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Waves/2-3D Problems

Problems
Figure 2.20: Graphical representation of the dispersion relation for shallow water waves in a river flowing in the $$x$$ direction. Units of frequency are hertz, units of wavenumber are inverse meters. Figure 2.21: Sketch of a beam expander for a laser.
 * 1) Point A is at the origin. Point B is $$3 \mbox{ m}$$ distant from A at $$30^{\circ}$$ counterclockwise from the $$x$$ axis. Point C is $$2 \mbox{ m}$$ from point A at $$100^{\circ}$$ counterclockwise from the $$x$$ axis.
 * 2) Obtain the Cartesian components of the vector $$\mbox{D}_1$$ which goes from A to B and the vector $$\mbox{D}_2$$ which goes from A to C.
 * 3) Find the Cartesian components of the vector $$\mbox{D}_3$$ which goes from B to C.
 * 4) Find the direction and magnitude of $$\mbox{D}_3$$.
 * 5) For the vectors in the previous problem, find $$\mbox{D}_1 \cdot \mbox{D}_2$$ using both the cosine form of the dot product and the Cartesian form. Check to see if the two answers are the same.
 * 6) Show graphically or otherwise that $$\vert \mbox{A} + \mbox{B} \vert \ne \vert \mbox{A} \vert + \vert \mbox{B} \vert$$ except when the vectors $$\mbox{A}$$ and $$\mbox{B}$$ are parallel.
 * 7) A wave in the $$x$$-$$y$$ plane is defined by $$h = h_0 \sin ( \mbox{k} \cdot \mbox{x} )$$ where $$\mbox{k} = (1,2) \mbox{ cm}^{-1}$$.
 * 8) On a piece of graph paper draw $$x$$ and $$y$$ axes and then plot a line passing through the origin which is parallel to the vector $$\mbox{k}$$.
 * 9) On the same graph plot the line defined by $$\mbox{k} \cdot \mbox{x} = k_x x + k_y y = 0$$, $$\mbox{k} \cdot \mbox{x} = \pi$$, and $$\mbox{k} \cdot \mbox{x} = 2 \pi$$. Check to see if these lines are perpendicular to $$\mbox{k}$$.
 * 10) Two deep plane ocean waves with the same frequency $$\omega $$ are moving approximately to the east. However, one wave is oriented a small angle $$\beta$$ north of east and the other is oriented $$\beta$$ south of east.
 * 11) Determine the orientation of lines of constructive interference between these two waves.
 * 12) Determine the spacing between lines of constructive interference.
 * 13) An example of waves with a dispersion relation in which the frequency is a function of both wave vector magnitude and direction is shown graphically in figure 2.20.
 * 14) What is the phase speed of the waves for each of the three wave vectors? Hint: You may wish to obtain the length of each wave vector graphically.
 * 15) For each of the wave vectors, what is the orientation of the wave fronts?
 * 16) For each of the illustrated wave vectors, sketch two other wave vectors whose average value is the illustrated vector, and whose tips lie on the same frequency contour line. Determine the orientation of lines of constructive interference produced by the superimposing pairs of plane waves for which each of the vector pairs are the wave vectors.
 * 1) Two gravity waves have the same frequency, but slightly different wavelengths.
 * 2) If one wave has an orientation angle $$\theta = \pi /4 \mbox{ radians}$$, what is the orientation angle of the other? (See figure 2.6.)
 * 3) Determine the orientation of lines of constructive interference between these two waves.
 * 4) A plane wave impinges on a single slit, spreading out a half-angle $$\alpha $$ after the slit. If the whole apparatus is submerged in a liquid with index of refraction $$n = 1.5$$, how does the spreading angle of the light change? (Hint: Recall that the index of refraction in a transparent medium is the ratio of the speed of light in a vacuum to the speed in the medium. Furthermore, when light goes from a vacuum to a transparent medium, the light frequency doesn't change. Therefore, how does the wavelength of the light change?)
 * 5) Determine the diameter of the telescope needed to resolve a planet $$2 \times 10^8 \mbox{ km}$$ from a star which is $$6 \mbox{ light years}$$ from the earth. (Assume blue light which has a wavelength $$\lambda \approx 4 \times 10^{-7} \mbox{ m} = 400 \mbox{ nm}$$. Also, don't worry about the great difference in brightness between the two for the purposes of this problem.)
 * 6) A laser beam from a laser on the earth is bounced back to the earth by a corner reflector on the moon.
 * 7) Engineers find that the returned signal is stronger if the laser beam is initially spread out by the beam expander shown in figure 2.21. Explain why this is so.
 * 8) The beam has a diameter of $$1 \mbox{ m}$$ leaving the earth. How broad is it when it reaches the moon, which is $$4 \times 10^5 \mbox{ km}$$ away? Assume the wavelength of the light to be $$5 \times 10^{-7} \mbox{ m}$$.
 * 9) How broad would the laser beam be at the moon if it weren't initially passed through the beam expander? Assume its initial diameter to be $$1 \mbox{ cm}$$.
 * 1) Suppose that a plane wave hits two slits in a barrier at an angle, such that the phase of the wave at one slit lags the phase at the other slit by half a wavelength. How does the resulting interference pattern change from the case in which there is no lag?
 * 2) Suppose that a thin piece of glass of index of refraction $$n = 1.33$$ is placed in front of one slit of a two slit diffraction setup.
 * 3) How thick does the glass have to be to slow down the incoming wave so that it lags the wave going through the other slit by a phase difference of $$\pi$$? Take the wavelength of the light to be $$\lambda = 6 \times 10^{-7} \mbox{ m}$$.
 * 4) For the above situation, describe qualitatively how the diffraction pattern changes from the case in which there is no glass in front of one of the slits. Explain your results.
 * 5) A light source produces two wavelengths, $$\lambda_1 = 400 \mbox{ nm}$$ (blue) and $$\lambda_2 = 600 \mbox{ nm}$$ (red).
 * 6) Qualitatively sketch the two slit diffraction pattern from this source. Sketch the pattern for each wavelength separately.
 * 7) Qualitatively sketch the 16 slit diffraction pattern from this source, where the slit spacing is the same as in the two slit case.
 * 8) A light source produces two wavelengths, $$\lambda_1 = 631 \mbox{ nm}$$ and $$\lambda_2 = 635 \mbox{ nm}$$. What is the minimum number of slits needed in a grating spectrometer to resolve the two wavelengths? (Assume that you are looking at the first order diffraction peak.) Sketch the diffraction peak from each wavelength and indicate how narrow the peaks must be to resolve them.

