hobz

I a spectrum employing this technique, and I didn't know why it was used. Have you ever seen it?

What are the advantages of normalizing a DFT with the frequency?

E.g.

A plot of DFT*frequency as a function of frequency.

As to the frequency, it's the same. If you drive electrons in a conductor at 1 kHz, you will get 1 kHz EM radiation.

That's pretty interesting. For if the creation of the photon happens to occur at some point, then the photon is, as such, unaware of the frequency of the jiggling at the point of its creation but still manages to have the "right" frequency property.

Steevy:

With one electron, there should be no leaping.

Hmm.. so any acceleration will give off photons.

Suppose I jiggle one electron at a certain frequency over a certain distance.

Will the number of emitted photons be related to the distance, and how?

How is the frequency of my jiggling related to the frequency (or wavelength) of the photon?

I'm not sure how you mean spontaneously here. The photons appear as a result of the acceleration of the charge, You can look at what happens to the field of the electron when this happens and see that it will radiate. The quantum nature means that the emission will not be continuous. Beyond that I really don't have much experience in the matter.

Suppose that I have one electron which I accelerate.

How far a distance must the electron move in order to cause the emission of one photon?

I.e. if the energy required to move the electron along a path must match the minimum energy of the emitted photon before the photon can be emitted?

Sort of like an energy-meter (a measuring device for energy) that has to be filled before a photon is emitted.

De-excitation of an atomic or nuclear system, acceleration of a charge, matter/antimatter annihilation.

Suppose I have an antenna where I drive the electrons back and forth using an alternating voltage source.

The acceleration will cause photons to emit. Do they spontaneously appear?

As photons carry momentum the accelerated charge will feel a force when the photon is created, right? Will this force accelerate the charge so that it will emit another photon?

Can the photon explanation be related to the wave explanation directly? As the electric(and magnetic) wave created by the acceleration does not propagate in the direction of acceleration, I assume that the wave function is zero in that direction.

How and when are photons created?

So is the tensor what takes a description in one coordinate system to the description in another?

Or is it the description it self?

Have a look at the Wikipedia entry.

According to this, "The upper indices are not exponents, but instead different axes. Thus, for example, x^2 should be read as "x-two", not "x squared", and corresponds to the traditional y-axis."

What is meant by "traditional y-axis"?

Anyway, the point is as Bignose also states that tensors have a very nice transformation law when changing local coordinates. It is so nice that any equations between tensors (physical laws) remain of the same form.

Could you point out a simple example of this?

Wow. Despite the friendly-sounding title, this book is quite the mountain. If you finish and understand this book, you will have an excellent understanding of fluids, but the learning curve of this book is unbelievably steep. This is a book over the heads of most graduate students.

 

Are you looking for an introduction to fluids or an introduction to tensors? I can recommend much more friendly books for either, that are still challenging without being overwhelming.

 

To answer the question, vector (which is really a tensor of rank 1) and tensor quantities follow certain rules when coordinate changes happen. These rules are in place to ensure that the principle that nature doesn't have preferred coordinate system remains in effect.

 

This is probably best explained by an example. Consider fluid flow in a (round) pipe. Because of the geometry, this is a problem that is very nicely described using the cylindrical coordinate system. However, nature doesn't know what a cylindrical coordinate system is. Nor what a Cartesian or spherical or bipolar or any of the other coordinate systems are. In the cylindrical coordinate system, we may describe the velocity at a certain point by [math] v_r, v_{\theta}, v_z[/math]. That velocity is the same as some other set of components in the Cartesian coordinate system [math] v_x, v_y, v_z[/math]. How you convert from one set to another is based on the rules of tensors. Shear stress in a fluid is a rank 2 tensor, and has components [math] \tau_{xx}, \tau_{xy}, \tau_{xz}, \tau_{yx}, \tau_{yy}, \tau_{yz}, \tau_{zx}, \tau_{zy}, \tau_{zz}[/math]. And, these components have some other values in another coordinate system. These components change from one to another coordinate system, following the rules of tensors.

 

They change values so that no matter what coordinate system you choose to describe the problem with, in the end you describe the same thing.

This is the most lucid explanation I have come across.

