Calculus Textbook for Home Education Student

[Linuxkid]

Hello all,


I'm considering Calculus 4th edition from M. Spivak for my Senior year Calculus book along with George Simmons Calculus with Analytic Geometry for a joint 18.01 MIT OCW and self-study curriculum.
I understand that his book involves quite some rigor and ruthless problems, but I'm up for it.

My algebra training is up to Hyperbolic functions, Binomial series, basic matrices, and some derivatives (and the rest of Precalculus, like sets and sequences, polynomial etc).


So, other than Calculus 4th edition; what supplemental -or not- Calculus textbook do you suggest for a prospective engineering/physics major?



Thanks for the help.


-Nikos

 

[chiro]

Hey LinuxKid and welcome to the forums.

Do you have books or resources in mind for Linear Algebra?

The whole calculus thing in multiple dimensions is based on linear algebra not only to do calculations, but also to understand what is going on.

Basically you can think of a matrix as a general linear object like Y or X is as a normal 1D variable. The object itself though behaves a bit differently than if you want to multiply say x*y since you are now dealing in an arbitrary space and not just a single scalar number.

Linearity is pretty much the most fundamental object in mathematics. All of differential and integral calculus depends on it (since derivatives are linear operators which means understanding linear operators helps you understand differentiation across any number of variables) and it's basically the foundation for looking at arbitrary vector spaces and bases for such vector spaces.

You can think of vectors as just being arrows: what the arrows correspond to is not so much important, in the same way that what a normal 1D variable x corresponds to isn't really important.

Now the ideas in Spivak use linear algebra since Spivak looks at generalized forms of mathematics and these forums require generalized structures and the generalized linear structure is a matrix.

A matrix provides the most basic way to have a general linear transformation not only on high dimensional spaces, but also in-between spaces as well (So instead of going from 3D to 3D you can go from 3D to 5D or 5D to 2D and so on). It does this in a linear way, but the structure is what is important.

So before you jump into Spivak, my recommendation is to understand linear algebra and what linearity means in the context of having a single object (a matrix) and what that object means in the context of algebraic identities involving matrices (like A*B, A+B up to more complicated algebraic expressions involving norms and other attributes).

You'll need to understand this when you look at how Spivak introduces the idea of a derivative in n-dimensions.

The real key is to get intuition for these abstract linear objects actually do and what they represent and it will take a little getting used to, but once you do it, you'll see all these multi-variable generalizations and they will make more sense.

 

[micromass]

Hello all,


I'm considering Calculus 4th edition from M. Spivak for my Senior year Calculus book along with George Simmons Calculus with Analytic Geometry for a joint 18.01 MIT OCW and self-study curriculum.
I understand that his book involves quite some rigor and ruthless problems, but I'm up for it.

My algebra training is up to Hyperbolic functions, Binomial series, basic matrices, and some derivatives (and the rest of Precalculus, like sets and sequences, polynomial etc).


So, other than Calculus 4th edition; what supplemental -or not- Calculus textbook do you suggest for a prospective engineering/physics major?



Thanks for the help.


-Nikos

You should know that Spivak is a very hard book and is usually not meant for a first encounter in calculus. Before tackling Spivak, you should know a bit of calculus already and you should be pretty well-versed in proofs.
If you want to be challenged mathematically, then Spivak is the ideal book however. It has very difficult exercises.

Hey LinuxKid and welcome to the forums.

Do you have books or resources in mind for Linear Algebra?

The whole calculus thing in multiple dimensions is based on linear algebra not only to do calculations, but also to understand what is going on.

Basically you can think of a matrix as a general linear object like Y or X is as a normal 1D variable. The object itself though behaves a bit differently than if you want to multiply say x*y since you are now dealing in an arbitrary space and not just a single scalar number.

Linearity is pretty much the most fundamental object in mathematics. All of differential and integral calculus depends on it (since derivatives are linear operators which means understanding linear operators helps you understand differentiation across any number of variables) and it's basically the foundation for looking at arbitrary vector spaces and bases for such vector spaces.

You can think of vectors as just being arrows: what the arrows correspond to is not so much important, in the same way that what a normal 1D variable x corresponds to isn't really important.

Now the ideas in Spivak use linear algebra since Spivak looks at generalized forms of mathematics and these forums require generalized structures and the generalized linear structure is a matrix.

A matrix provides the most basic way to have a general linear transformation not only on high dimensional spaces, but also in-between spaces as well (So instead of going from 3D to 3D you can go from 3D to 5D or 5D to 2D and so on). It does this in a linear way, but the structure is what is important.

So before you jump into Spivak, my recommendation is to understand linear algebra and what linearity means in the context of having a single object (a matrix) and what that object means in the context of algebraic identities involving matrices (like A*B, A+B up to more complicated algebraic expressions involving norms and other attributes).

You'll need to understand this when you look at how Spivak introduces the idea of a derivative in n-dimensions.

The real key is to get intuition for these abstract linear objects actually do and what they represent and it will take a little getting used to, but once you do it, you'll see all these multi-variable generalizations and they will make more sense.

Spivak's calculus doesn't require any knowledge of linear algebra. So you can study Spivak and linear algebra in any order you want.

 

[chiro]

The point was that linear algebra is a good thing to know to understand calculus in many dimensions. Did you read my comment? I elaborated on this and some of the reasons in detail.

Also when you see the modern definition of a derivative as opposed to other definitions, then you'll see just how important understanding linear algebra can be.

 

[micromass]

The point was that linear algebra is a good thing to know to understand calculus in many dimensions. Did you read my comment? I elaborated on this and some of the reasons in detail.

Spivak doesn't do calculus in many dimensions, so no, I don't see the point.

 

[PKDfan]

Maybe he's referring to Calculus On Manifolds? I haven't read that book but from what I've heard linear algebra is necessary for understanding it.

 

[micromass]

Maybe he's referring to Calculus On Manifolds? I haven't read that book but from what I've heard linear algebra is necessary for understanding it.

Aaah, yes, that might be the case. I forgot Spivak had calculus on manifolds as well. And indeed, you absolutely need to know linear algebra to understand that.

 

[chiro]

Yeah that's the one I meant: should have clarified!

 

[Linuxkid]

Thanks for the replies guys,


I practiced some matrices and concepts in Linear Algebra, but only briefly as part of precalculus. But as clarified above, shouldn't be a problem with Calculus 4th edition.