title: Waves/2-3D Problems


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Template:Waves/GopticsTOC

Waves : Geometrical Optics 1 - 2 - 3 - 4 - 5 - 6 Problems

 * This template is part of the Waves book. This serves as a partial TOC for pages in the "Geometrical Optics" section. This template should appear at the top and bottom of every page in that section.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Template:Waves/GopticsTOC


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Waves/Total Internal Reflection

Total Internal Reflection
When light passes from a medium of lesser index of refraction to one with greater index of refraction, Snell's law indicates that the ray bends toward the normal to the interface. The reverse occurs when the passage is in the other direction. In this latter circumstance a special situation arises when Snell's law predicts a value for the sine of the refracted angle greater than one. This is physically untenable. What actually happens is that the incident wave is reflected from the interface. This phenomenon is called total internal reflection. The minimum incident angle for which total internal reflection occurs is obtained by substituting $$\theta_R = \pi /2$$ into equation (3.2), resulting in
 * $$ \sin ( \theta_I ) = n_R / n_I \quad \mbox{(total internal reflection)} . $$ (4.3)

title: Waves/Total Internal Reflection


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Waves/Anisotropic Media

Anisotropic Media
Notice that Snell's law makes the implicit assumption that rays of light move in the direction of the light's wave vector, i. e., normal to the wave fronts. As the analysis in the previous chapter makes clear, this is valid only when the optical medium is isotropic, i. e., the wave frequency depends only on the magnitude of the wave vector, not on its direction.

