Learn Vector Calculus With This Book

There are not that many books that are completely devoted to the subject of Vector Calculus. In this video I will show you a book that is devoted to this beautiful area of mathematics which you can use for self-study or to supplement your current Calculus course. The book is titled Vector Calculus and it was written by Susan Jane Colley.
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Thank you:)

[@rithikkhanna1053]

I don't think you mentioned the best part of this book: She fully covers functions from m-space to n-space, leading her to define the derivative as a Jacobian Matrix, use the Hessian to classify critical points, and state the Change of Variables Theorem in full generality. This is my favorite book for multivariable differential calculus; for integral calculus, I prefer Bray's Multivariable Calculus.

[@eswyatt]

Yes! And the so-called "bordered Hessian" in constrained optimization problems! Unheard of in any calculus or linear algebra book! [He actually landed on it @ 6:41 !!]

[@ivan5187]

I love Bray's work too. My mid-life re-learning has started with it.

[@jonmartin3026]

You have no idea how much you've made me spend on books this month. I need a new bookshelf already and this looks like another text I need to get :)

[@benhill3098]

I have this exact edition. It was 6.69 U.S.D. when I got it, but It was free for me because I applied some reward points from the website where I ordered this book. Indeed, I couldn't find other vector calculus books at the time. I'm glad to see this is a great resource for learning vector calculus when I'm ready for it. Thanks!

[@arims8346]

tysm!! i'm a mechatronics engineering major and just now i'm (finally) in calculus 3 and i was looking for book recommendations. Truly thankful to give the spotlight to hidden treasures :D

[@themessiahcodemovement4540]

Thank you sir I really appreciate you sharing your knowledge it is helping me grow in my mathematics

[@vnever9078]

Hey Math Sorcerer, great review as usual. Another great book on the same topic is _Vector Analysis_ by Homer E. Newell Jr. Great text with many physical applications and yet a thorough mathematical treatment!

[@madhavpr]

This is a beautiful book. The fourth edition has an additional chapter on differential forms. Great stuff!! :D I keep going back to this book for refreshing my knowledge about anything related to multivariable calculus (especially the three big theorems of vector analysis- Green, Gauss and Stokes) and physics (mechanics or electromagnetism).

[@Taylor-rx4yb]

Wow differential forms! I don't think I've ever seen those discussed in a calculus text (well explicitly at least). Awesome book!

[@minimalrho]

I've always stopped at Green's theorem when teaching Calc 3, never getting into surface integrals. Maybe because I'm teaching at a community college, but the pacing is hard to get any more material in that class beyond that. The idea of going through all of vector calculus in that class boggles my mind.

[@TheMathSorcerer]

Agreed, it would have to be insanely fast lol.

[@vade248721]

I got a C in multivariable calculus because of Greens and Stokes. Things were not too bad until we got there. Now I feel cheated. This is the first time I ever heard this opinion about stopping before Green's theorem. I feel cheated but actually feel better now that you said it. I learned it at a community college too.

[@ThuyNguyen-lf2sm]

Where can I find a good Calc BC tutor? Thanks.

[@alexgian9313]

Good call, this book! As a refresher, but also as a preparation for DG I found it hit the spot. Thanks. Thinking of doing Morgan's Riemannian Geometry after this.

[@AceOfHearts001]

Love this type of layout. Considerate to the learner.

[@evionlast]

This book is very good for self learning you'll have a sequence of exercises that is very instructive from practical calculation to theory development, I'll say you'd want to do more after this naturally it's very introductory.

[@evionlast]

@@Hector-jj1jg Yes some linear Algebra is needed the book introduces most the of Linear Algebra you'll need, but it would be helpful to have a linear algebra course in your repertoire, you'll need to know about euclidean vector spaces and their algebra, matrix operations, determinants, transformations, basis and eigenvalues and eigenvectors, however the treatment is very accessible, The excercises are really helpful some are direct development of the theory but split in parts, if you don't know much about proofs it will guide you, then you can move to more advanced texts, a text I would recommend as a follow-up is Advanced Calculus a differential forms approach, Harold Edwards very good book a little bit more difficult but worthy, for a full modern analysis book: A modern introduction to mathematical analysis, Alessandro Fonda, Italians write very good math books In swear, beware though this is a different game... it has no excercises it's a full theory book you'll be forced to understand the theory and do it like excercises, this is what I did however he recommends this excercises book Solving Problems in Mathematical Analysis by T. Radozycki, but I don't know it.

