Machine learning is largely based on the following chain of mathematical topics

Calculus (through Vector, could perhaps leave out a subsequent integration techniques course)

Linear Algebra (You are going to be using this all, a lot)

Abstract Algebra (This isn't always directly applicable but it is good to know for computer science and the terms of groups, rings, algebras etc will show up quite a bit)

General Topology (Any time we are going to deal with construction of a probability space on some non trivial manifold, we will need this. While most situations are based on just Borel sets in R^n or C^n things like computer vision, genomics, etc are going to care about Random Elements rather than Random Variables and those are constructed in topological spaces rather than metric ones. This is also helpful for understanding definitions in well known algorithms like Manifold Training)

Real Analysis (This is where you learn proper constructive formulations and a bit of measure theory as well as bounding theorems etc)

Complex Analysis (This is where you will get a proper treatment of Hilbert Spaces, Holomorphic functions etc, honestly unless you care about QM / QFT, P-chem stuff in general like molecular dynamics, you are likely not going to need a full course in this for most ML work, but I typically just tell people to read the full Rudin: Real and Complex Analysis. You'll get the full treatment fairly briefly that way)

Probability Theory (Now that you have your Measure theory out of the way from Real Analysis, you can take up a proper course on Measure Theoretic Probability Theory. Random Variables should be defined here as measurable functions etc, if they aren't then your book isn't rigorous enough imho.)

Ah, Statistics. Statistics sits atop all of that foundational mathematics, it is divided into two main philosophical camps. The Frequentists, and the Bayesians. Any self respecting statistician learns both.

After that, there are lots, and lots, and lots, of subfields and disciplines when it comes to statistical learning.

A sample of what is on my reference shelf includes:

Real and Complex Analysis by Rudin

Functional Analysis by Rudin

A Book of Abstract Algebra by Pinter

General Topology by Willard

Machine Learning: A Probabilistic Perspective by Murphy

Bayesian Data Analysis Gelman

Probabilistic Graphical Models by Koller

Convex Optimization by Boyd

Combinatorial Optimization by Papadimitriou

An Introduction to Statistical Learning by James, Hastie, et al.

The Elements of Statistical Learning by Hastie, et al.

Statistical Decision Theory by Liese, et al.

Statistical Decision Theory and Bayesian Analysis by Berger

I will avoid listing off the entirety of my shelf, much of it is applications and algorithms for fast computation rather than theory anyway. Most of those books, though, are fairly well known and should provide a good background and reference for a good deal of the mathematics you should come across. Having a solid understanding of the measure theoretic underpinnings of probability and statistics will do you a great deal--as will a solid facility with linear algebra and matrix / tensor calculus. Oh, right, a book on that isn't a bad idea either... This one is short and extends from your vector classes

Tensor Calculus by Synge

Anyway, hope that helps.

Yet another lonely data scientist,

Tim.

hey nerdinthearena,

i too find this area to be fascinating and wish i knew more on the upper end myself. i'm just going to list off a few resources. in my opinion, graduate school will concentrate a lot on progressing your technical knowledge, but will likely not give you a lot of time to hone your intuition (at least in the first few years). so, the more time you spend in undergraduate school doing so, the better.

helpful for intuition and basic understanding

• geometrical vectors by gabriel weinreich - why there's more to vectors and how to draw them. this basically shows visual examples of differential forms and the like.
  
• advanced calculus: a differential forms approach by harold edwards - a fantastic approach to differential forms. it's a treasure, so just read it. :) it is actually well built for skimming and skipping around.
  
• vector calculus, linear algebra, and differential forms: a unified approach - this is a big book, but you could just jump ahead to the later sections covering differential forms for an interesting approach and technical skill building.
  
• div, grad, curl are dead by william burke - pure intuition and similar to geometrical vectors. don't look for anything else here, but definitely read it.
  
• applied differential geometry by william burke - a more complete work than the above and showcases a lot of the connections between geometry and physics.
  
  

  more advanced but still intuitive
  

• an introduction to manifolds - this is at the same level and covers essentially the same material as lee, but in my opinion, it is much better. it is much more concise and to the point with more focused exercises.
  
• differential forms in algebraic topology by raoul bott and loring tu - the sequel to the above, but i haven't read beyond the first chapter. it is well known as a must read classic, but having some topology background will help with this book.
  
• the laplacian on a riemannian manifold: an introduction to analysis on manifolds - an interesting introduction to geometric analysis.
  
