Maths needed to advance my physics understanding

[rudransh verma]

I am a 10(high school)+ 2 years student. I have a basic knowledge of calculus, trigonometry, algebra, arithmetic. I need to advance as an undergraduate to start my physics journey. I want a mathematics book for physics which if it will cover some basics in early chapters will be of much help.
Thank you!
By the way I was looking at mathematical methods for physics and engineering.

 

[Mayhem]

A solid grasp on "applied" calculus and linear algebra will get you quite a long way.

 

[rudransh verma]

A solid grasp on "applied" calculus and linear algebra will get you quite a long way.

Can you suggest me some books?

 

[Mayhem]

Can you suggest me some books?

Mary Boas' book is pretty good.

 

[rudransh verma]

Mary Boas' book is pretty good.

A bit costly! How is this book in the OP?

So after high school people in colleges read Mary boas for physics or is there something in between that?

 

[jasonRF]

here are a few suggestions.

For calculus, a used old edition of a reasonable book will do fine. In college I learned from
https://www.amazon.com/dp/0201163209/?tag=pfamazon01-20
But other options can work, too.

For linear algebra, used copies of old editions of Elementary Linear Algebra by Anton are reasonable. There is also a good free book
https://hefferon.net/linearalgebra/

For a pretty good (free) book on math methods for physics at the undergrad level see
http://www.physics.miami.edu/~nearing/mathmethods/

 

[rudransh verma]

@jasonRF @Mayhem how is linear algebra Gilbert strang ?

 

[Mayhem]

While not "physics", The Chemistry Maths Book is fairly cheap and also covers a lot of these subjects.

 

[jasonRF]

@jasonRF @Mayhem how is linear algebra Gilbert strang ?

Strang has at least two linear algebra books.

Introduction to Linear Algebra is reasonable if this is your first time learning the subject. Linear Algebra and It’s Applications is not as good for that purpose.

 

[Falgun]

I don't have a specific book I'd recommend but Mary Boas is the standard for undergraduates and Arfken for graduates. The Nearing book is also very easy to use. I really like Riley , Hobson and Bence Mathematical methods book which is more comprehensive than Boas or Nearing but it is very dense and difficult to start with.

If you are just starting out the courses at MIT OCW are great. They are also labelled with the appropriate prerequisites so you won't have much problem navigating through.

Mathematics Department

They also have a lot of physics courses.
Physics Department

It might be a little hard to work through them all but they make for a great way to learn.

I recommend that you start out from 18.01 Calculus and work your way up.

 

[rudransh verma]

@Mayhem @jasonRF Where should I start? Should I first polish high school stuff and then start with undergraduate books from later chapters or should I start directly and it will automatically get polished in the way of learning higher stuff? If I should polish then should I do it from highchool books or from these books?

 

[Mayhem]

@Mayhem @jasonRF Where should I start? Should I first polish high school stuff and then start with undergraduate books from later chapters or should I start directly and it will automatically get polished in the way of learning higher stuff? If I should polish then should I do it from highchool books or from these books?

Definitely polish your HS math first. Get comfortable with calculus up to and including u-sub, integration by parts before moving on.

 

[vanhees71]

I think high-school math books are not very helpful nowadays. I'd rather recommend to start with beginning-undergraduate textbooks but with something titled like "Mathematics for physicists (or engineers)" than a pure-math textbook, which are pretty formal, and as a physicist for the first steps it's more important to have some hands-on applicable calculational techniques at hand than all the formal proofs although also this becomes important at a more advanced level for the physicist to a certain extent.

 

[mathwonk]

From what I could gather from a quick perusal of courses on their website, I do not especially recommend the MIT OCW notes posted online. (The calculus with applications notes were ridiculously non rigorous and the calculus with theory notes were equally ridiculously formally rigorous.) I would much more strongly recommend the book suggested by JasonRF, actually written by the MIT professor most famous for engineering calculus books, the late George B. Thomas Jr. This (7th) edition is revised by Ross Finney, but I believe he does a good job. Later ones with names like Hass and Weir attached are not at all recommended.

Here is an old cheap used copy of Thomas and Finney 7th ed:
https://www.abebooks.com/servlet/BookDetailsPL?bi=30739058642&searchurl=spo=30&sortby=17&tn=calculus&p=2&an=george+b.+thomas&sp=1&cm_sp=snippet-_-srp2-_-title1

and if you look hard enough you can also find a few older ones just by Thomas.

To be more precise about MIT OCW, there is an enormous variety of offerings there and no doubt some are wonderful, but not all are in my opinion.