Certain kinds of crystals, such as those made of calcite, are not isotropic -- the speed of light in such crystals, and hence the wave frequency, depends on the orientation of the wave vector. As an example, the angular frequency in anisotropic medium might take the form
 * $$ \omega = \left[ \frac{c_1^2 ( k_x + k_y )^2}{2} + \frac{c_2^2 ( k_x - k_y )^2}{2} \right]^{1/2}, $$ (4.4)

where $$c_1$$ is the speed of light for waves in which $$k_y = k_x$$, and $$c_2$$ is its speed when $$k_y = - k_x$$.

Figure 3.3: The right panel shows the fate of a light ray normally incident on the face of a properly cut calcite crystal. The anisotropic dispersion relation which gives rise to this behavior is shown in the left panel.

Figure 3.3 shows an example in which a ray hits a calcite crystal oriented so that constant frequency contours are as specified in equation (3.4). The wave vector is oriented normal to the surface of the crystal, so that wave fronts are parallel to this surface. Upon entering the crystal, the wave front orientation must stay the same to preserve phase continuity at the surface. However, due to the anisotropy of the dispersion relation for light in the crystal, the ray direction changes as shown in the right panel. This behavior is clearly inconsistent with the usual version of Snell's law!

It is possible to extend Snell's law to the anisotropic case. We will not derive the equations here, but they are
 * $$ \frac{dx_i}{dt}=\frac{\partial \omega}{\partial k_i} \quad

\frac{dk_i}{dt}=-\frac{\partial \omega}{\partial x_i}$$

Later we will see a suggestive resemblance between these equation of geometrical optics and the equations of classical mechanics.

For now, though, we will assume that isotropic optical media are used, and discuss optical instruments.

title: Waves/Anisotropic Media


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Waves/Optical Instruments

Thin Lens Formula and Optical Instruments
Given the laws of reflection and refraction, one can see in principle how the passage of light through an optical instrument could be traced. For each of a number of initial rays, the change in the direction of the ray at each mirror surface or refractive index interface can be calculated. Between these points, the ray traces out a straight line.

Though simple in conception, this procedure can be quite complex in practice. However, the procedure simplifies if a number of approximations, collectively called the thin lens approximation, are valid. We begin with the calculation of the bending of a ray of light as it passes through a prism, as illustrated in figure 3.4.

The pieces of information needed to find $$\theta $$, the angle through which the ray is deflected are as follows: The geometry of the triangle defined by the entry and exit points of the ray and the upper vertex of the prism leads to

 * $$ \alpha + ( \pi / 2 - \theta_2 ) + ( \pi /2 - \theta_3 ) = \pi, $$ (4.5)

which simplifies to

 * $$ \alpha = \theta_2 + \theta_3 . $$ (4.6)

Snell's law at the entrance and exit points of the ray tell us that
 * $$ n = \frac{\sin ( \theta_1 )}{\sin ( \theta_2 )} \qquad n = \frac{\sin ( \theta_4 )}{\sin ( \theta_3 )}, $$ (4.7)

where $$n$$ is the index of refraction of the prism. (The index of refraction of the surroundings is assumed to be unity.) One can also infer that
 * $$ \theta = \theta_1 + \theta_4 - \alpha . $$ (4.8)

This comes from the fact that the sum of the internal angles of the shaded tetrahedron in figure 3.4 is $$( \pi /2 - \theta_1 ) + \alpha + ( \pi/2 - \theta_4 ) + ( \pi + \theta ) = 2 \pi$$.

Combining equations (3.6), (3.7), and (3.8) allows the ray deflection $$\theta $$ to be determined in terms of $$\theta_1$$ and $$\alpha $$, but the resulting expression is very messy. However, great simplification occurs if the following conditions are met:

 * The angle $$\alpha \ll 1$$.
 * The angles $$\theta_1 \ll 1$$ and $$\theta_4 \ll 1$$.

