54

Some of your students will become engineers, and engineers use complex numbers all the time, e.g., to represent impedance. This kind of thing is by far the most common application. Complex numbers are also used in quantum mechanics. after algebra II, they never use complex numbers until pretty much complex analysis. I assume you mean "they never use ...


32

I have worked with a lot of students coming out of courses such as yours who: passed the course by blindly memorising proofs, theorems, and algorithms; learnt nothing (lasting) except solving some calculating exercises, a very vague idea of some terminology; had no idea what they just learnt, why they learnt it, and how it relates too their field of study; ...


26

We owe students a presentation of the Fundamental Theorem of Algebra -- that every nonconstant polynomial has a root; or, equivalently, the marvelous fact that every polynomial of nth degree has precisely n roots (including multiplicities). When made, it serves as a capstone and culmination of all the work that the student has done in elementary algebra. Of ...


18

At my University, there are four different first-semester Linear Algebra courses taken by Undergraduates: Math 214, Applied Linear Algebra, is "an introduction to matrices and linear algebra... The emphasis is on concepts and problem solving. The sequence 214-215 is not for math majors. It is designed as an alternate to the sequence 215-216 for engineering ...


15

I am surprised that nobody has mentioned differential equations. If you know about complex numbers and Euler's formula then there is a beautiful unified theory of linear differential equations with constant coefficients in terms of the characteristic equation. Without complex numbers the theory becomes somewhat ad-hoc, with different solutions depending on ...


15

I believe you need to listen beyond what your student is saying. Your student is not saying "I want to do some applications in class." What your student is really saying is "I'm bored and lost and this is rapidly becoming a waste of time for me, so I'm making this suggestion because I care enough about my own learning and trying to connect with you is how I ...


13

I think the fundamental tenet of this question is simply false. Here are some of the many encounters that undergraduate college students have in my classes: In Calculus II, as an application of power series, we discuss Euler's identity: $$e^{i\theta} = \cos(\theta)+i\sin(\theta).$$ Still in Calculus II as an application of the preceding example, we derive ...


13

There's a reason why you can't find a good non-geometric example: when the dimensions of the axes on a graph are distinct, perpendicularity is units-dependent. There is thus no natural aspect ratio with which to draw the graph. If you change the scale for just one axis, you destroy any perpendicularity that was present, thus demonstrating that it was an ...


12

Consider something besides an "all or nothing" approach. Here's what I did a couple of times when the topic was optional and I didn't have much time, but I still wanted to give students an introduction to the method. Simply restrict yourself to introducing the method by defining the Laplace transform, computing it for some simple examples, explaining what ...


9

Symmetries give profound insight into structure. This is demonstrated by Noether's theorem, proved just under 100 years ago by Emmy Noether (probably the greatest female mathematician in history). Noether's theorem states that every symmetry of a physical system is there because something is being conserved. Examples: Translational symmetry -> ...


9

There is Methods of Modern Mathematical Physics by Reed and Simon, which is a 4-volume book which teaches functional analysis, with a focus on operators in Hilbert spaces. Its main aim is to provide a sound mathematical background for the methods used in quantum mechanics, but it serves well as a textbook on functional analyis (for example, it covers some ...


8

People tend to use the presence of symmetry in a phenomenon to simplify a model of it. For example, in physics, systems with certain symmetries are the easiest to model (e.g., an infinite plane, an infinitely long cylinder, a sphere, etc.). But we have to be careful here. Ian Stewart has a very nice article on symmetry-breaking. (The article is from his ...


8

At the introductory level, Erwin Kreysiz's Introductory Functional Analysis with Applications is excellent. See the reviews at amazon.com. It's probably a bit elementary for you at this point (still, it could be very suitable for others reading this thread for recommendations), but I recommend at least looking through a copy at the library. I've actually had ...


8

Just another question from a math guy showing his ignorance of math as a service course. 2nd order diffyQ with constant coeffiecients (most important diffyQ for applications) has complex roots in the characteristic equation. [IOW very standard part of standard ODE course; OFTEN part of even a second semester calculus course during the diffyQ survey...was ...