[@billmorrigan386]

I have the 4th edition. It's an excellent book. Standard calculus books have 1000+ pages but vector calculus and multiple variables are not as deeply and comprehensively covered as in this book. Multiple variables are of course a much more advanced part of calculus. This book nails it providing comprehensive coverage. It's just like ODE are not covered deeply either in calculus books. You need a separate book for that. Unfortunately, they are not that many good books on vector calculus. This is a must-have book for those who want to gain a superior and advanced knowledge of calculus. It blows out of the water any standard calculus book when it comes to multiple variables and vector calculus (Calc 3 and 4). This book is my top recommendation for those who want to master calculus very deeply at a serious level. However, there's no ODE in it but Boyce DiPrima's book solves that. By the way, Colley has some good book recommendations at the end of her _Vector Calculus._ It's not just a haphazard list of books. It's a well structured recommendation of important math books. It is worth mentioning that analytic geometry has been severely reduced. Its remnants are now part of calculus. However there's a good separate book on analytic geometry too. It's by Douglas. It's also very expensive. Analytic geometry should precede vector calculus. Luckily, there are electronic versions of books nowadays and ridiculous price tags can be avoided.

[@blizzsoft5910]

do you have any thoughts about Hubbard's vector calculus? it seems maybe you likely know this book being known that it is must-have book as well.

[@billmorrigan386]

@@Hector-jj1jg No, but I consider the _Elementary Algebra and Trigonometry_ by Stewart (or alternatively the same title by Swokowski or Larson) to be absolute prerequisites of doing college math. Regular HS books, precalc books and various remedial courses just don't cut it in my opinion. They have too many gaps here and there. Still, they are a good remedy for the poor level of HS math. They are just not as comprehensive as _Elementary Algebra and Trigonometry,_ which itself is far from being comprehensive. Now then, of course Calculus 1 and 2 are also prerequisites for Colley's text. Linear algebra is not but you would be much better off knowing the basics of linear algebra and analytic geometry. She does cover the necessary topics in both linear algebra and analytic geometry but that's not the same as having good separate intro books on those topics. She basically plugs some heinous holes. Of course, she knows students know nothing. She knows the most glaring holes need to be plugged. So she does plug terrible gaps and holes students have. A lot of kudos to her for that but then again, such stuff is not a replacement for the basics of analytic geometry (nor is it a good replacement for the basics of linear algebra because she provides, like, one chapter on the very basics of matrices and determinants). Knowing analytic geometry and linear algebra beyond one chapter is gonna be beneficial. I mean working through and figuring out the logic (proofs and derivations) in the first several chapters in a good book on analytic geometry would very beneficial for Colley's text. The same applies to linear algebra but analytic geometry is more important for Colley's text. There's not much linear algebra in her text. A superb textbook on analytic geometry is by Douglas. A superb text on linear algebra is by Kolman: _Elementary Linear Algebra with Applications._ I do think knowing the core material, the core proofs and derivations in the first chapters of such books are very beneficial for Colley's text and for gaining good math knowledge at large. Most students do end up rote-learning most of the stuff, including proofs and derivations without really understanding the logic behind. That's a huge problem, a scourge of math education. Like I said, there's not much linear algebra in Colley's text but in terms of analytic geometry, I think all the material in Douglas's book will come in handy. I mean not just the first chapters (they are on plane geometry!) but the second part of Douglas's book that deals with the 3D space. To recap, Colley's text doesn't require analytic algebra or linear algebra. She does cover the basics students need to know. However, it's some sort of makeshift remedy for the more detailed knowledge/coverage a proper book on analytic geometry provides. The same applies to linear algebra but to a lesser extent. There's not much of linear algebra there but determinants and matrices are there. But they are even in the book _Elementary Algebra and Trigonometry._

[@billmorrigan386]

@@Hector-jj1jg Nowadays, analytic geometry is spread around in _Calculus_ and _Elementary Algebra and Trig._ So, Douglas's text can be used as a good detailed treatment of geometry in one place. Similarly, Colley gives more explanations and details on Calc 3,4. I mean it's not necessary to finish one book before the other: comprehensive books can be consulted (some material can be studied in them) whenever you feel your current book doesn't have enough material or doesn't reveal necessary points/logic. There's also a short rigorous _Calculus_ by Apostol. It's around 1000 pages but contains calculus, analytic geometry and even linear algebra. Apostol provides extra explanations which are often missing in other books. He gives more proofs. I think it is a good idea to feel free to consult and delve into his book along with Colley's and Douglas's texts whenever you feel necessary. It'll help expand on the material and it'll help to understand things more clearly. IMO, it's good to go through Stewarts' books _Elementary Algebra and Trig_ and then to work through his calculus while delving deeper whenever if feels like that through books by Apostol, Colley and Douglas. That's because a lot of material is the same. It's just that books by Apostol, Colley and Douglas will expand on the material and will clarify stuff. The problem with college education is they press too much material into short timeframes while students lack the prerequisites. Then they skip tons of stuff in books and make do with student's rote-learning. The books aren't really the problem. But having the best books (and alternative texts) and more time to work though them properly is the most efficient way of studying math IMO.

