baxtrom

Newton et al used series expansions to calculate logarithms. For example like this. And, if you've calculated logarithms of numbers from say 1 to 100, you can obtain log(500) as log(5) + 2*log(10), and log(13.2) as log(66/5) = log(66) - log(5) and so on. In used books stores you can still find old books with tables loaded with logarithms, probably calculated by some poor PhD student.

In reality there is of course no such thing as a point load. In structural analysis it is however often convenient to use a load model which is statically equivalent to the true load, i.e. a point load could be applied instead of a load with a complex distribution over a relatively small area. If a finite force is applied to a single node in a finite element model consisting of soild elements, stress levels in the immediate vicinity of the load would be "imaginary" and increase without bound if the mesh is refined, while some distance away the computed stress will approach the true stress. This fact is referred to as Saint-Venant's principle. In contrast, if classical Euler-Bernoulli beam theory is used, stress singularities are not captured and thus a point load will only produce global loads such as bending moments at the clamped end of a cantilever beam. In many cases local stress concentrations are not interesting unless fatigue or crack propagation is considered, so beam or shell formulations are powerful tools for simplifying a model.

In structural analysis, an applied bending moment [math]M[/math] can be thought of as two opposite point forces [math] F [/math] a distance [math] d [/math] apart. If there is a negative force at say [math]x=0[/math] and a positive force at [math]x = -d[/math], one then has [math] q = F \delta(x+d) - F \delta(x) [/math] where [math]q[/math] is the load intensity ("force per length"). Since [math]M = F d[/math] from elementary statics, we get [math]F = M/d [/math] and [math] q = M \frac{\delta(x+d) - \delta(x)}{d} [/math]. Now, if we let [math]d \to 0[/math] (i.e. applying a point moment to a structure) this looks very much like the derivative of a function. Is this a mathematically correct treatment of the delta "function" or is it just working anyway in the world of structural analysis?

There are commonly accepted orders of operation. Different calculators, for example, may use different rules. Correct use of brackets should eliminate the problem. The last example you give is definitely wrong in that aspect. Check out this wiki article.

Depends on distance between jet cars? If the jet cars drive at a distance (front-to-front) of say 100 ft apart then the total length of the cue is ca 360,000 x 100 ft = 6818 miles. The last jet car will pass the end of the bridge after driving 6818 miles + 3.4 miles ~ 6822 miles and will have done so in 1 hour. Since the cars are assumed to be spaced 100 ft apart they all have the same speed. If instead the jet cars drive in 360,000 parallel lanes, the average speed will be 3.4 miles per hour.

Ouch, bad, bad mathematician. English is not my native tounge - I live in Sweden - and thus it is highly improper to complain about minor mistakes of spelling! And, I recall in this very thread seeing horrible freaks of spelling, like "distributuins" and "whjat". And what is an "are"? Are you perhaps referring to the area measure equal to 100 m2? Come, come. Do not take everything personally, dear DrRocket. I am sure you have a lot of engineering qualification. Good for you! Remember, likewise, that not all engineers are occupied with soulless numbercrunching in MS Excel, or playing with 3D Cad softwares. Personally I work in the field of strength of materials, which has been advanced by engineers like Henri Tresca and mathematicians like Richard von Mises. Perhaps it is a fundamental lemma of psychology - a mechanism of defense - that mathematicians often feel a need to shall we say look down on engineers, since engineers get all the chicks! Mmm.. I have the feeling this thread is getting OT. To get back into tracks, I put forward a question for you math gurus out there; are there cases in applied sciences where the basic principles of the delta "function", i.e. sifting property, integrates into Heaviside unit step function et c, will not suffice and a more rigourus definition in terms of "measures" or similar is needed? Seems to me that in most textbooks on physics where the delta "function" is introduced the fact that it is not a function in the true sense is at best only mentioned. Don't get upset now, it's just a humble question!

So they worked with mathematical tools for solving technical problems? As did Archimedes, Leibniz, Newton et al. Sounds like engineering to me. They were probably all engineers deep inside but too shy to admit it. "Engineer", btw, from the latin word ingenium, meaning cleaver. I bet they all were very engineerius!