I was really looking into tensors, but please do recommend books for fluid introduction as well. Thank you.

Fix a smooth manifold. A tensor is a special kind of geometric object.

 

Definition: A geometric object (at a point) consists of

 

1. with respect to any allowable coordinate system there is one and only one ordered system of functions called components with respect to the given coordinate system.
   
2. a law which allows the representation of the components in an allowable coordinate system in terms of the components in any other allowable coordinate system, the corresponding coordinate transformations, the Jacobian matrix and their derivatives.
   

 

Characteristics of a geometric object are the number of coordinates, the highest order of derivatives in the transformation law and the particular representation of the groupoid of coordinate transformations the object forms.

 

 

Definition Tensors are first order geometric objects that form a linear representation.

 

(For supermanifolds you will want to be a bit more relaxed on the representation, but this is another story.)

 

So, the components of a tensor in a given coordinate system look like

 

[math]T^{A_{1} A_{2} \cdots A_{p}}_{B_{1} B_{2} \cdots B_{q}}(x)[/math].

 

Then under a coordinate change like [math]x^{A} \rightarrow x^{A'}(x)[/math] the components transform as

 

[math]T^{A_{1}^{\prime} A_{2}^{\prime} \cdots A_{p}^{\prime}}_{B_{1} ^{\prime}B_{2}^{\prime} \cdots B_{q}^{\prime}}= \left(\frac{\partial x^{B_{1}}}{\partial x^{B_{1}^{\prime}}} \right)\left(\frac{\partial x^{B_{2}}}{\partial x^{B_{2}^{\prime}}} \right)\cdots \left(\frac{\partial x^{B_{q}}}{\partial x^{B_{q}^{\prime}}} \right)T^{A_{1} A_{2} \cdots A_{p}}_{B_{1} B_{2} \cdots B_{q}} \left( \frac{\partial x^{A^{\prime}_{1} }}{\partial x^{A_{1}}} \right)\left( \frac{\partial x^{A^{\prime}_{2} }}{\partial x^{A_{2}}}\right) \cdots \left( \frac{\partial x^{A^{\prime}_{p} }}{\partial x^{A_{p}}} \right)[/math],

 

where I have used the Einstein summation convention. (I have not worried about any ordering here as everything is commutative, again on supermanifolds you would need to be a little more careful.)

 

Now, the important thing about these transformation laws is that they preserve the form of any tensor identity. Thus they are naturally suited to physics where nothing should really depend on the details of how you decide to present things. Other geometric objects, particularly densities, that is objects that pick up powers of the Jacobian are also very important in physics.

 

To stress things, in this local description using coordinates the transformation law is the vital thing when describing tensors or more general geometric objects. You can do things globally without local coordinates using natural vector bundles. That is in terms of sections of a vector bundle built entirely from the data of a smooth manifold (or supermanifold). In more sophisticated language a natural vector bundle is a functor from the category of smooth manifolds to vector bundles such that local changes of coordinates become vector bundle automorphisms. The canonical example here is the tangent bundle whose sections are vector fields.

 

Hope I have not confused you more than you already were.

Perhaps, but then again, I bet tensors (and their related background) are not taught as single-post forum replies

I am having trouble with your notation (or the summation notation). What does it mean?

Also," Tensors are first order geometric objects that form a linear representation", what does it mean to form a linear representation?

What is a tensor and why is it useful?

I have grabbed "Vectors, Tensors and the Basic Equations of Fluid Mechanics" by Rutherford Aris, but it is not a gentle introduction (some of the notation used is not explained at all!).

You can compare the piezoelectric material to a capacitor where, instead of charging it with a voltage, it is charged by mechanical stress.

So applying stress to the capacitor creates a voltage across the material. If you were to use this voltage (by allowing charge to flow) the voltage would drop until equilibrium was reached. Removing the rock would re-create a potential difference although reversing the sign.

As to how it produces the voltage, I suspect that there would have to be some electric dipole moment of the material that is altered by the application of stress.

To my understanding, the wave-nature of energy can explain a lot of phenomena with the exception of interaction with certain other energy manifestations e.g. light interacting with an electron.