• differentiable manifolds: forms, currents, harmonic forms by georges de rham - a classic by one of the greats and also one of the only sources i've found that covers de rham currents (besides research papers that just use them).
  
• riemannian manifolds: an introduction to curvature by john lee - the sequel to lee's smooth manifolds book.
  
  hopefully this helps. if i were to revisit geometric analysis, i would basically use the above books to help bone up my understanding, intuition, and technical skill before moving on. these are also mainly geometry books, so learning analysis (like functional analysis) would be good as well. i mainly have three suggestions there.
  
  

  three general analysis favorites
  

• introductory functional analysis with applications by erwin kreyszig - used everywhere.
  
• calculus of variations by gelfand and fomin
  
• linear differential operators by cornelius lanczos - a geometrical look at the subject. also take a look at the variational principles of mechanics by him as well.

let me give you a shortcut.

You want to know how partial derivatives work? Consider a function with two variables: f(x,y) = x^2 y^3, for a simple example.

here's what you do. Let's take the partial derivative with respect to x. What you do, is you consider all the other variables to be constant, and just take the standard derivative with respect to x. In this case, the partial derivative with respect to x is: 2xy^3. That's it, it's really that easy.

What about taking with respect to y? Same thing, now x is constant, and your answer is 3x^2 y^2.

This is an incredibly deep topic, but getting enough of an understanding to tackle gradient descent is really pretty simple. If you want to full on jump in though and get some exposure to way more than you need, check out div curl and grad and all that. It covers a lot, including a fair amount that you won't need for any ML algorithm I've ever seen (curl, divergence theorem, etc) but the intro section on the gradient at the beginning might be helpful... maybe see if you can find a pdf or something. There's probably other good intros too, but seriously... the mechanics of actually performing a partial derivative really are that easy. If you can do a derivative in one dimension, you can handle partial derivatives.

edit: I misread, didn't see you were a junior in highschool. Disregard div curl grad and all that, I highly recommend it, but you should be up through calc 3 and linear algebra first.

To change my advice to be slightly more relevant, learn how normal derivatives work. Go through the Kahn Academy calc stuff if the format appeals to you. Doesn't matter what course you go through though, you just need to go through a few dozen exercises (or a few hundred, depending on your patience and interest) and you'll get there. Derivatives aren't too complicated really, if you understand the limit definition of the derivative (taking the slope over a vanishingly small interval) then the rest is just learning special cases. How do you take the derivative of f(x)g(x)? f(g(x))? There's really not too many rules, so just spend a while practicing and you'll be right where you need to be. Once you're there, going up to understanding partial derivatives is as simple as I described above... if you can take a standard derivative, you can take a partial derivative.

Also: props for wading into the deep end yourself! I know some of this stuff might seem intimidating, but if you do what you're doing (make sure you understand as much as you can instead of blowing ahead) you'll be able to follow this trail as far as you want to go. Good luck, and feel free to hit me up if you have any specific questions, I'd be happy to share.

You're English is great.

I'd like to reemphasize /u/Plaetean's great suggestion of learning the math. That's so important and will make your later career much easier. Khan Academy seems to go all through differential equations. All of the more advanced topics they have differential and integral calculus of the single variable, multivariable calculus, ordinary differential equations, and linear algebra are very useful in physics.

As to textbooks that cover that material I've heard Div, Grad, Curl for multivariable/vector calculus is good, as is Strang for linear algebra. Purcell an introductory E&M text also has an excellent discussion of the curl.

As for introductory physics I love Purcell's E&M. I'd recommend the third edition to you as although it uses SI units, which personally I dislike, it has far more problems than the second, and crucially has many solutions to them included, which makes it much better for self study. As for Mechanics there are a million possible textbooks, and online sources. I'll let someone else recommend that.

What level E&M? If it is intro physics 2 then look for AP physics B/C stuff in addition to what you would normally look for since that's the same level.

If it is an upper division E&M class then I will recommend a book you can probably find in most of your professors offices somewhere: Div, Grad, Curl, and All That. Older editions are much cheaper even and archive.org has a PDf of the 3rd edition. I have no idea what the differences are, but I have the 4th and it is just great.

I have yet to find an E&M textbook I like. Griffiths is alright and when paired with Div, Grad, Curl and maybe a Schaum's outline on E&M it forms what I think should just be one textbook.

As for online resources I think The Mechanical Universe about Maxwell does a great job at covering Maxwell's laws, especially the bit starting around 15 minutes in

I've never used this site but it looks like it has a bunch of solved problems as well.

