Legendary Calculus Book

foreign in this video want to show you one of my books it is calculus volume 1. introduction with vectors and analytic geometry by Tom apostol California Institute of Technology Blaisdell Publishing Company very nice book this book feels amazing to hold like it just feels like it's quality it's really well made which honestly it needs to be because this book and this is this is probably the worst thing about this book this book is expensive like it's not cheap you can find it on the internet I'll try to find it and leave link in the description but like it's not cheap and that's that's the worst thing about this book I'm pretty sure there's copies out there for like over 200 I've seen them so it's it's up there and so Tom Apostle he was math teacher where he taught mathematics and he believe he did number Theory he worked at the California Institute of Technology and he lived to be 92 years old so he lived long life he was the son of Greek immigrants and he had his name changed his original name believe was something like apostolopolis and when his parents became citizens they change their name to Apostle and then so Tom inherited that last name from his father so actually he's kind of just an interesting background on Apostle and have picture of him here if you want to take look at what he looks like here's The Man Behind The Legend behind this very expensive book if he only knew how expensive his book was you know he knew but he lived long life 90 92 years old believe long time time so let's take look at this book I'm going to open it up we're just going to look at it and explore it it's very good book it was it's definitely harder than your regular calculus book definitely recommend this book think it's great book nothing negative to say other than the price of the book right it's just it's expensive yeah nice just have to smell it just got to give it whip can already smell it this one has particular like old comic book smell yeah yeah it smells like used used to be an old comic used to collect Amazing Spider-Man comic books like was really into that and so that's what it smells like so let's look at the contents briefly and then we'll just look at some of the mathematics think that would be more fun let's just look at some of the actual examples and see if we can make sense of them it starts with integral calculus calculus from the page here and then it goes into more Concepts here and then differential calculus yeah the logarithm differential equations got some first order stuff second order stuff Vector stuff lines planes dot products curves and surfaces the cylinder circles parabolas the mean value theorem and the general generalization some applications of it infinite Series this is calc 2 stuff infinite series and that's it and this is volume one so there is another volume right so it's not everything let's just open up to random page and see what we find so let's see this is bolzano's theorem and the intermediate value theorem so little bit more it's not computational right nothing computational there inverses of the trigonometric functions this is interesting this is something that people have hard time with Intrigue is Computing the values of the inverse trig functions because they think you know you have you have to pay attention to the range right you have to pay attention to the range of the trig function for example here you have of equals sine if is between negative pi over 2 and pi over 2. so they're defining the sine function on that particular interval they're restricting the domain of the sine function and the reason you do that is because by restricting the domain you're able to take the inverse because then your function is invertible otherwise it's not invertible you know it won't even pass the horizontal line test you see here here's the picture of the restricted sine function and then over here you see the inverse function you see both graphs sine on the left and arc sine are the inverse sine on the right so and you learn that in trig and that restriction of the domain and all that in trig class think it's overwhelming to people it's just something that they're not used to seeing so again here's already he's talking about the integral of 1 over the square root of 1 minus squared is arc sine yeah and then over here he has the integral of sine he looks he's going through something there are some of the exercises derive the differentiate differentiation formulas and exercises one through five okay yeah that's pretty simple it's not too hard for for number four is is little bit interesting because you have to pay attention to the domain and the range and stuff in order to justify that absolute value it's little bit more delicate there that's that's number that's number four right in the exercises so that's pretty hardcore mean mean that example is significantly harder than like the first one right in my opinion yeah nice nice problems and he has answers so if you're wondering is this good for self-study sure this is this is collector's book this is this is good for everything this is this is what it's all about right here this is one of those books it's like yeah this is the Apostle book it's it's one of my best books like it's one of my you know as collector this is this is pretty nice this is pretty nice yeah so you see he has most of the solutions three four five six seven eight nine ten elevens there's couple missing it the the smell of the book just it's it's intoxicating just have to smell it again I'm sorry it's just wow wow just wow wow wow someone left comment once about that so maybe you know about this there's there's something with they even left link and and the link work usually when you leave links in in comments YouTube like Auto deletes lot of stuff so someone left link once saying that these old books something with the chemicals in the books that they produce is smell the chemicals that have sense of like vanilla and don't know if it was lavender and something else there's chemicals in the paper that decompose and it was like chemistry thing it was showing that these chemicals produce these scents which are generally liked by people for example I'm pretty sure that like most people but there's been studies think where they say like most people think vanilla is you know scent that's pleasing that's why there's so many like vanilla sprays like you know