# What do I study in calculus beyond the minimum required for undergraduate engineering?

I am majoring in aerospace engineering. I love math a lot, especially calculus. I have a lot of free time right now and I want to learn more stuff in calculus that would be helpful for my major but I am confused where to start. Do you have any recommendations with good books or resources?

The things I have studied in mathematics:

• limits in single variable
• differentiation single variable
• integration single variable
• some series and sequences
• linear algebra
• complex analysis
• Are you saying that you want to learn more calculus than what you are being required to take in your undergraduate course? Nov 25 '19 at 12:26
• Yes. But the part that would be actually useful in research fields in engineering Nov 25 '19 at 12:31
• I have edited the question with the intention of making it clearer. If have inadvertently altered what you mean, please roll back my edit or make other changes using the edit button below the question.
– J W
Dec 13 '19 at 15:26
• Have you also studied multivariable calculus? It is not in your list, but would usually come before complex analysis.
– J W
Dec 13 '19 at 15:37
• No I have done basic DE like very basic Dec 13 '19 at 16:26

There is a big old textbook Advanced Engineering Mathematics by Kreyszig. Maybe looking at its table of contents HERE will show you what mathematics he thought was useful for engineering students.

(copied from the web site)

WILEY

Chapter 1: First-Order ODEs
Chapter 2: Second-Order Linear ODEs
Chapter 3: Higher Order Linear ODEs
Chapter 4: Systems of ODEs. Phase Plane. Qualitative Methods
Chapter 5: Series Solutions of ODEs. Special Functions
Chapter 6: Laplace Transforms
Chapter 7: Linear Algebra: Matrices, Vectors, Determinants. Linear Systems
Chapter 8: Linear Algebra: Matrix Eigenvalue Problems
Chapter 9: Vector Differential Calculus. Grad, Div, Curl
Chapter 10: Vector Integral Calculus. Integral Theorems
Chapter 11: Fourier Series, Integrals, and Transforms
Chapter 12: Partial Differential Equations (PDEs)
Chapter 13: Complex Numbers and Functions
Chapter 14: Complex Integration
Chapter 15: Power Series, Taylor Series
Chapter 16: Laurent Series. Residue Integration
Chapter 17: Conformal Mapping
Chapter 18: Complex Analysis and Potential Theory
Chapter 19: Numerics in General
Chapter 20: Numeric Linear Algebra
Chapter 21: Numerics for ODEs and PDEs
Chapter 22: Unconstrained Optimization. Linear Programming
Chapter 23: Graphs. Combinatorial Optimization
Chapter 24: Data Analysis. Probability Theory
Chapter 25: Mathematical Statistics

Are you an undergrad majoring in aerospace engineering, or are you interested prerequisites from graduate programs in aerospace engineering?

For an undergrad program, note the following syllabus.

For great references on aerospace engineering, see Penn State's list here.

• The opening question you ask might be better as a comment under the OP's question. Just a thought. As for the helpful links, could you summarize or supply small quotations perhaps?
– J W
Dec 23 '19 at 8:39

You do not mention differential equations, but this is a very useful topic in engineering, aerospace included, and the list of topics/courses you have studied mean that you would be well prepared for it.

A relevant book is Goodwine's Engineering Differential Equations:

This book is a comprehensive treatment of engineering undergraduate differential equations as well as linear vibrations and feedback control. While this material has traditionally been separated into different courses in undergraduate engineering curricula. This text provides a streamlined and efficient treatment of material normally covered in three courses. Ultimately, engineering students study mathematics in order to be able to solve problems within the engineering realm. Engineering Differential Equations: Theory and Applications guides students to approach the mathematical theory with much greater interest and enthusiasm by teaching the theory together with applications. Additionally, it includes an abundance of detailed examples. Appendices include numerous C and FORTRAN example programs. This book is intended for engineering undergraduate students, particularly aerospace and mechanical engineers and students in other disciplines concerned with mechanical systems analysis and control. Prerequisites include basic and advanced calculus with an introduction to linear algebra.

[emphasis mine]

Here is the stereotypical undergraduate engineering math curriculum in the US. Of course every person/school is different, but this is a useful reference point for initial awareness. (The last sentence is actually covered by the single word "stereotypical" but some of the caveat Na...police need to be told twice.)

SEM 1: limits, differential calculus, start of integral calc

SEM 2: more integral calc (trickier methods), chapter on series, chapter on diffyQs. YOU ARE HERE.*

SEM 3: multivariable calculus

SEM 4: Ordinary differential equations (some repetition of chapter from SEM 2, but more detail). Typically most (even basic/applied texts) have a bit too much to cover in a semester and later topics (especially systems of equations, Laplace transform, and series methods) will get an abbreviated (even omitted) coverage.

SEM 5: "Engineering Mathematics" or "Methods for Engineers". This is the most vague, but a typical course would be about a half semester of linear algebra (matrices) and a half semester of PDEs/Fourier series. Both of these are topics that could easily get a full semester each, even for an applied engineer look...but few schools give that much time. [You do have a huge slew of foundational mechE courses like statics, dynamics, fluids, thermo before you even get into your aero majors courses where you learn lift and drag--there is just limited time in the schedule.] Although some will do a separate formal linear algebra requirement of a semester, in which case, the engine math course may go deeper into PDEs. I have also sometimes seen physicists (but not engineers) doing a two semester "methods for physicists" which is basically like two semesters of engine math. Typically undergrad engineers (especially non EEs) will not have complex analysis, even "for engineers" as part of their required sequence.

