about math majors programs. These needs more than 4 years and a betters sequence of classes. The same applies to physics. Mathematics requires exhaustive study and every theorem proven in detail without skipping steps.
Calculus needs four courses.
Ordinary differential equations needs 2 courses.
Tensor calculus needs three courses.
Combinatorial techniques needs two courses.
Discrete math needs 3 courses.
Proof techniques needs 3 courses.
Abstract algebra needs 4 courses.
Special functions needs 3 courses.
Differential geometry needs 4 courses where geometry is included.
Probability and Statistics needs 4 courses.
Advanced calculus needs 3 courses.
Logic 2 courses.
Topology 2 courses.
This is the very basics to do something with math such as physics, programming and simulation, actuarial science, rocketry and aerospace. People may carry on studying math much deeper if they want to exploring topics in metamathematics.
This video gives a gentle introduction. But this needs more detail and depth. Serious practitioners dedicates 8 to 10 years forming this skill.
Mathsorcery, please bring graduate mathematics books
where does measure theory come in ?
How long it should take to learn each course?
Depends how deep you go into each subject, but in general textbooks are planned around a 1 - 3 semester course. So anywhere from 4 - 12 months per book if you follow that format. Most authors discuss course planning in the intro. Depends how quickly you learn as well, obviously.
What about Set theory?
And the geometries?
Can anyone recommend a pre-intermediate algebra book or beginner algebra?
Elementary Algebra by Sullivan, Struve and Mazzarella
What's Your Height Bro (Math Sorcerer)
I have autism and I wanted to know if I can succeed by getting a PhD in physics at the university of Cambridge. But I’m having doubts if my IQ is high enough to master the concepts because I know that IQ is important when getting into these sort of fields
You might be getting an email from me soon.
Math => LOVE,,,,