For your calculus brush-up, I would wholeheartedly recommend Calculus Made Easy by Silvanus P. Thompson. Available as pdf here or a newer, revised edition from Amazon here in which Martin Gardner has updated terminology, notation and such, as well as adding some excellent introductory chapters that help with the intuition. It is a deceptively small book with around 300 A5 sized pages, but it delivers most everything you need to know about calculus, including many handy tricks, in a intuitive down to earth style. Each chapter has a bunch of problems of varying degree along with solutions in the appendix. To top it off, Richard Feynmann was introduced to calculus from this book too...

In my opinion, a solid and intuitive understanding of calculus is one of the most important aspect of understanding much of physics, and the book has certainly helped me a great deal.

Another important aspect is of course vectors, for which I enjoyed the slightly unusual treatment in About Vectors by Banesh Hoffmann, although I'm unsure if it is fitting for revisiting.

I'm preparing to go from a pure maths/stats background to an applied maths graduate program in the fall, and I bought both of these books:

1. Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, by H.M. Schey, and
   
2. A Student's Guide to Maxwell's Equations, by Daniel Fleisch.
   
   Hoping they'll help me when I get home to read them, maybe they'll help you too? The Amazon feedback seems pretty positive. Good luck!

same here. In the rush to get everyone ok with matrix arithmetic, I feel like most students never realized that the matrix was just a nifty way to do bookkeeping. And it's not a bad thing; it's a great tool to use and if you're on the STE side of STEM majors there might not be a lot of reason to go above it. But it does make for a wild transition when/if you ever go up a level in abstraction.

I, for example, had no idea at all. Then I was helping a friend with some programming and we stumbled onto quaternions for 6DoF modeling. Talk about a 180. I feel like I'm still re-wiring all of my algebraic understanding from that point backwards. This book was fun to read when I made that turn too.

Geometric algebra sounds extremely interesting. I definitely have heard that it makes a lot of vector calculus more intuitive, and apparently the results come more naturally from the framework. Most people I talk to haven't heard about it, and I'm surprised to see it being so applicable to so many fields. Especially interesting was when they said the theory isn't exactly equivalent in GR, leading to different calculations. Kind of crazy to see that in GA, curved spacetime isn't a thing. I'm not sure how that would work, since isn't the big picture in GR about particles moving through geodesics in curved space?

As an undergrad, I would definitely love the possibility of taking a class on it. I've seen a book that introduces linear algebra and geometric algebra together, though I haven't really gone through it that much. The author even made a textbook to teach vector calculus and geometric calculus as a natural generalization. Maybe one day I'll sit down and go through it.

As SDogwood said you need an intro in proofs. Unfortunately I know of no better way to do this than to sit in on a class such as "intro to abstract algebra", I took it as a junior level course. I can't even tell you what book to use, cause the prof wrote their own ~100 page book and sold it as notes for like $5. The most difficult part of the class was actually having the will power to show up. I did almost no work outside of the class. One of the things that the department required the class do is make the students present a proof, normally 3 students at the beginning of class and you would rotate through the roster. Shockingly it was one of the more fun classes I took. If you can do this it is probably your best option. I know engineering curriculums can be tight, but you really should see if maybe there is a night version or something, its worth it.

I would also suggest just picking out some dover books like this one and working through it on your own. Stuff like that you won't need proofs for and depending on what type of engineer you are may also be of help.

Not sure if they qualify as "beautifully written", but I've got two that are such good reads that I love to go back to them from time to time:

• One Two Three . . . Infinity
  
• Div, Grad, Curl, and All That

For single variable calculus, like everyone else I would recommend Calculus - Spivak. If you have already seen mechanical caluculus, mechanical meaning plug and chug type problems, this is a great book. It will teach you some analysis on the real line and get your proof writing chops up to speed.

For multivariable calculus, I have three books that I like. Despite the bad reviews on amazon, I think Vector Calculus - Marsden & Tromba is a good text. Lots of it is plug and chug, but the problems are nice.

One book which is proofed based, but still full of examples is Advanced Calculus of Several Variables - Edwards Jr.. This is a nice book and is very cheap.

Lastly, I would like to give a bump to Calculus on Manifolds - Spivak. This book is very proofed based, so if you are not comfortable with this, I would sit back and learn from of the others first.