[@satyavivekanandbattula1091]

Thank you so much sir for sharing.

[@thomasblackwell9507]

“Vector Calculus” by Marsden and Tromba is absolutely great!

[@douglasstrother6584]

Indeed! I still have my 2nd Edition. Tony Tromba was my Calc 1 & 2 professor at UC Santa Cruz; we used "Calculus" by Marsden & Weinstein, still have that, too.

[@John_Smith__]

Teach what you can and teach well ...wise words no doubt.

[@guidichris]

Again, look for the 2nd or 3rd editions. 2nd edition can be had for under $35, whereas the current edition is 145 used. Nice thing about self study is you don't have to have the most current edition.

[@davidmata9952]

I recommend "Introduction to Vector Analysis" by Harry F. Davis and Arthur David Snider. It's a book designed for upper-division courses and covers topics not normally covered in Calc. 3 courses, such as tensors and orthogonal curvilinear coordinates. It's also heavy on physics applications.

[@ottoomen5076]

Stewart has a vector calculus book too. Very similar to his traditional calculus book. Also, did you know Zill has a book titled Advanced Engineering Mathematics?

[@eswyatt]

Is Stewart's book called "Multivariable Calculus"? I have it and it's just the lopped off end of his calculus book.

[@billmorrigan386]

@@eswyatt Yeah, Stewart has no book on vector calculus but his Calculus can be purchased in either a one volume edition or two-volume edition. They are the same though. Stewart's text is excellent but Colley blows it out of the water in terms of how comprehensively she covers multiple variables and vectors. Not everyone needs it, and Stewart should suffice for most folks. Yet those few who want a higher level knowledge and better understanding should study Colley's text. It has a high price: a lot of time is needed to do that. And yeah, the physical book also costs.... 🙂

[@pablov.viteri9345]

Thank you so much for your videos professor, Could you show us some video related to tensorial calculus? i need it to understand einstein relativity

[@timduncankobebryant]

I teach my classes using Shifrin multivariable calculus. It's a bit more rigorous.

[@hikaryagravity]

This book cover looks like so much like an amazing science-fiction book !

[@davidhowe6905]

Yes, when I saw it very small in my YT suggestions, it reminded me of a ribcage and the Alien films!

[@HussainFawzer-ct2ts]

Question: So recently we've been going over integration in Calculus. In subjects like integration by substitution and differential equations she said that you had to cross multiply dy/dx to isolate either dy or dx for integration. I always thought dy/dx was a function like a sine or cosine and I don't see why you can break up dy/dx by cross multiplying. What do dy and dx represent? I've always thought dy/dx represented a type of function like logs or trig functions. Obviously, you can't break up sin(x) into s * in(x). I thought the same rule applied to dy/dx. I don't really understand what dy and dx are individually and why when divided, you get the derivative of a function. Integration by substitution: INT[sin(x^2)2x dx] Let u=x^2 du/dx=2 2x dx =du The Cross-Multiplying step I don't understand dx= du/2x INT[sin(x^2)2x dx] INT[sin(u)2x du/2x) INT[sin(u) du] -cos(u) -cos(x^2) Differential Equation: dy/dx = 2x/y y dy = 2x dx The Cross-Multiplying step I don't understand INT[y dy] = INT[2x dx] y^2/2 = x^2 + C y^2 = 2(x^2) + C y = sqrt(2(x^2) + C) I understand the thought processes behind substitution and differential equations. I have no problem with doing the homework problems. I just have a problem understanding what dy and dx are individually why dividing them gets the derivative of a function.