With these approximations it is easy to show that
 * $$ \theta = \alpha (n - 1) \quad \text{(small angles)} . $$ (4.9)

Generally speaking, lenses and mirrors in optical instruments have curved rather than flat surfaces. However, we can still use the laws for reflection and refraction by plane surfaces as long as the segment of the surface on which the wave packet impinges is not curved very much on the scale of the wave packet dimensions. This condition is easy to satisfy with light impinging on ordinary optical instruments. In this case, the deflection of a ray of light is given by equation (3.9) if $$\alpha $$ is defined as the intersection of the tangent lines to the entry and exit points of the ray, as illustrated in figure 3.5.

A positive lens is thicker in the center than at the edges. The angle $$\alpha $$ between the tangent lines to the two surfaces of the lens at a distance $$r$$ from the central axis takes the form $$\alpha = C r$$, where $$C$$ is a constant. The deflection angle of a beam hitting the lens a distance $$r$$ from the center is therefore $$\theta = C r (n - 1)$$, as indicated in figure 3.5. The angles $$o$$ and $$i$$ sum to the deflection angle: $$o + i = \theta = C r (n - 1)$$. However, to the extent that the small angle approximation holds, $$o = r/d_o$$ and $$i = r/d_i$$ where $$d_o$$ is the distance to the object and $$d_i$$ is the distance to the image of the object. Putting these equations together and cancelling the $$r$$ results in the thin lens formula:
 * $$ \frac{1}{d_o} + \frac{1}{d_i} = C(n - 1) \equiv \frac{1}{f} . $$ (4.10)

The quantity $$f$$ is called the focal length of the lens. Notice that $$f = d_i$$ if the object is very far from the lens, i. e., if $$d_o$$ is extremely large.

Figure 3.6 shows how a positive lens makes an image. The image is produced by all of the light from each point on the object falling on a corresponding point in the image. If the arrow on the left is an illuminated object, an image of the arrow will appear at right if the light coming from the lens is allowed to fall on a piece of paper or a ground glass screen. The size of the object $$S_o$$ and the size of the image $$S_i$$ are related by simple geometry to the distances of the object and the image from the lens:
 * $$ \frac{S_i}{S_o} = \frac{d_i}{d_o} . $$ (4.11)

Notice that a positive lens inverts the image.

An image will be produced to the right of the lens only if $$d_o > f$$. If $$d_o < f$$, the lens is unable to converge the rays from the image to a point, as is seen in figure 3.7. However, in this case the backward extension of the rays converge at a point called a virtual image, which in the case of a positive lens is always farther away from the lens than the object. The thin lens formula still applies if the distance from the lens to the image is taken to be negative. The image is called virtual because it does not appear on a ground glass screen placed at this point. Unlike the real image seen in figure 3.6, the virtual image is not inverted.

A negative lens is thinner in the center than at the edges and produces only virtual images. As seen in figure 3.8, the virtual image produced by a negative lens is closer to the lens than is the object. Again, the thin lens formula is still valid, but both the distance from the image to the lens and the focal length must be taken as negative. Only the distance to the object remains positive.

Curved mirrors also produce images in a manner similar to a lens, as shown in figure 3.9. A concave mirror, as seen in this figure, works in analogy to a positive lens, producing a real or a virtual image depending on whether the object is farther from or closer to the mirror than the mirror's focal length. A convex mirror acts like a negative lens, always producing a virtual image. The thin lens formula works in both cases as long as the angles are small.

title: Waves/Optical Instruments


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Waves/Fermat's Principle

Fermat's Principle
An alternate approach to geometrical optics can be developed from Fermat's principle. This principle states (in its simplest form) that light waves of a given frequency traverse the path between two points which takes the least time. The most obvious example of this is the passage of light through a homogeneous medium in which the speed of light doesn't change with position. In this case shortest time is equivalent to the shortest distance between the points, which, as we all know, is a straight line. Thus, Fermat's principle is consistent with light traveling in a straight line in a homogeneous medium. Fermat's principle can also be used to derive the laws of reflection and refraction. For instance, figure 3.10 shows a candidate ray for reflection in which the angles of incidence and reflection are not equal. The time required for the light to go from point A to point B is
 * $$ t = [ \{ h_1^2 + y^2 \}^{1/2} + \{ h_2^2 + (w - y)^2 \}^{1/2} ]/c $$ (4.12)

where $$c$$ is the speed of light. We find the minimum time by differentiating $$t$$ with respect to $$y$$ and setting the result to zero, with the result that
 * $$ \frac{y}{ \{ h_1^2 + y^2 \}^{1/2}} = \frac{w - y}{ \{ h_2^2 + (w - y)^2 \}^{1/2}} . $$ (4.13)