A list of several GR books (mostly lecture notes from classes)

Einstein's 1920 relativity book (notation is a bit old fashioned)

Or Pauli's book (written at age 21 the bastard), which can be bought for ~$4 on amazon used which appears to be under copyright still though google search seems to show available for free download.

I have Pauli/Einstein books and find them less useful than modern books.

I learned GR as an undergrad using F&N and liked it for being short and to the point + Schaum's book on Tensor Analysis, which can be obtained for ~$35 and ~$5 on amazon used.

> Second and third semester calculus

Is this vector calc? If so I enjoyed this book as it's very geometric, not at all rigorous and has lots of worked examples and exercises. Sorry it seems to be so expensive -- it wasn't when I bought it, and hopefully you can find it a lot cheaper if it's what you're looking for.

In general Stewart's big fat calculus book is a nice thing to have for autodidacts.

Obviously what you describe might include analysis, which these books won't help with.

>Formal logic theory (Think Kurt Godel)

I've heard Peter Smith's book on Godel is good, but haven't read it. Logic is a huge field and it depends a lot on what your background is and what you want to get out of it. You may need a primer on basic logic first; I like this one but again it's quite personal.

I thought of some books suggestions. If you're going all in, go to the library and find a book on vector calculus. You're going to need it if you don't already know spherical coordinates, divergence, gradient, and curl. Try this one if your library has it. Lots of good books on this though. Just look for vector calculus.

Griffiths has a good intro to E&M. I'm sure you can find an old copy on a bookshelf. Doesn't need to be the new one.

Shankar has a quantum book written for an upper level undergrad. The first chapter does an excellent job explaining the basic math behind quantum mechanics .

For multivariable calculus I cannot recommend Div, Grad, Curl and All That enough. It’s got wonderful physically motivated examples and great problems. If you work through all the problems you’ll have s nice grasp on some central topics of vector calculus. It’s also rather thin, making it feel approachable for self learning (and easy to travel with).

1. Vector Calculus isn't just a required math course, and the often-suggested supplementary textbook Div, Grad, Curl, and All That has a terribly misleading title - VC's not just a temporary annoyance, you'll actually need this stuff later.
   
   
2. Same for probability. If you skate thru probability hoping you can forget it right away, you're gonna have a bad time in your Signals classes and your Communications classes later. Stochastic Processes will strangle you and urinate on your corpse.
   
   
3. During your internship(s), do your best to befriend the engineers you work around & with. They have much to teach you and can give you excellent advice after your internship is over. Plus they can write letters of reference that are a lot more influential than your Logic Design professor can write.
   
   
4. No matter how much you enjoyed your Chemistry classes, and no matter how well you did in them, it turns out that Chemistry is 99% irrelevant to EE. Sorry.
   
   
5. Programming and software are a fact of EE life. Become a good coder and don't let your skills atrophy. Learn Linux or at least UNIX or at least the UNIX underpinnings of MAC OSX. Learn command line tools.
   
   
6. Often the best EEs are the ones with the most bravery, the least afraid of the unknown. "I've never done that before" is a reason to jump in and try something, NOT an excuse to back away.
   
   
7. Analysis Paralysis really does exist. Avoid it.

I looked at the free pages on Amazon and it does seem a bit wordier than the physics books I remember. It could just be the chapter. Maybe it reads like a book; maybe it's incredibly boring :/

If money isn't an issue (or if you're resourceful and internet savvy ;) you can try the book by Serway & Jewett. It's fairly common.

http://www.amazon.com/Physics-Scientists-Engineers-Raymond-Serway/dp/1133947271

As for DE, this book really resonated with me for whatever reason. Your results may vary.

http://www.amazon.com/Course-Differential-Equations-Modeling-Applications/dp/1111827052/ref=sr_1_2?s=books&ie=UTF8&qid=1372632638&sr=1-2&keywords=differential+equations+gill

If your issue is with the technical nature of textbooks in general, then you'll either have to deal with it or look for some books that simplify/summarize the material in some way. The only example I can come up with is:

http://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161/ref=sr_1_1?s=books&ie=UTF8&qid=1372632816&sr=1-1&keywords=div+grad+curl

Although Div, Grad, Curl, and all That is intended for students in an Electromagnetics course (not Physics 2), it might be helpful. It's an informal overview of Calculus 3 integrals and techniques. The book uses electromagnetism in its examples. I don't think it covers electric circuits, which are a mess of their own. However, there are tons of resources on the internet for circuits. I hope all this was helpful :)

For vector calculus: Div, Grad, Curl, and All That: An Informal Text on Vector Calculus

For complex variables/Laplace: Complex Variables and the Laplace Transform for Engineers - Caution! Dover book! Slightly obtuse at times!