lot of women think they have like products that smell like vanilla and lavender is another popular one and think those are things that human beings perhaps are naturally drawn to and it's like the smell of decaying paper just it's just it's just wonderful it's just amazing look at these curves these are the interval curves of the differential equation yeah lately I've I've you know graphing is interesting it's something that lot of people don't like was was looking at book the other day and it had lot of really interesting graphs in it and was thinking like yeah you know someone mentioned that there's entire books fill the graphs like there's there's books of graphs think that's interesting maybe I'll have couple of those should mention one of those one day and show you they're really cool solve the differential equations in exercise one through twelve by separating the variables okay yeah all right so like number one Prime equals cubed squared so you would write that as and then separate the variables and integrate both sides that's not so bad these are some of these are pretty good this is class someone wrote in it don't think this copy is signed don't think it is have bunch of books that are signed found book the other day that was written by by the author made the book and he gave new copy to his colleague and his colleague was also famous mathematician so it's like really cool you have book written by famous mathematician and he gives it to his friend and signs it like when sincere gratitude and like gives it to him it's kind of cool that you know it's like piece of History some stuff with vectors straight lines as before we shall denote the vector from the origin to an arbitrary Point in space let naught be given point anywhere in space and let be given non-zero vector if the vector minus not joining now to is parallel to and there's scalar such that yes that's true or what amounts the same thing as yep yeah that's that's useful that's very useful it's useful equation that's cool the scalar is called parameter and this equation is called vector parametric equation or simply vector equation yeah so this is the set of points satisfying this equation is called the straight line through naught parallel to yeah cool scalar triple product wow look at the diagrams wonder how they did those in the old books like how they how they actually did that that's kind of interesting here we have looks like determinant of three by three Matrix yeah because you can use that to compute the cross product of two vectors which they do here right here's the cross product of two vectors using the determinant formula so basically what you do here is you like for example for the first piece here you cross out this row and this column you get negative eight three four three and there you see it you have the and the pattern is plus minus plus and then for the second one you cross out the jth column well rather the second column and the first row you get two zero three three the checks and for the last one you cross out the last column in the first row so two negative eight zero four yep it checks no typos in the book so far so I'm sure it has typos haven't found any but I'm sure it has mistakes Quality quality pages yeah yeah it's nice really clean copy so think this is good book for anyone who if you're wondering like should you buy this book is it good book lot of people wonder my opinion is this book is good for anyone who wants math book like it's just awesome it's classic right it's one of those books you have forever hopefully you have all of your books forever that's that's my goal at least keep them forever guess what's the prereq for this book well lot of effort lot of the more math you know the better if you already know some calculus that's better mean it's going to take lot of work also having some knowledge of proof writing would be highly beneficial before reading this book mean there's lot of things that aren't proof you could certainly skip those focus on the computational aspects mean you could do that but it's it's certainly better think to to have more the more math you know the better don't think that should be reason for person to not explore it you know think lot of times people are curious about the prereqs or something because they're wondering you know do have enough to explore it and lot of times the answer is no right lot of times the honest answer is no even if you already have some skills maybe you're just not ready for it right maybe your math is just not at the level of the level that's presented in the book right you're just not there and that's okay but think you could still jump into it and you can still try and you can still do your best mean that's how you get really strong you know by by jumping in in the deep end remember one time was out in the ocean and that's when was first learning to surf on it and paddled out on my board and it was these two guys and they look really rough looking guys they were out there and paddled out and was like Hey guys and wanted to you know wanted to Surf with them and they said to me said hey said hey what's up and fell off my board and told him I'm just new I'm just learning and the guy looked at me and he goes you come out here you're going to learn really quick and was like yeah and it was perfect day mean perfect waves just one of those amazing days you never forget but you know jump in the deep end so you can do the same thing with mathematics right jump in the deep end and it's lot safer than surfing so no sharks you're not going to drown the board's not going to hit you on the head but it's certainly lot harder it's lot harder lot harder and takes lot more time circular helix cool right yeah it's just an intoxicating smell with this book the general equation ax squared plus bxy plus squared plus plus plus equals zero cool yeah good look at this book rotations of axis on the plane here's some more exercises identify the locus represented by the given equation when the locus is Iconic it sketch the curve and indicate the locations of important features such as Foci vertices Center asymptotes Etc yeah conics conic sections are really interesting really cool took studied cryptography long time ago and there was actually we were doing some stuff with conics in in cryptography so mean conics are very they you know they go pretty far and hear little curvature curvature