However, we note that the left side of this equation is simply $$\sin \theta_I$$, while the right side is $$\sin \theta_R$$, so that the minimum time condition reduces to $$\sin \theta_I = \sin \theta_R$$ or $$\theta _I = \theta _R$$, which is the law of reflection.

A similar analysis may be done to derive Snell's law of refraction. The speed of light in a medium with refractive index $$n$$ is $$c/n$$, where $$c$$ is its speed in a vacuum. Thus, the time required for light to go some distance in such a medium is $$n$$ times the time light takes to go the same distance in a vacuum. Referring to figure 3.11, the time required for light to go from A to B becomes
 * $$ t = [ \{ h_1^2 + y^2 \}^{1/2} + n \{ h_2^2 + (w - y)^2 \}^{1/2} ]/c . $$ (4.14)

This results in the condition
 * $$ \sin \theta_I = n \sin \theta_R $$ (4.15)

where $$\theta _R$$ is now the refracted angle. We recognize this result as Snell's law. Notice that the reflection case illustrates a point about Fermat's principle: The minimum time may actually be a local rather than a global minimum -- after all, in figure 3.10, the global minimum distance from A to B is still just a straight line between the two points! In fact, light starting from point A will reach point B by both routes -- the direct route and the reflected route.

Figure 3.13 illustrates a rather peculiar situation. Notice that all the rays from point O which intercept the lens end up at point I. This would seem to contradict Fermat's principle, in that only the minimum (or maximum) time trajectories should occur. However, a calculation shows that all the illustrated trajectories in this particular case take the same time. Thus, the light cannot choose one trajectory over another using Fermat's principle and all of the trajectories are equally favored. Note that this inference applies not to just any set of trajectories, but only those going from one focal point to another.

title: Waves/Fermat's Principle


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Waves/Geometrical Problems

Problems
Figure 3.14: Refraction through multiple parallel layers with different refractive indices. Figure 3.15: Refraction through a $$45^{\circ }$$-$$45^{\circ }$$-$$90^{\circ }$$ prism. Figure 3.16: Focusing of parallel rays by a parabolic mirror. Figure 3.17: Refraction through a wedge-shaped prism.

 * 1) The index of refraction varies as shown in figure 3.14:
 * 2) Given $$\theta_1$$, use Snell's law to find $$\theta_2$$.
 * 3) Given $$\theta_2$$, use Snell's law to find $$\theta_3$$.
 * 4) From the above results, find $$\theta_3$$, given $$\theta_1$$. Do $$n_2$$ or $$\theta_2$$ matter?
 * 5) A $$45^{\circ }$$-$$45^{\circ }$$-$$90^{\circ }$$ prism is used to totally reflect light through $$90^{\circ }$$ as shown in figure 3.15. What is the minimum index of refraction of the prism needed for this to work?
 * 6) Show graphically which way the wave vector must point inside the calcite crystal of figure 3.3 for a light ray to be horizontally oriented.
 * 7) The human eye is a lens which focuses images on a screen called the retina. Suppose that the normal focal length of this lens is $$4 \mbox{ cm}$$ and that this focuses images from far away objects on the retina. Let us assume that the eye is able to focus on nearby objects by changing the shape of the lens, and thus its focal length. If an object is $$20 \mbox{ cm}$$ from the eye, what must the altered focal length of the eye be in order for the image of this object to be in focus on the retina?
 * 8) Show that a concave parabolic mirror focuses incoming rays which are parallel to the optical axis of the mirror precisely at a focal point on the optical axis, as illustrated in figure 3.16. Hint: Since rays following different paths all move from the distant source to the focal point of the mirror, Fermat's principle implies that all of these rays take the same time to do so (why is this?), and therefore all traverse the same distance.
 * 9) Use Fermat's principle to explain qualitatively why a ray of light follows the solid rather than the dashed line through the wedge of glass shown in figure 3.17.
 * 10) Test your knowledge of Fermat's principle by finding the value of $$y$$ for which $$t$$ is a minimum in equation (3.14). Use this to derive Snell's law.