For the finite difference stuff I would wait until you have a damn good reason to learn it, because there are a hundred books on it and none of them are that good. You're better off waiting for a problem to come along that really requires it and then getting half a dozen books on the subject from the library.

I can't help with the measurement text as I'm a physicist, not an engineer. Sorry. Hope the rest helps.

I recommend Tensors Demystified

Think of them as bras and kets. The downsies indices are bra like, they eat kets and spit out c-numbers. The upsies are ket like, they correspond to the original vector space. You can combine them to give two indices in the same manner as taking the tensor product of two or more kets to get the full state.

I've heard that, while Spivak's Calculus may be difficult because of proofs, it is good. However, his Manifolds is basically a graduate level reference book, and isn't the best multivariable calculus book for rebuilding/reteaching the basics of it. I've read that this is good in that regard.

I'd also hope to find a book that goes into the physics side. I've heard this is good for that.

Have you heard anything on these? Have other suggestions?

This is the best damn self study book I have ever seen on the subject and think it does better than the latter half of Math 53 in setting up many of the key concepts.

It is short, to the point, and from the outset makes the connections to EM abundantly clear. It is not difficult to find copies of that text online.

I thought that Marsden and Tromba was a pretty good book. It does a lot of stuff in n-dimensions, which you wouldn't need for E&M, but everything is there and it is computation oriented, rather than "proofy". You know it's good for physics because it has a picture of Newton on the front!

Read this for the basic algebriac perspective (really only need the super short first chapter on tensors), then this for the application to general relativity, which is, to a good approximation, just tensor analysis on manifolds (mainly chapter 2 and 3).

Precalculus is not an actual field, so I do not know what exactly is taught in the class, but the best book I know of on analysis would be A Course of Pure Mathematics - G.H. Hardy

As others have mentioned, there are a lot of good books on Math Methods of Physics out there (I used Hassani's Mathematical Methods: For Students of Physics and Related Fields).

That said, if you're having trouble with calculus, I'd recommend going back and really understanding that well. It underlies more or less all the mathematics found in physics, and trying to learn vector calculus (essential for E&M) without having a solid understanding of single-variable calculus is just asking for trouble.

There are a number of good books out there. Additionally, Khan Academy covers calculus very well. The videos on this page cover everything you'd encounter in your first year, and maybe a smidge more.

Once you move on to vector calculus, Div, Grad, Curl and All That is without equal.

Favorite Book Ever

http://www.amazon.com/Geometrical-Vectors-Chicago-Lectures-Physics/dp/0226890481

Vector Calculus & Geometry. This is the clearest, most enjoyable & enlightening math book I have ever read.

Try this.

But really it comes with practice, the more you use it, the better you get at reading it and comfortable with it. In my case at least.

There's a book called Div, Grad, Curl and All That, here is an Amazon link. It's an informal approach to vector mathematics for scientists and engineers and it's pretty readable. If you're struggling with the math, this is for you :) All their examples are EM too.

It's also a good idea to get a study group together. The blind leading the blind actually do get somewhere. :)

I found the book Div Grad Curl and All That to explain it pretty well. The book is short enough to read through in a couple hours.

I don't think you'll "spoil" what you'll learn later. If anything, seeing the material before will help you understand cooler stuff during the class next year. There's a lot of remarks and subtle examples I missed the first time I went through the standard undergrad math topics, that I only learned later.

But if you still want to avoid the topics you'll see in class, you could try some point-set topology (e.g. Munkres Topology). It would be beneficial for the real analysis class too. For differential geometry, I'd recommend Jänich Vector Analysis, which says it only needs calculus and linear algebra as prereqs.

For vector calculus, you might enjoy the less formal British text Div, Grad, Curl, and All That by H. M. Schey; for group theory in brief, consider the free textbook Elements of Abstract and Linear Algebra by Edwin H. Connell.

Alternatives to Schey's book include the much more formal Calculus on Manifolds by Michael Spivak, which does have more exercises than Schey but uses most of them to develop the theory, rather than as the mindless drills that fill an ordinary textbook; Michael E. Corral's free textbook Vector Calculus isn't huge but is written closer to an ordinary textbook.

http://www.amazon.co.uk/Vector-Calculus-Jerrold-Marsden/dp/0716749920/ref=dp_ob_title_bk Is supposed to be a good one and is recommended at uni.

https://www.amazon.com/Vector-Calculus-Jerrold-Marsden/dp/1429215089

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Someone has it listed here on reddit for free

I don't know about adsfgk, but I recommend Lovric.