Is the reason that the wave-nature cannot explain this that the superposition principle of a wave simply will add the two wave-forms of the photon and electron, and not cause a change in either's momentum?

If so, could the wave-theory be modified to somehow exclude superposition under some circumstances?

How does a digital input work?

I suspect that it's a flip-flop with a clock input, so that the value of the digital input is sampled at the rising (or falling) edge of the clock.

I see a problem with this though. If I wanted the digital input to be a counter, then, if the input is another clock running faster than the system (sampling) clock, I would not be able to count accurately.

So how are digital inputs usually implemented?

Ah yes, and then L'Hopitâl doesn't apply since the limit does not converge to 0/0.

Whilst you can use implicit differentiation, I think the OP is more interested in how you obtain this from the limit definition of the derivative. Having said this, I'm not 100% sure how to do it directly from the limit. The way you might approach it is to evaluate

 

[math]\lim_{x\to 0} \frac{e^x-1}{x} = e^x[/math].

 

This follows immediately from the Taylor series definition of [imath]e^x[/imath]. Then re-write [math]a^x = e^{x \ln a}[/math] and apply the chain rule.

Interesting. I will try and follow your suggestion, when I get some free time.

I've used

[math]

e = \lim_{x\rightarrow 0}\left(1+n\right)^{\frac{1}{n}}

[/math]

and substituted [math]a[/math] with that definition of [math]e[/math] in

[math]

\frac{\mathrm{d}}{\mathrm{d}x}\,a^x = a^x \cdot \lim_{h\rightarrow 0}{\frac{a^h-1}{h}}

[/math]

which eventually results in

[math]

\frac{\mathrm{d}}{\mathrm{d}x}\,e^x = e^x \cdot \lim_{h\rightarrow 0}{\frac{h}{h}}

[/math]

Edit:

Btw. Does the limit of [math]\lim_{h\rightarrow 0}{\frac{h}{h}}[/math] exist? Applying L'Hopitâl seems to yield 0/0 which is undefined.

In your derivation you use [math]f'(n)= a^n(\ln(a))[/math], but that is the very thing I am trying to prove. So how can the proof contain the original problem?

For a large part, I think measured by hand. Although the 'nice' values for the trig functions can be proven from the geometric axioms.

 

Logarithms only came up in the past couple of centuries, they'd have been done by (and I'd imagine still are to an extent) by approximate Riemann sums.

Interesting. Logarithms are far older than Riemann sums, as far as I know.

the calculator requires the use of javascript enabled and capable browsers This script will calculate any combination of logarithms, trigonometric functions, and the logarithms of trigonometric functions. Enter the angle value you want to apply the function to in decimal degrees.

Although the ancient Greeks are generally credited for alot of things, I doubt that javascript is one of them.

Thanks!

What is the limit of this thing

[math]

\lim_{n\rightarrow 0}\frac{a^n-1}{n}

[/math]

It has come up in an attempt of mine to find the derivative of [math]a^x[/math].

According to my derivatives table, the derivative of [math]a^x[/math] should be [math]a^x \ln(a)[/math], suggesting that the limit should converge to [math]\ln(a)[/math]. The question is, how do I arrive at this?

The opposite could result in overflow, if y and z are sufficiently far apart, and x is large enough.

I wonder if these rules of thumb are summarized somewhere?

I think we're all focussing a bit too much on Khaled's terrible choice of approximation. There is a valid point to be found in the fact that given finite precision, the accuracy of a calculation does depend on the way that calculation is performed as well as the inputs themselves.

So the point is, if you can come up with an approximation that is accurate down to the precision of the computer, then it would be worth examining whether one calculation method would perform better than another.

I see! So the implicit requirement for a function to be smooth in order to be differentiable, can be used "in reverse" to find out if a function is smooth.

This also implies that all smooth functions can be represented as a Taylor series?

I agree, but I think you misunderstood my question. I am asking how these things were calculated. Not how the need for them (or their names) came to be.

Merged post follows:

(Sorry for the "How was", but I appended logarithms at the last moment, and forgot to change to "were")

I was wondering how the ancient Greeks calculated sine, cosine, etc.

My guess is that they measured, and put the values in a table.

Likewise, how was the first logarithms (or inverse functions in general) calculated?