title: Waves/Geometrical Problems


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Template:Modern Physics/AspecrTOC

Special Relativity

1 - 2 - 3 - 4 - 5 - 6

title: Template:Modern Physics/AspecrTOC


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Modern Physics/Math/Four Vectors

Math Tutorial - Four-Vectors
We have seen that a wave is described by four numbers, the components of the spatial vector k, and the frequency ω

In special relativity these four numbers form a four-vector

It is called a four-vector because it has 3 spacelike components, forming a vector, and one timelike component when there are 3 space dimensions. It is called a four-vector because of the way it behaves when we change reference frames.

The spacelike component of the wave four-vector is just $$\mathbf{k}$$ when there are 3 space dimensions, while the timelike component is $$\omega/c$$ where the c is in the denominator to give the timelike component the same dimensions as the spacelike component.

Let us define some terminology. We indicate a four-vector by underlining and write the components in the following way: $$\underline{k} = (k, \omega /c )$$, where $$\underline{k}$$ is the wave four-vector, $$k$$ is its spacelike component, and $$\omega /c$$ is its timelike component. For three space dimensions, where we have a wave vector rather than just a wavenumber, we write $$\underline{k} = (k, \omega /c )$$.

Another example of a four-vector is simply the position vector in spacetime, $$\underline{x} = ( x, ct )$$, or $$\underline{x} = ( \mathbf{x}, ct )$$ in three space dimensions. The $$c$$ multiplies the timelike component in this case, because that is what is needed to give it the same dimensions as the spacelike component.

In three dimensions we define a vector as a quantity with magnitude and direction. Extending this to spacetime, a four-vector is a quantity with magnitude and direction in spacetime. Implicit in this definition is the notion that the vector's magnitude is a quantity independent of coordinate system or reference frame. We have seen that the invariant interval in spacetime is

 * $$I = \sqrt{x^2 - c^2 t^2}$$,

so it makes sense to identify this as the magnitude of the position vector. This leads to a way of defining a dot product of four-vectors.

Given two four-vectors

 * $$\underline{A} = ( \mathbf{A} ,A_t ) \quad

\underline{B} = ( \mathbf {B}, B_t )$$ then the dot product is
 * $$\underline{A} \cdot \underline{B} = \mathbf{A} \cdot \mathbf{B} -

A_t B_t \quad \mbox{(dot product in spacetime)} $$

This is consistent with the definition of invariant interval if we set
 * $$\underline{A} = \underline{B} = \underline{x}$$,

since then

 * $$\underline{x} \cdot \underline{x} = x^2 - c^2 t^2 = I^2$$.

Now, the key point about dot products for 3-vectors is that they are scalars, independent of the observer. They do not change if the axes are rotated, as was proven earlier.

For our definition of the dot product of four-vectors to be useful, it should also be independent of the observer. In particular, it should not depend on the observers velocity, else it would violate the principle of relativity.

We can easily check that our definition does satisfy this criterion.

It's clear that its independent of rotation, since it is the difference between a dot product and the product of two scalars, both of which terms aren't affected by rotating the coordinates.

Is it also frame velocity independent?