 

[vanhees71]

This seems to become a disease: Some new authors modernize good textbooks from the past. Why the heck don't they print the good textbooks in their original form and let new textbook writers write their own textbooks? Some years ago they even dared to "modernize" Hilbert and Courant!

 

[jasonRF]

By the way I was looking at mathematical methods for physics and engineering.

Just some friendly advice here. Since you are in high school you may not realize that there are many books with identical titles. That title could refer to a book by Blennow (who is on PF!), or the one by Riley, Hobson and Bence, the sequence of books by deLyra, or perhaps some other books. In the future you should include both the name of the author(s) and the title if you want people to understand what book you are talking about. Sometimes you only need the name of the author if they only wrote one book on the subject, but if they wrote more than one book on the subject (like Strang and linear algebra) then leaving out the title leaves the readers unable to determine what book you are talking about.

Jason

 

[mathwonk]

One advantage of the original book by Thomas is its inclusion of techniques for using centers of mass to make some calculations easier, such as work. He does not give exactly the following one, but it illustrates the power of the technique and is similar to his explanation of how to calculate work and potential energy ( see below). Problem is to find the 4 dimensional volume of the unit 4-ball, i.e. the interior of the set with equation x^2+y^2+z^2+t^2 = 1, in 4 space.

By integral calculus, and the usual method of slicing taught in most of these books, one is led to integrate the 3 diml volume of the slice at height t, namely (4π/3)(1-t^2)^3/2 dt, from t=-1 to t=1, not so easy for the average student.

But using centers of mass, and the ancient technique named after Pappus, one can compute the volume by considering the 4-ball as obtained by revolving half a 3 -ball around a 2-diml axis in 4 space, (just as one can represent a 3 -ball by revolving a half 2-disc around a line in 3 space).

It seems we need to know the center of mass of the half 3 - ball, to compute how far it travels under the rotation, but we can finesse that since by Archimedes, we know the volume of half a 3-ball equals the difference between the volume of a cylinder and that of a cone, and even Archimedes knew their centers of mass. I.e. the center of mass of the cylinder of height 1 and base radius 1, is at height 1/2, and the center of mass of the (inverted) cone of height 1 and base radius 1, is at height 3/4.

So we just revolve a cylinder of volume π around a circular path of radius 1/2, getting as product: volume times length of path = π^2, and then subtract the result of revolving a cone of volume π/3 around a circular path of radius 3/4 getting a product of π^2/2. The difference is the volume of the unit 4 - ball, namely π^2/2.

The point is that by teaching students the wisdom of the ancients, calculations are sometimes made far easier than forcing them to use more difficult methods in circumstances where they are not the most appropriate. Thomas' original book was written with a sensitivity to doing things more naturally, using appropriate methods, and I noticed that some of this is lost in the rewrites of his book.

You may ask why a physicist wants to know about 4 dimensional volume, since only a mathematician does such exotic examples! But if you write down the integral for the work done by raising a half 3-ball through a certain distance, e.g. the work done by pumping out the water from a swimming pool shaped like half a 3-ball, you will see that the integral is the same as that for computing the volume generated by revolving a half 3-ball around a 2 diml axis in 4 space, using the method of cylindrical shells, except for a factor of 2π. I.e. work computations done when moving 3 dimensional objects, or potential energy calculations on solid objects situated at a certain height, are exactly the same, mathematically, as computing 4 dimensional volumes!

Now just as the example above shows how to simplify such a volume computation using centers of mass, Thomas explains, in his original book, how to simplify computing work using centers of mass. Now I may well be wrong, since I no longer have access to the later books, but my memory is that the later books teach work computations first, using more difficult methods, and then teach centers of mass only later, and then do not explain how to use them to simplify work computations. This criticism is at least consistent with the fact that in the table of contents I found online, work occurs earlier than centers of mass in the 14th edition of this book. Maybe someone who has access to a copy, can check whether after teaching centers of mass, the 14th edition then uses that concept to simplify work calculations, or for mathematics students, whether they teach Pappus’ theorem to do volumes of solids of revolution. Indeed I hope so. Ok, end of rant. peace. and “to be fair” one can learn something even from later editions, just not as much, and at a much higher price. (I seem to recall now the first version of these Thomas rewrites that was forced on me, and inspired this rant, was called "University Calculus", by Hass, Weir, and Thomas (and published the year he died. It is now described on amazon as a "streamlined" version of Thomas, at only 960 pages.)

what frustrates me as a teacher, is why we deny students knowledge of concepts that were known centuries ago, and instead teach them to use modern ideas that actually make some computations harder. hello? isn't the point of education to make solving problems easier?

