# Does One Need Linear Algebra 'before' Spivak's Calculus?

1. Jun 24, 2009

So I am attempting to reteach myself Calculus with some rigor. So do I need Linear Algebra before Spivak's Calculus or along with? (Or at all?)

Thanks,
Casey

Last edited: Jun 24, 2009
2. Jun 24, 2009

### Astronuc

Staff Emeritus
That's an interesting question. I've seen advanced calculus taught with LA, with elements of LA and without LA. In my experience, Linear Algebra should be taught before or early in conjunction with calculus.

Based on the other thread, I looked at some of my old (1970's) texts on advanced calculus: Wilfred Kaplan's (2/E) and Francis Hildebrand's Advanced Calculus texts. Kaplan's book has linear algebra in the first chapter, whereas Hildebrand's does not. Perhaps Hildebrand assumed that a science or engineering student at that point would have had a course in LA. I took a course in linear algebra during my second year at university.

Wilfred Kaplan, University of Michigan

1. Vectors and Matrices.

Introduction.
Vectors in Space.
Linear Independence
Lines and Planes.
Determinants.
Simultaneous Linear Equations.
Matrices.
Scalar Times Matrix.
Multiplication of Matrices.
Inverse of a Square Matrix.
Gaussian Elimination.
*Eigenvalues of a Square Matrix.
*The Transpose.
*Orthogonal Matrices.
Analytic Geometry and Vectors n-Dimensional Space.
*Axioms for Vn.
Linear Mappings.
*Subspaces
Rank of a Matrix.
*Other Vector Spaces.

2. Differential Calculus of Functions of Several Variables.
3. Vector Differential Calculus.
4. Integral Calculus of Functions of Several Variables.
5. Vector Integral Calculus.
6. Infinite Series.
7. Fourier Series and Orthogonal Functions.
8. Functions of a Complex Variable.
9. Ordinary Differential Equations.
10. Partial Differential Equations.

Francis B. Hildebrand, Massachusetts Institute of Technology
http://www.pearsonhighered.com/educ...-Calculus-for-Applications/9780130111890.page

1. Ordinary Differential Equations.
2. The Laplace Transform.
3. Numerical Methods for Solving Ordinary Differential Equations.
4. Series Solutions of Differential Equations; Special Functions. Boundary-Value Problems and Characteristic-Function Representations.
5. Vector Analysis.
6. Topics in Higher-Dimensional Calculus.
7. Partial Differential Equations.
8. Solutions of Partial Differential Equations.
9. Solutions of Partial Differential Equations of Mathematical Physics.
10. Functions of a Complex Variable.
11. Applications of Analytic Function Theory.

Compare those two books with Spivak's book and Tom Apostol's Mathematical Analysis: A Modern Approach to Advanced Calculus, 2/E
(Tom M. Apostol, California Institute of Technology)

1. The Real and Complex Number Systems.
2. Some Basic Notions of Set Theory.
3. Elements of Point Set Topology.
4. Limits and Continuity.
5. Derivatives.
6. Functions of Bounded Variation and Rectifiable Curves.
7. The Riemann-Stieltjes Integral.
8. Infinite Series and Infinite Products.
9. Sequences of Functions.
10. The Lebesgue Integral.
11. Fourier Series and Fourier Integrals.
12. Multivariable Differential Calculus.
13. Implicit Functions and Extremum Problems.
14. Multiple Riemann Integrals.
15. Multiple Lebesgue Integrals.
16. Cauchy's Theorem and the Residue Calculus.

3. Jun 24, 2009

### fluidistic

At my university Calculus I and II are taught in the first year along with Linear Algebra. (I should say Mathematical Analysis since it's a proof based Calculus course) Calculus II and Linear Algebra are taught the same semester, so that when we start to study Calculus III we already know some Linear Algebra which is indeed important because you have to deal with determinants, vector spaces, etc.

But if you're studying single variable calculus then I don't think Linear Algebra will be of any use. If I remember well, Spivak's Calculus is of one variable, so sincerely I don't think Linear Algebra to be of any use.
Basically you'll prove all the theorems up to infinite series one. There is absolutely no need of Linear Algebra knowledge up to this, in my opinion.
It's a different story if you're looking to study vector based calculus though.

4. Jun 24, 2009

Thanks for the great responses guys.

Should one study Spivak before 'vector-based' Calculus or are they independent of each other? What is the difference?

There seem to be too many different types of calculus. What order does one preferably study them in?

5. Jun 24, 2009

### fluidistic

Spivak's Calculus is Calculus I and II if I remember well.
Yes you should first learn well this and later move on to vector based calculus.
Usually Calculus I consist of continuity of functions, limits of functions and sequences, derivatives, injective/bijective function, even/uneven functions, etc.
Calculus II deals with integral calculus, infinite series, Taylor series and probably more.
All this is with one variable functions.
While calculus III is basically very similar to calculus I and II but with several variables. Yes, there's a big difference and in order to tackle this course I think it's very nice to have a Linear Algebra course under your belt.
That's my personal experience. (I just finished my calculus 3 course)

Thus if you plan to study the whole Calculus sequence, I suggest you to study Linear Algebra along with Spivak's Calculus and when you're done with them, start with Vector Calculus.

6. Jun 24, 2009

### Nabeshin

Often you see universities label their calculus classes I-III(IV), each corresponding to a different subject matter:
Calc I: Differential calculus of a single variable
Calc II: Integral calculus of a single variable + taylor series
Calc III: Differential and integral calculus of multiple variables, vector calculus
Calc IV (sometimes): Ordinary differential equations

Certainly it makes sense to study I -> II -> III, as it is a logical progression, but IV does not necessarily come after III. Linear algebra plays a part in several proofs from calc III, but like fluidistic says it is largely irrelevant for I, II.

So, I think the best place for LA would be between calc II and III, assuming you do want the level of rigor in proving most of the theorems.

7. Jun 24, 2009

Okay. Well as odd as this may sound, I have already taken Calculus 1-4. We just never called Calculus 3 "Vector Calculus."

I am in Engineering, so it was from a more applied standpoint. I wish to reteach myself calculus from a more theoretical standpoint.

Spivak has been highly recommended to me for this purpose. I am looking for some rigor and to gain some experience with proofs.

I feel like some mathematical maturity is in order and that it will benefit me greatly in my self-study of more advanced physics topics.

8. Jun 25, 2009

### g_edgar

No

Spivack's "Calculus" neither presupposes nor uses linear algebra.