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I'm just finishing up a graduate course in computational topology which could be adapted very effectively for this purpose. We're focusing on topological data analysis and computational homology. All the topology in the course has been self-contained, meaning that essentially no previous experience in topology was required.
The book we're using is ...
9
I am unaware of standard mathematical texts for engineers at a level beyond that of the usual advanced engineering mathematics texts. Perhaps the closest thing in some sense might be Strang's classic Introduction to Applied Mathematics or his more recent Computational Science and Engineering. While the aforementioned books do overlap with typical advanced ...
9
Of course it depends on how much time you're willing to spend on this. If the answer is "very little" then no chance that you can say something more than "in the future this will be useful for you"...
So I start from the idea that you can devote at least one hour to this. And it better be a well prepared one, otherwise time will not be enough.
Codes: I do ...
8
Two Four ideas:
(1) "composing linear transformations": Use rotation, scaling, and shearing. If you extend to homogenous coordinates, you can include translations. Fundamental to all computer graphics. Explore which combinations of these transformations always commute, and which sometimes do not commute. In $\mathbb{R}^2$ and in $\mathbb{R}^3$. E.g., ...
8
Wow, thanks for the recent shout-out. I hope this is the right place for me to add a few references that might be useful and haven't already appeared in the answers.
Rob Ghrist has just written a fantastic new book called Elementary Applied Topology which provides a soaring and current overview of the field. There are no exercises (yet!) but the figures are ...
7
First: there seems to be a traditional belief that "pure" math fusses over tiny uninteresting details that "applied" math takes for granted, etc. Sure, we can operate this way, and make "pure math" as irrelevant as we want, or caricaturize it as such. Oppositely, if we try, we can caricaturize "applied math" as slip-shod fuzzy-thinking. :)
As I've ranted ...
7
I would start with a discussion with the Engineering department about what it is they want the students to learn. Teaching different skills will need different methods, so you might as well start with learning what will help you most in the short term. Personally, my experience suggests proofs are likely to be very low priority, with a focus on calculation ...
7
Not certain this satisfies your set of criteria, but...
This is a challenging but wonderful book. Arnold emphasizes
the geometry of manifolds throughout:
differential forms,
Riemannian geometry, symplectic geometry, Lie groups and Lie algebras, dynamical systems, integrable systems.
Arnold, Vladimir Igorevich. Mathematical Methods of Classical Mechanics. ...
7
In our department, large introductory math courses, such as calculus, linear algebra, and discrete mathematics, come together with little satellite courses called "advanced investigations in *", where * is the main course.
Students who want to explore the subject in depth register for the main course and the satellite course. Example: in main Calculus I, ...
7
It occurs to me that maybe you can approach this from the viewpoint of approximations. Briefly, I’m thinking of the kinds of approximations they likely use in practice, and which are often given in an appendix at the back of texts, such as the following:
For $x \approx 0$ we have $\frac{1}{1-x} \approx 1+x$ (multiply numerator and denominator of left side ...
7
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
Kreyszig:Advanced Engineering Mathematics, 10th Edition
Table Of Contents
Chapter 1: First-Order ODEs
Chapter 2: ...
7
I wouldn’t feel bad about leaving it out, but I think it’s a valuable conceptual example for understanding matrix algebra. Computing the QR decomposition is equivalent to applying Gram-Schmidt orthogonalization to the columns, and I think it’s really instructive to see how this corresponds exactly to the fact that Q is orthogonal and R is upper triangular (...
6
Here are two activities I have used. The second does not fit in an hour, however.
(1) Give the students maybe two different sizes and thickness of paper.
Ideally, the two are quite different.
Their goal is to design a paper airplane that flies the longest distance
when launched from a standard height.
Depending how ambitious you are, you could even brush
up ...
6
Geometric Methods and Applications for Computer Science and Engineering, by J. Gallier, does the job for what concerns classical differential geometry.
Elasticity and Geometry by Audoly-Pomeau addresses a classical problem with many important applications with quite advanced math techniques.
As for Linear Algebra, I'd like to mention Applied linear algebra ...