In a pipe which is closed in one end (x = 0) and open in the other end (x = L), the boundary conditions become [math] u = 0 [/math] at [math] x = 0 [/math], and [math] p = 0 [/math] at [math] x = L [/math], where [math]u[/math] is the particle velocity and [math]p[/math] the acoustic pressure. The first condition is natural since the wall prohibits motion. The second condition is perhaps less intuitive but can be seen as "grounding" the acoustic pressure. The pressure wave is so to say short circuited into the void. And, as it happens the physics of wave propagation imply that if at [math] x=L [/math] there is a pressure node, then there must be a velocity antinode at the same location. This follows from the fact that the pressure in a channel can be written as the composition of one wave traveling in the +x direction and another traveling in the -x direction. Using complex notation, [math] p = e^{i \omega t} \Big[ A e^{-i k x} + B e^{i k x} \Big] [/math], where [math] \omega [/math] is the frequency and [math] k [/math] is the "wave number". The particle velocity becomes [math] u = \frac{1}{\rho c} e^{i \omega t} \Big[ A e^{-i k x} - B e^{i k x} \Big] [/math], where [math] \rho [/math] is the density and [math] c [/math] the speed of sound. Note the change of sign for the B-wave, which is due to the fact that it is traveling in the -x direction. Note also that if [math] u = 0 [/math] then also the derivative of [math] p [/math] is zero and thus the amplitude of [math] p [/math] is an extreme. The boundary condition at [math] x=L [/math] is actually only approximate and more complicated expressions are found in the litterature.

Yet sometimes it is engineers that push the frontiers of mathematics forward. "Mathematics is an experimental science, and definitions do not come first, but later on." Oliver Heaviside

One example of a sequence of functions that satisfies the basic properties of the [math]\delta(x)[/math] "function" in the limit [math]n \to \infty[/math] is [math]g_n(x) = \frac{n}{\pi (1 + n^2 x^2)} [/math] with properties [math] \lim_{n \to \infty} g_n(x) = 0 [/math] for [math] x \neq 0 [/math] and [math]\int_{-\infty}^\infty g_n(x) \mathrm{d} x = 1[/math] Since I'm an engineer I tend to use mathematics in a less rigorous way and have only faint clues about properties like "Lebesgue integrable" and "compact support" and similar stuff often discussed within the subject of generalized functions. We engineers use math like hammers and prying bars.

Ah, that was what the editor meant by "unencyclopedic". It's an "original idea"! Well, perhaps "fascism" is a better topic for wikipedia than "original" thoughts on the relationship between mathematical functions. Good point! Still I can't imagine such a simple idea to be original.

Oui, that's the classical relationship using complex math. I just thought it was interesting to note that there is also a very simple first order DE with obvious similarities to the exponential DE which results in a trig solution. Just look at them, they belong together [math] \frac{d y}{d x} = y(x) [/math], which results in the exponential solution, and [math] \frac{d y}{d x} = y(-x) [/math], which gives a trigonometric solution. (I don't remember how to get the d:s straight in LaTeX ). Perhaps it's just the romantic in me - a cold, rational mathematician would hardly give this pair of DEs a second thought. Still, a function class [math] y_{\nu}(x) [/math] which satisfies [math] y'(x) = y(\nu x ) [/math] is, if I'm not mistaken, only convergent for [math] \nu \in [-1 \ldots 1] [/math] and includes the trigonometric function above for [math] \nu = -1 [/math] and the exponential for [math] \nu = 1 [/math]. Obviously it's possible to construct a function class which contains any arbitrary pair of otherwise unrelated functions, yet still I find this apparant simple relationship between the exp and the trig functions in the real domain interesting. But wikipedia editors thought otherwise!

Hello fellow hobby-scientists, the normal differential equation-based definition of the trigonometric functions sin and cos are as solutions to the well-known second order DE y''(x) = -y(x). It occured to me that they can also be defined by a simple first order functional differential equation on the form y'(x) = y(-x), which for y(0) = 1 has the solution y(x) = sin(x) + cos(x). This obviously means it is possible to construct sin(x) as 0.5*(y(x) - y(-x)) and cos(x) as 0.5*(y(x) + y(-x)). As an attempted short section discussing this fact was swiftly and mercilessly rejected from the wikipedia article on trigonometric functions I'd like to ask you guys what you think. Does this definition add to the understanding of the nature of the sine and cosine functions and their apparent relationship to the exponential function (which is defined by y'(x) = y(+x), y(0) = 1..) ? As this is my first entry, try to be nice