To check, first we need to be able to write down our four-vectors in the new reference frame. We know how to do this for the position vector — use the Lorentz transform. It can be shown that the same transform must hold for all vectors, so the components of a four vector in the new reference frame, moving at velocity v along the x-axis with respect to the previous one, are

 * $$\begin{matrix}

A'_x & = & \gamma \left( A_x - \frac{v}{c} A_t \right) \\ A'_y & = & A_y \\ A'_z & = & A_z \\ A'_t & = & \gamma \left( -\frac{v}{c} A_x + A_t \right) \end{matrix}$$

The dot product in this frame is

 * $$\underline{A'} \cdot \underline{B'} =

A'_x B'_x + A'_y B'_y + A'_z B'_z - A'_t B'_t $$

Simplifying, we get

 * $$\begin{matrix}

\underline{A'} \cdot \underline{B'} & = & \gamma^2 \left( A_x B_x -\frac{v}{c} (A_x B_t + A_t B_x) + \frac{v^2}{c^2} A_t B_t \right) \\ & & + A_y B_y + A_z B_z &\\ & & - \gamma^2 \left( \frac{v^2}{c^2}A_x B_x -\frac{v}{c} (A_x B_t + A_t B_x) + A_t B_t \right) \\ & = & \left(1- \frac{v^2}{c^2} \right)^{-1} \left( A_x B_x + \frac{v^2}{c^2} A_t B_t \right) \\ & & + A_y B_y + A_z B_z \\ & & - \left(1- \frac{v^2}{c^2} \right)^{-1} \left( \frac{v^2}{c^2} A_x B_x - A_t B_t \right) \\ & = & A_x B_x +  A_y B_y + A_z B_z   - A_t B_t \end{matrix}$$

which is just the dot product in the original frame, exactly as we wanted.

We now know that the dot product of two four-vectors is a scalar result, i. e., its value is independent of coordinate system. This can be used to advantage on occasion.

In the odd geometry of spacetime it is not obvious what perpendicular means. We therefore define two four-vectors $$\underline{A}$$ and $$\underline{B}$$ to be perpendicular if their dot product is zero, in the same way as with three-vectors.

 * $$\underline{A} \cdot \underline{B} = 0$$

Because the dot product is a scalar, if vectors are perpendicular in one frame, they will be perpendicular in all frames.

We can also consider the dot product of a four-vector $$\underline{A}$$ which resolves into $$(A_x, A_t )$$ in the unprimed frame. Let us further suppose that the spacelike component is zero in some primed frame, so that the components in this frame are (0, At' ) The fact that the dot product is independent of coordinate system means that

 * $$\underline{A} \cdot \underline{A} = A_x^2 - A_t^2 = - A_t'^2$$

This constitutes an extension of the spacetime Pythagorean theorem to four-vectors other then the position four-vector. Thus, for instance, the wavenumber for some wave may be zero in the primed frame, which means that the wavenumber and frequency in the unprimed frame are related to the frequency in the primed frame by $$k^2 - \omega^2 / c^2 = - \omega'^2 / c^2$$.

Proper time
Classically, the temporal derivative, d/dt acts like a scalar so we can multiply a vector by it, and get another vector.

In relativity t is part of a four-vector, which means d/dt also is, so we can't simply differentiate vectors with respect to t and expect to get vectors.

For example, the position of a stationary particle is (0, ct).

Viewed from a frame moving at v to the right, its position becomes (-vτ, cτ), where τ=γt is the time as measured in the moving frame.

If we differentiate with respect to τ the velocity would be (-v, c)

If we differentiate with respect to t, we get (0, c) in the stationary frame, which would be (using the Lorentz transform) (-γv, -γc) in the moving frame, if this were a four vector.

These two expressions differ by a factor of γ, when measured in the same frame, so this can not be a four vector.

However, if the moving observer divides by γ, which is the time dilation, they will get the same vector as the stationary observer.

Doing this is equivalent to differentiating by the time in the particles own rest frame. Since this works for the position vector, we can expect it to work for all vectors.

The time measured in a particle's rest frame is called its proper time.