[@billmorrigan386]

They simply factor in and out, dy, dx, etc. There are also mistakes/typos in what you wrote. For enhanced understanding, consider these books on calculus and ODE: Apostol's calculus, Courant Calculus (very old book), ODE by Boyce DiPrima. Apostol often explains what is left out by other authors. I guess you have to start with elementary stuff though, like, _Elementary Algebra and Trig_ by Stewart. Note that dy/dx is usually *not* treated as a fraction (to avoid division by zero). Using the limit of delta y/delta x is preferred. For example the chain rule is not proved like this du/dt = du/dx dx/dt where dx in the numerator and denominator cancels out and the formula becomes obvious even for a small kid. That's because if dx =0, you will have a separate case to be proved and your initial proof is then gonna be incomplete. We no longer prove the separate case, we tend to just roll everything into one and go through differentials to produce one sleek proof that bypasses this problem. You have to study differentials too to avoid confusion. Differentials let you bypass this problem. Like I said, some good books discuss such things and explain them. Yet some misunderstanding may remain even after all these books. I'm sorry but math books just don't go out of their way to show all subtleties. For example, in separable differential equations authors seem to forget what they wrote before and start to treat dy/dx as a fraction without a good explanation for the poor student who doesn't even know elementary algebra properly. In terms of rigor, we now can treat dy/dx as a fraction of two differentials or as the limit of a fraction (i.e. as y'). That's gonna be rigorous and absolutely correct. That's how it's done to bypass division by zero. Yeah, so much fuss.... 🙂I guess, it's inevitable when you treat the differential dx as dx=Δx on the one hand, and on the other hand you have dx under definite integral signs where they are not like Δx anymore but the limits where Δx approaches zero and there they are denoted dx. It gotta be very confusing to a lot of folks.

[@severenr]

Differential forms are one way of giving the dy and dx in dy/dx meaning, which you can read about in the linear algebra + vector calculus book by Hubbard and Hubbard, which is excellent, but requires more mathematical maturity than a typical calculus book.

[@oraz.]

This looks like a really good calc 3 book

[@Info-God]

Can you, please, present a similar book but for Fractional Differential Calculus.

[@GetOnTheSession]

“Exercise is the best instrument in learning.” - Robert Recorde (1510 - 1558), mathematician.

[@davidhowe6905]

- who invented the equals sign!

[@richardgray8593]

I have a couple of vector calculus books buried somewhere, but I've never even heard of this one. I have a Scribd subscription, and I can read the 4th edition there online, but of course the downside is that I can't smell it.

[@argonwheatbelly637]

My Calc III covered Stoke's and Gauss'. No Calc IV in my day. That was 1984. Made Physics a breeze, though.

[@TheMathSorcerer]

that is so awesome:)

[@ivan5187]

I've had great fun studying Vector Calculus with Clark Bray's "Multivariable Calculus" and his accompanying KZbin lectures. I'll definitely check out this one, thank you for the recommendation! BTW, can you recommend a book for studying tensors?

[@chaoshengzhe]

check the channel @MathTheBeautiful by Dr. Grinfeld, and his textbook. Dr. Pavel's advisor is the legendary Gilbert Strang, and you may have seen Dr. Grinfeld in Strang's last lecture kzbin.info/www/bejne/ooa4pZhopJeGpJo For coordinate independent ("abstract nonsense") treatment of tensors, of course we should just read Multilinear Algebra by Werner Hildbert Greub. There are channels trying to reproduce the content in Greub's fashion, but the book's content really needs time to get used to.

[@comradeedwin1006]

Another good book on vector calculus is "Div grad curl and all that" by H. M. Schey.

[@douglasstrother6584]

MS reviewed this as well.

[@sohybali2696]

May I suggest that you compare this textbook with Davis and snidar introduction to vector analysis?? I suppose the later is the best book in its field.

[@dimitrioskalfakis]

essential math for engineers. good stuff. rough seas ;-)

[@kani_sama]

May I interest you with Multivariable Analysis by Duistermaat?

[@rubyistarla6306]

Sir could you please suggest best logarithmic and exponential books for beginners.

[@StaticBlaster]

I have the first edition of this book. It's a dark blue book. It's a pretty good book.

[@TheMathSorcerer]

nice:)

[@re1ence633]

that's neat. is there a book with a more rigorous treatment than this?

[@TheMathSorcerer]

Advanced Calculus by Fitzpatrick or Advanced Calculus by Buck treat these topics but more rigorously.

[@economicist2011]

4:12 The number of professors who pride themselves on doing exactly that is rather disturbing.

[@rawslash7548]

Sir,I am quite interested in knowing the history mathematical symbols like pi,zeta,omicron,etc.Would plz suggest me a math symbol history book? I would be very happy to hear your suggestion.Thank you.

[@wescraven2606]

Math Sorcerer, have you looked at the Vector Calculus book by Paul C. Matthews?

[@Mahmood42978]

I have the 3rd edition of this text.

[@kristianwichmann9996]

The latter part of this, I was only taught as a tool when learning electromagnetics.

[@jmsaucedo]

How about the Hwei hsu Vector analysis?