8

Another classic is the following: A rectangular floor measures $300 \text{ cm} \times 195 \text{ cm}$. What is the largest square tiles that can be used to cover the floor exactly?


8

Going back to Euclid, I have found questions such as `Given a large supply of rods of length $15$ and $21$, what lengths can be measured?' can appeal to students. This also motivates the result $\gcd(a,b) = ra+sb$ of the (extended) Euclidean Algorithm.


8

Some nice geometrical applications arise in the analysis of periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Concrete experience with implementations in toys like Spirograph also provides excellent motivation for more abstract concepts such as cyclic groups.


8

Given a set of sites determine which points are closer one of the sites than any of the others? If closer is measured by Euclidean distance then given the distinct points A and B the points equidistant from A and B lie on the perpendicular bisector of the segment A and B. The regions one obtains are known as the Voronoi diagram associated with the sites, and ...


6

Peter Lax: Functional Analysis. It is sometimes difficult to use because of its cryptic style, but it is a great source of applications and a great source of historical references on applications which motivated functional analytic concepts.


6

I'm fond of the Gini index, a useful and interesting measure of the "fairness" of income distribution and requires the ability to integrate.


6

One possible class of examples of high dimensional systems is discretizations of infinite dimensional ones. Differential equations, as suggested in the comments, are one option, but let me propose another one that hopefully seems interesting and useful to students. This is a linear problem that is solved several times every day in hospitals around the world....


6

Also having a short period of time to introduce my DE students to Laplace Transforms I began with two 'Axioms': (1) $\mathcal{L}\{c_1y_1(t)+c_2y_2(t)\}=c_1Y_1(s)+c_2Y_2(s)$ (2) $\mathcal{L}\{y^\prime\}=sY(s)-y(0)$. From these two we derived the Laplace transforms for polynomial, exponential and sinusoidal functions and proceeded to use those results and ...


6

Most people are not going to use most of what they use in high school. High school (in the US, at least) is about building a broad foundation for students to be able to jump into any specialization when they go to college. Some examples of things I haven't had to do since high school: Name the 5 (I think 5, maybe it changed?) biological kingdoms Diagram a ...


6

Here's a word problem for the greatest common divisor: 12 boys and 15 girls are to march in a parade. The organizer wants them to march in rows, with each row having the same number of children, and with each row composed of children with the same gender. What is the largest number of children per row that satisfies these constraints? There should be $\...


6

Builders use plumb lines and levels to determine vertical and horizontal directions (which are perpendicular) in order to ensure that floors are horizontal and walls are vertical. When dealing with vectors (in the plane or in space), which are used constantly in statics and dynamics, one often represents each vector as the sum of a pair of perpendicular ...


6

I love this example. Suppose you want to cut out an irregular triangle from the center of a piece of paper. You can do it with one straight cut, first folding flat along angle bisectors:           And of course the angle bisectors meet at the center of the incircle (Proposition 4, Book IV of Euclid):         &...


6

I challenge the assertion that students need to see applications in everything. When I first started teaching I labored under the delusion that I should explain connections to physics whenever I could (in calculus, DEQns, linear algebra etc.). Now, I think my efforts do have an audience, but not the main audience that I find in my classes. Personally, I've ...


6

I'd like to expand a bit on The Chef's answer. Specifically, there's no need to require any kind of emphasis on "real world" applications. That is: Generally "real world" applications refers to some kind of industry mechanism in which some piece of mathematics is directly used (circuits, GPS, etc.). A student uninterested in a particular aspect of ...


5

What is Mathematical Modeling? You might think this to be a simple, straightforward answer, but unfortunately we have no such luck. The definition of mathematical modeling varies depending on the author unlike other more clearly defined mathematical terms like prime or group. Pollak Definition Arguably, one leading mathematician in the field of modeling ...


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