6
This may not be a direct hit, but since you mentioned "emphasis on visualization," may I suggest you investigate Tristan Needham's Visual Complex Analysis. E.g., see this MSE answer:
5
I strongly agree with Douglas Zare that it is highly dependent on what incoming students are prepared for in terms of their mathematical background beforehand. As far what is the minimum calculus knowledge that engineers need, this too is dependent on what field of engineering they go into. I know several engineers that use no more than basic trigonometry ...
5
Surprisingly not yet mentioned is
Advanced Mathematical Methods for Scientists and Engineers by Carl M. Bender and Steven A. Orszag
Below are some comments about this book I made in a 10 January 2007 sci.math post archived at Math Forum:
I had a course out of this book 23 years ago. If someone or some company had approached me (or I had known who to ...
5
I think you need to build a support network locally. Even if it means going outside the university to a nearby university, you will need mentors, fellow TA's, administrators, and others to help you not just understand and solve the problems you currently see, but to anticipate problems and improve your efforts as a teaching assistant.
Even going outside ...
5
Since "epsilon-delta" has been mentioned a couple of times by the question asker, I just thought I'd add my opinion that the epsilon-delta business should not be seen as a divider between pure and applied math. Understanding how sensitive the outputs of a function are to the inputs must certainly be important in real-world applications. Also, one cannot do ...
5
(edited)
I have used Kreyszig, Advanced Engineering Mathematics, fifth edition, as an undergrad, a practicing engineer, and in grad school. I love it.
(1) It is in English, composed in English, good English, but written by a German native speaker mathematician. Not sure that this matters but might be positive.
(2) One of high repute, check out his books ...
5
Let me suggest the following story: if we consider $z = x+iy$ as a point in the complex plane then we have $z = (x,y)$ in the usual Cartesian coordinate notation. Since
$$ z=x+iy = x(1)+y(i) = x(1,0)+y(0,1)$$
we have $1=(1,0)$ and $i=(0,1)$ in this approach. Viewing $z=(x,y)$ as a vector based at the origin we have:
$$ \boxed{z = r\hat{r}} $$
where $\hat{r}...
5
Are you an undergrad majoring in aerospace engineering, or are you interested prerequisites from graduate programs in aerospace engineering?
In either case, the following syllabus may help you consider courses as an undergrad so that you can round out strictly calculus coursework..
For an undergrad program, note the following syllabus.
For great ...
5
Solving least squares problems by QR factorization is much more numerically stable than solving them by Cholesky factorization of the normal equations. This can easily be demonstrated on an ill-conditioned test problem.
4
"What is my mission when I teach math for engineering students?"
I think it is possible to zig-zag between applications and appreciation of
the history and the beauty of the mathematics underlying those applications.
An example is the "simple suspension bridge," which follows
a catenary.
There is interesting history here, in that Galileo realized the curve ...
4
I learned linear algebra (decades ago) as an "adjunct" to "engineering mathematics."
The focus of most engineering students is "systems of equations." As long as you remember that, you're fine. So engineers are most focused on those parts of linear algebra that provide solutions to systems of equations, including lower echelon row reduction and eigenvalue ...
4
In computer science, specifically in combinatorics (much used in algorithm analysis) one important task is to derive asymptotic behaviour of sequences, which in turn are easiest to get in form of a generating function. The techniques used are heavily based on complex analysis. See for example Flajolet and Sedgewick's "Analytic Combinatorics" (Cambridge ...
4
I'd suggest having a look at Geometric Mechanics by Darryl Holm.
The theme of developing mechanics in the framework of differential geometry has a considerable record at the graduate level, as indicated in Dave Renfro's comment. At a level suitable for undergraduates, though, the pickings may be slim. Somewhat more accessible than Abraham and Marsden ...
4
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 ...
3
I'm currently looking at Steinmann, Geometrical Foundations of Continuum Mechanics (Springer 'Lecture notes in Mathematics and Mechanics' series), as a way of making differential forms relevant and palatable to senior engineering undergraduates.
3
Dave Renfro's advice [below] in his comment to your question is eminently sensible. Familiarize yourself with this particular course and what is expected of the student so you can help him/her best.
I would also add that many engineering mathematics courses, but certainly not all, have less emphasis on proofs than courses on the same topic intended for ...
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