You might enjoy Parry Moon’s Theory of Holors. There’s a whole cornucopia of obscure names for number-like things, https://www.amazon.com/Theory-Holors-Parry-Hiram-Moon/dp/0521019001 https://en.wikipedia.org/wiki/Parry_Moon#Holors

Instead of “-ions” though, we get “merates”.

Although it has already been answered, I recommend the book "Tensors, Differential Forms and Variational Principles" by Lovelock and Rund. From what I gather, you are looking for a more analytic approach and this is exactly what that book offers. It's a Dover publication, hence it is very cheap (currently under $10).
Link to its American amazon page

Calc 3 was series for us, 4 was multivariable. We were quarters with summer quarter being optional so it was really trimesters for most people. Vector calc was basically taught from the book Div, Grad, Curl and All That. So it was useful prior to going into electrodynamics, which was also 4th year.

​

EDIT: Added link.

I found this book quite useful for an intro to group theory.

Div, Grad, Curl, and All That is a good way to shore up your knowledge of vector calc.

It's useful for Electromagnetic physics. Surface integrals are used for finding the flux through a Gaussian surfaces so you can use Gauss' Law on non-symmetrical surfaces. Line integrals are used with Ampere's Law to find the magnetic flux. Once you learn the mechanics of working with multivariable calculus, you should read "Div, Grad, Curl and All That"

When I took EM in addition to Cheng the professor suggested getting Div, Grad, Curl and all of that. I found that to be alot of help in solidifying the math and intuition needed.

• Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach
  
  
• Div, Grad, Curl, and All That: An Informal Text on Vector Calculus
  
  
• Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds
  
  
• Advanced Calculus of Several Variables (Dover Books on Mathematics)
  
  
• Calculus of Several Variables (Undergraduate Texts in Mathematics)

Late, but here are undergrad books on the subject: geometric algebra, geometric calculus.

A grad-type book that has both and their applications to physics would be this one

I'm currently learning the geometric algebra undergrad book. It's a good read so far, and the author keeps up with book errors.

I used this book for when I took the course:

https://www.amazon.com/Introduction-Vector-Analysis-Harry-Davis/dp/0697160998

It's a book you really need to read every word to get an understanding of all the topics. It explains the Frenet formulas well as well as the cross product. These bits of exposition may be lost on someone's first course in vector analysis. We also covered the optional sections on tensor calculus. If the students have had a proof based linear algebra course, then they will eat up tensor notation.

Haven't used it myself, but you might want to check out Div,Grad,Curl by Schey.

A commonly used book for this exact purpose is Div, Grad, Curl by Schey.

Most of the books by John Stillwell, Klaus Jänich and John Conway. In particular, The Four Pillars of Geometry ,Numbers and Geometry, Vector Analysis, Topology

A History of Vector Analysis: The Evolution of the Idea of a Vectorial System by Michael J Crowe

https://smile.amazon.com/History-Vector-Analysis-Evolution-Vectorial-ebook/dp/B00VO68LBE/ref=sr_1_1?ie=UTF8&qid=1492740635&sr=8-1&keywords=vector+history

Are you familiar with Div, Grad, Curl, & All That. If not you'd probably enjoy it.

http://www.amazon.com/dp/0471725692/ref=wl_it_dp_o_pd_S_ttl?_encoding=UTF8&colid=2UCFQZHNW5VVF&coliid=I1RPWVCSMOOV09 is one good suggestion, I've seen around here. It's on my wishlist and the book that I intend to work from.

Now I always struggled with vector calculus and its motivations. So I have this one waiting for me as well http://www.amazon.com/dp/0393925161/ref=wl_it_dp_o_pC_nS_ttl?_encoding=UTF8&colid=2UCFQZHNW5VVF&coliid=I20JETA4TTSTJY since I think it covers a lot of the concepts that I had the most trouble with in calc 3

http://www.amazon.com/Introduction-Tensors-Group-Theory-Physicists/dp/0817647147/ref=sr_1_1?ie=UTF8&qid=1370289295&sr=8-1&keywords=jeevanjee

This one is probably the most intuitive description of tensors and group theory I've read, but it's not free =(

I have been trying to study tensor calculus in my own time. I have this and this book. I'm finding it a bit difficult. Any suggestions?