[@fernandoramirez1917]

This book has and spanish editions (cuarta edicion Pearson)

[@eswyatt]

I'm working through a review this very material right now! I see from the photo she draws arrows over her vectors. Teachers who don't do this really irritate me! (Especially with this material where ds , the differential of surface, is different from ds [arrow over s], the vector.)

[@billmorrigan386]

Good point. But why is it so bad? It's pretty standard not to draw arrows but use, say, bold face. How am I gonna insert vectors into a comment, for example. Well, I would just put them in bold, like, for example in a triangle you have *a=b+c* and squaring that as scalars produces the law of cosines. I fully agree though that Colley's text must be commended for going through very good editing and for providing nice illustrations.

[@eswyatt]

@Bill Morrigan bold is fine, too. But hard to do on a board

[@azimuth4850]

"...and that's why self-study was invented" LOL

[@mdabusaqibalamansari9979]

Hi sir. Please make a review on the book "Chalanges and thrills of pre college mathematics" its a quite popular book here in india

[@Podzhagitel]

I got a BIG FAT F in my vector calculus class last semester so this will be a big help when I retake it in a few weeks LOL

[@TheMathSorcerer]

Yeah this will help a ton!

[@alejrandom6592]

That's one trippy cover

[@eduardoponce8671]

Hi, do anyone know where I can find multivariable epsilon Delta limit exercises?

[@BlackbodyEconomics]

Every time you smell a book ... I can almost smell it myself. I freakin' love book smell :P

[@TheMathSorcerer]

Oh me too!! LOVE IT!!!!!!

[@surrealistidealist]

I can't stop collecting all these books!!! 🤣❤️🤣❤️🤣❤️

[@TheMathSorcerer]

LOL me neither. I have soooooo many books!

[@SEGA-0]

haznos el video de este libro en español porfavor.

[@masoomparwej5650]

Hlo..I am from Nepal... How can I buy this book?

[@faithlesshound5621]

On Amazon UK, the 4th edition (2011) costs 198 Pounds Sterling hardback and £115 paperback: not cheap! There's also a 5th edition (2022) for £115 which is described as a "rental edition." So it has gone up by more than inflation from 90 US Dollars twenty years ago. There's also a "Study Guide" for this book which is out of print just now. Amazon India has three formats of the 4th edition: paperback for 11,000 rupees, hardback for 23,000 and loose leaf for 26,000. Do you have to buy the latest edition? Maths at the undergraduate level can't have changed very much over the last century, but teaching methods may have improved since teachers had to give up their canes.

[@argonwheatbelly637]

Umm... just one thing: Gauss is pronounced, "gowss," not, "gahss." Excellent video!

[@TheMathSorcerer]

Oh wow, yes you are right. I used to pronounce it "gowss" and now I've switched to "gahss". I wonder why I did that. THANK YOU!

[@cladimarce]

Excelente Book ...and expensive 🤑 at Amazon

[@mustafamursheed1878]

Let me tell you a fun fact....in my country, "Bangladesh",,,,,, we as high school students also learn vector calculus.. Gradient, divergence, curl, partial differentiation, nabla operator and so on......🤣🤣but matter of disgrace is that physics & math textbooks in our country are not written with standard rigor.

[@vade248721]

This book can be found online for free. And solutions for free. These people probably got too much money from broke college students already. No need to pay.

[@TheLuminousOne]

I learnt Calculus from Professor Calculus of Tín Tín.

[@TheHeartbeast]

Just commenting because of magic the gathering:)

[@kummer45]

People MUST understand that VECTOR calculus is simply GEOMETRY of moving points and the choreography of change. VECTOR, TENSOR, SPINOR calculus IS fundamental for the full comprehension of computer graphics.

[@dont-want-no-wrench]

i wish i was smart enough to learn this stuff, and had some actual application for learning it. nope on both counts

[@ahmadhabil7933]

But the book is without proofs

[@TheMathSorcerer]

Yeah the classic book by Fleming would offer a more rigorous treatment:)

[@billmorrigan386]

That's not true! There are tons of proofs and derivations. The author has them in exercises too. This book requires some maturity. It doesn't often state here's the proof, here's the theorem. Some stuff is treated as derivations and propositions. They are proven in the book, of course. But some are left as exercises.

[@JaskoonerSingh]

The book is expensive

[@richardgray8593]

That is a 20-year-old price. It's not unusual for new math textbooks to cost nearly $300.

[@bernard2735]

We really need KZbin + smell (SmellTube? YouSmell?) - I, too love the smell of old books 😊