nice that tells you how sharply curve bends so the greater the curvature the sharper the bend that's the idea behind this new intuition behind curvature looks like here they're Computing some things yeah there's formula for curvature that's one of the formulas for curvature it's one of them zoom in so you can see it there's there's couple others it says here and practice it is usually easy to apply this formula here yeah that's true you know all other books say that too the steward book says that as well the Larson calculus book does that as well lot of books say that this is more easily done and it's true it's true there are some times though when the other formula is is also good so recently did problem with the other formula and was thinking about that comment and was like yeah well didn't use it but that's just one example there are other examples where you know the formula that they recommend is is the one that the one that you're going to use curvature is something you study in at least in the U.S if you took calculus 3 class that's when you would study curvature same thing with cylindrical coordinates this would be calc 3 topic applications to planetary motion yeah pretty cool right you can use math to describe what happens in space that's just you know the more you know I've always been pure math person you know was pure math major and took lot of abstract algebra analysis and stuff and topology like that stuff but the more math learned the more start to appreciate certain things you know you're you're your tastes change over time as you learn more math as you grow mathematically you discover new interests and appreciate certain applications not all you know different people like different things some people hate statistics mean some people hate statistics what's this use equation 710 to establish the following inequalities wow cool Taylor's formula with remainder you look like they were underlining this very carefully they must have had ruler there yeah so this is the this is the tangent line isn't it it says here for example the linear polynomial has the value of and the derivative Prime of at the point of geometrically this means the graph of is tangent line yep it's the tangent line yeah right so he mentions that that's the first order Taylor polynomial also for function center.x equals same thing as the linear approximation or the tangent line approximation or just the equation of the tangent line they all mean the same thing pretty much so yeah calculus beautiful stuff and here's the formula let be function with derivatives of order at the point equals zero then there exists one and only one polynomial of degree less than equal than which satisfies the plus one conditions namely so it's Unique so it's unique expansion it exists and there's and it's Unique yeah cool and then here's here's where the center is not zero or just in general it could be anything yeah when you look at the series when you let Go to Infinity you get the series it's called the Taylor series and then this one when goes to Infinity it's called the McLaren series the McLaren series is just Taylor series Central Net Zero let's see if we can find Taylor series here is it is it in this section does he make that leap let's see don't think it moves that quick no not yet notations extreme values proof of roles rawl's theorem other tests for extreme at stationary value someone really worked through this book right look at all that writing that wasn't me someone maybe perhaps in the 60s you wonder who they were and where they are now always wonder when find books with names in them always Google the person and it's really interesting I've found couple people just you know yesterday and think one today just because I've been looking at some books and it's interesting here's some limits this is lopital's Rule yeah there's all kinds of ways to spell it by the way I'm pretty sure you can spell it hospitals I'm pretty sure that's correct spelling someone explained it to me once knew someone who spoke French and they were explaining it to me and but forgot so he just mess around look at number one the answer There Is over so it's pretty easy yeah you could do lot of these in your head but like that should be three number four is zero number three looks really interesting right that's cool problem that looks fun some cool problems in this book when you have answers too so yeah just got whiff of it again just like it's just rising from the pages and it's just like so so until just I'm sorry gotta give it whiff here just one more wow wow wow Tom apostle yeah the legend great book though think it's excellent whoops it's an excellent book we should take the dust jacket off and look at the book without the dust jacket let's try that just to see what it looks like so you can get good look there think they put like plastic on the dust jacket so appreciate whoever did that that's good look at that calculus Apostle volume one beautiful yeah wow the weight anyways probably should end this video thanks for being subscriber just wanted to sit down here and show you book that lot of people so lot of people have been asking about this book for long time for years and just always just forget there's other books like that too that really need to to mention because people have been asking so whenever you leave comment by the way and you asked me to record you asked me about book and Heart It read your comment right because only heart comments that read so basically that's what do just that way you know if don't heart your comment and it's good comment it means probably didn't read it so and don't have time to reply to all of them so don't always reply but just try to like I'm behind haven't read all my comments there's too many but do appreciate all the book recommendations because that's how learned to get books so someone was recommending philosophy books to me earlier because bought bought philosophy book earlier because have another book which let me see if have it here let me just check no don't have it put it away it's another book wanted to show you it's really cool it's it's book that they used in the Army in the 40s to train army people and so yeah so bought philosophy book for the Army after reading that book is the story basically was inspired by the Army book so yeah anyways until next time just wanted to show you this book good luck and take care