It is possible to get specialized texts for all of the semester 3 to 5 topics. However, the Kreyszig text covers them very well. It was written by a mathematician and actually teaches math, tricks and concepts/derivations (not just formulas in passing while learning some physical topic), but by someone from the prehistoric era when mathematicians liked engineers and physicists. It's a very good middle ground between the sort of unsatisfying explication you get within a technical topic (by people not that great at math OR teaching math) and the proof-centric (but not caring about tricks/techniques) approach of the pure math types. Within it, you will have a clear "about semester" coverage of the following:

• Multivariable ("vector") calculus
• ODEs (K reverses the order and has it first...nothing wrong with that)
• Fourier series and PDEs
• Linear algebra
• Complex analysis [This is "extra" above the minimum normally required for aero.]

If you pound all that out (and I mean do it like a martial artist, learning skill after skill, doing each drill), you'll have a very, very strong basic coverage. Better than the normal engineer. Of course you can go to even greater applied depth with specific texts and/or do theory side trips. But I would start with this baseline FIRST. Can build on the foundation later, if needed/time. [You do need to learn slip and drag and Reynolds number and such sometime!] I would just use Kreysig since it is a single book, to lay this baseline. At least it is self consistent. While still also being very easy to pick certain sections to work independently. [Other competitor books probably fine too. But, stay the hell away from Arfkin and Weber...it's a mess--see the Amazon reviews (and I have it)...physicists trying to write math methods and reads like a laundry list, not an integrated teaching book, really.]

Some other math topics, not on the main calculus path, but very useful for engineers. I have listed them in priority order. (Also I would prioritize (1) and (2) below BEFORE complex analysis, for an aero engineer.):

1. Probability and statistics ("probes and shafts"). Do a basic survey course. K has a chapter on this, if you want the fast version. But maybe worth it to do a simple, applied text/semester on its own. This has some of the "ODE problem" in that the "texts that sell" have more like 1.5-2 semesters of work (yes, even in a simple applied introduction) and teachers will skip many sections to breeze through the topic. And it is possible to go into much more depth on TOP of what a basic intro would do--people do Ph.Ds in it! (C'est la vie.)

2. Design of Experiments, Quality Methods, etc. May have a non-intuitive name. But the key concept is (grossly simplifying) how to design and evaluate partial factorial examinations of complex, poorly understood phenomena (multivariable and nonlinear, on steroids...i.e. real life, engineering, business, farming, social topics like crime and engineering, etc.). The classic text here is Box, Hunter, Hunter but there are other good ones as well. May be better to take an actual class here, versus self study, as you get access to a lab and perhaps university copy of helpful statistical computer program. If you are going to go get a job in industry, this might even be more useful than the basic probes and shafts...and there is a little overlap.

3. Numerical methods. This is probably overkill for the average engineer to do a specialization in. But if you are going into grad school would be nice to actually spend a concentrated time on the different methods and understanding their error characteristics and computation efficiency. K has a chapter or two here, but I would probably look at a course or self standing text. But again, if you never get to it, you can still probably hack away as needed ad hoc.

4. Perturbation methods. Bender and Orzag is the classic text. Really not needed unless you are a very high end (mathematically inclined) aero Ph.D. But it is sort of "more calculus".

Few other text recommendations.

A. I am a huge fan of Schaum's Outlines. Very clearly written (not too much text...get to the point, simple wording). Lots of drill problems AND solutions. And implicit "teach by do" approach. They are economical, not like the normal textbook market. Order one whenever you have an issue with a regular text, hard to understand teacher, or just want more drill problems. Can do it ad hoc as needed by topic.

B. I am a big fan of the Michael Lindberg engineer license prep text books. The EIT reference manual essentially covers an entire undergrad engineer curriculum (all the statics/fluids mechE crap, but even basics in circuits, chemistry, physics, math, econ etc.) WITH DRILL PROBLEMS [spend the extra few bucks and buy the complete solutions manual also]. All in one book (with thin paper admittedly). I have "literally" run circles around engineers (despite lacking a BSE) because of all the quick reference info in this book. (OK, figuratively running the circles, but I was extremely tempted to warhoop around their desks.) I don't know about aero, but the Lindberg mechE prep book (I have it) is of similar quality to the EIT volume.

*From your question (i.e. not knowing the landscape), comment answers, and questions on the main math site, it looks like you are about at the end of semester 2.

• Thank you very much Dec 22 '19 at 14:51
• It would be helpful if you would link the Kreyszig text Dec 22 '19 at 14:52
• Read this link (especially the story of the clerk and the encyclopedia entry for Correggio): nato.int/nrdc-it/about/message_to_garcia.pdf Dec 22 '19 at 15:01

I recommend learning multivariable calculus. I think the book Div, Grad, Curl and All That is a pretty good short book on this topic. I also think that the book Introduction to Electrodynamics by Griffiths has a good intuitive explanation of vector calculus. If you want a more rigorous exposition, Folland's book Advanced Calculus is good.

I'm going to plug my own book, "Calculus from the Ground Up," for several reasons:

1. It spends quite a bit of time showing you how to use calculus to create new formulas, which is important for engineering.
2. It shows how to use the intuition behind the rules to build up new rules as needed.
3. It shows how the intuition behind the math can be helpful for problem-solving in general.
4. It focuses on differentials instead of derivatives, which is what physicists/engineers tend to do.

It will be a review of the concepts for you, but I think it will also show it to you in a new, helpful way, that will be especially useful as you go forward.

• The OP wants to learn something new, not review what they already know.
– user507
Dec 13 '19 at 22:12
• I've often found in my life that the best new thing to know is to know what I already know at a deeper level or from a new perspective. Dec 14 '19 at 21:15
• Tangential comment, but it might be nice if on the Amazon page for your book, the table of contents and preface were accessible by clicking on the image of the book. Dec 20 '19 at 13:21