 

[jasonRF]

I see that the OP is in India.

Here is a version of the 6th edition
https://www.amazon.in/dp/818501552X/

Thomas' original book was written with a sensitivity to doing things more naturally, using appropriate methods, and I noticed that some of this is lost in the rewrites of his book.

My copy of the 7th edition is at work - I'll have to check next week to see if the center of mass approach is covered the same way you describe. I took the classes with that book 30+ years ago so I cannot remember!

jason

 

[mathwonk]

Here is a specific example of what I am complaining about, in a free onine book with Gilbert Strang's name on it, from MIT.
https://d3bxy9euw4e147.cloudfront.net/oscms-prodcms/media/documents/CalculusVolume2-OP.pdf

On p. 195 they show how to calculate force on a submerged plate due to the water surrounding it, as an integral. They do not observe that this is mathematically identical to a volume calculation for a solid of revolution, hence they do not explain how to greatly simplify this calculation using centers of mass and Pappus theorem, and the concept of center of mass is only introduced in the next section. I.e. notice that since the centroid of the triangle in the example on p. 195 is at depth x = 1, the calculation using centers of mass is just: area of triangle times depth of centroid times the factor 62.4, or 12x62.4 = 748.8. This trivial computation is instead rendered more difficult as an integral problem.

In the next section, they do at least explain how to use centroids and Pappus to do a volume calculation, p.216. but never explain the connection with work, force, or potential energy.

 

[vanhees71]

I don't see, why this is so bad. It's a straight-forward calculation. What's a real sin is the use of non-metric units though ;-)).

 

[mathwonk]

Indeed that is true. My philosophy is however, that since math in general and calculus in particular is hard for students, not to give them easy methods when they exist, is in fact a sin against honest teaching. In this example, although using calculus as you say is a straight forward calculation, it is nonetheless entirely unnecessary, as one can make the calculation instantly using the much simpler idea of centers of mass. I would support giving the integral calculation as is, and then adding the remark that in this case, or any case where one knows the center of mass, that the integration is unnecessary.

And in fact the only reason this particular example is a straight forward integral calculation, is that the integral calculus method is so complicated in general that they can only give examples in cases where the computations are easy.

E.g. suppose they asked the student to do the force calculation when the end of the triangular trough is no longer isosceles, i.e. the upper base of the triangle remains the same, but the lower vertex is translated 37 inches (forgive me) to the right, and 3 feet deeper. How long would it take the student using calculus to deduce that the answer has quadrupled? (The altitude of the triangle has doubled, the base is the same, and the depth of the centroid has doubled.)
Or suppose this triangular piece were embedded in the end of a larger trough, with the surface of the water 3 feet higher, and the given triangle were rotated 20 degrees counterclockwise about the point currently having coordinate x=1 on the x axis. The calculus method seems then much more difficult, but using centroids the answer is just multiplied by 4. (The depth of the centroid is now 4 instead of 1, and the area of the triangle has not changed.)

Or suppose they gave a problem to compute the force only given the area of the triangular end of the trough, plus the depth of its centroid? With current instructions, the student could not readily do the problem at all. But with the centroid method all these calculations can be done in your head.

So I would support giving the explanation as presented, but then supplementing it by saying that in fact all that is needed is that last information, i.e. the area of the end piece, and the depth of its centroid, and hence advising the student that when that information is already at hand, then doing the integral is a waste of precious time. And as all engineers must know, time is money.

I.e. I believe the point of teaching is to equip the student with more than the ability to do the trivial examples in the book; they need to know how to do the ones they may meet in the real world, for which they need to know as much as possible. In the case, of the Thomas rewrites, there are many other things to complain of as I recall, but this is one specific one that I remember annoyed me, since they reordered the presentation of Thomas so as to remove a bit of valuable insight that was provided in the original. And nothing was gained, since they did teach centers of mass in the next section, but by reversing the order of presentation they made the concept unavailable to clarify the previous section! I.e. they taught centers of mass, but not why they are useful! I fear math courses are notorious for this, teaching abstract concepts that the student sees no use for.

Ok, apologies, I know you know this,... getting off stump now. I support also your right to disagree completely, which is why I gave a specific example of my objections. If i just say such and such a book is "bad" but don't say why, then you have no chance to see if you agree or disagree based on the evidence. In this case, since the Strang book does this topic exactly the same way as the complained of Thomas rewrites, it motivates me to wonder whether Strang et al may be just rewriting those rewrites!

cheers!