Differentiating a vector with respect to proper time gives another vector, which is the relativistic equivalent of the temporal derivative.

title: Modern Physics/Math/Four Vectors


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Modern Physics/Principle of Relativity Applied

An application of relativity to waves
Returning to the phase of a wave, we immediately see that

 * $$\phi = \mathbf{k} \cdot \mathbf{x} - \omega t =

\mathbf{k} \cdot \mathbf{x} - ( \omega /c)(ct) = \underline{k} \cdot \underline{x} .$$

Thus, a compact way to write a wave is $$A( \underline{x} ) = \sin ( \underline{k} \cdot \underline{x} ) .$$(6.8)

Since $$\underline{x}$$ is known to be a four-vector and since the phase of a wave is known to be a scalar independent of reference frame, it follows that $$\underline{k}$$ is indeed a four-vector rather than just a set of numbers. Thus, the square of the length of the wave four-vector must also be a scalar independent of reference frame

 * $$\underline{k} \cdot \underline{k} =

\mathbf{k} \cdot \mathbf{k} - \omega^2 / c^2 = const .$$

Resolution of a four-vector into components in two different reference frames.

Let us review precisely what this means. As this figure shows, we can resolve a position four-vector x into components in two different reference frames, e.g (X,T) and (X′,T′), but these are just different ways of writing the same vector.

This is exactly the same as the way a three-vector has different components in a rotated frame.

Similarly, just as a three-vector has the same magnititude in all frames, so does a 4-vector; i.e,

 * $$X^2-c^2 T^2 = X^{\prime 2} - c^2 T^{\prime 2}$$

Applying this to the wave four-vector, we infer that

 * $$k^2-c^{-2} \omega^2 = k^{\prime 2} - c^{-2} \omega^{\prime 2}$$

where the unprimed and primed values of k and ω refer to the components of the wave four-vector in two different reference frames.

Up to now, this argument applies to any wave. However, waves can be divided into two categories, those for which a special reference frame exists, and those for which there is no such special frame.

As an example of the former, sound waves look simplest in the reference frame in which the gas carrying the sound is stationary. The same is true of light propagating through a material medium with an index of refraction not equal to unity. In both cases the speed of the wave is the same in all directions only in the frame in which the material medium is stationary.

If there is no material medium, then there is no unambiguous way of finding a special frame so the waves must fall into the second category. This includes all waves in a vacuum, such as light.

In this case the following argument can be made. An observer moving with respect to waves of frequency ω and wave number k sees waves of frequency ω′ and wave number k′. If the observer can tell in any way that they come from a source moving with respect to them, then they can use this to identify a special frame for those waves, so the waves must look just like ones from a stationary source of frequency ω

This forces us to conclude that for such waves

 * $$ \omega^2= c^2 k^2 + \mu^2 \,$$

where μ is a constant. All waves in a vacuum must have this form, a much more restricted choice than in classical physics.


text:
Modern Physics/Characteristics of Relativistic Waves

In classical physics, ω and k for light are related by

 * $$\omega = c k \,$$

In relativistic physics, we've seen that for waves with no special reference frame, such as light, ω and k are related by

 * $$\omega^2 = c^2 k^2 + \mu^2 \,$$

If μ=0 then the relativistic equation reduces to the classical, so we can assume that, for light, μ does equal zero.

This means that light does not have a mimimum frequency.

If μ is not zero then the wave being described are dispersive. The phase speed is
 * $$u_p = \frac{\omega}{k} = \sqrt{c^2 + \frac{\mu^2}{k^2}} $$

This phase speed always exceeds c, which at first may seem like an unphysical conclusion. However, the group velocity of the wave is
 * $$u_g = \frac{d\omega}{dk} = \frac{kc^2}{\sqrt{k^2c^2+\mu^2}}

= \frac{kc^2}{\omega} = \frac{c^2}{u_p}$$

which is always less than c. Since wave packets and hence signals propagate at the group velocity, waves of this type are physically reasonable even though the phase speed exceeds the speed of light.

Another interesting property of such waves is that the wave four-vector is parallel to the world line of a wave packet in spacetime. This is easily shown by the following argument.

The spacelike component of a wave four-vector is k, while the timelike component is ω/c. The slope of the four-vector on a spacetime diagram is therefore ω/kc. However, the slope of the world line of a wave packet moving with group velocity is c/ug, which is also ω/kc.

Note that when we have k is zero we have ω=μ. In this case the group velocity of the wave is zero. For this reason we sometimes call μ the rest frequency of the wave.


text:
Modern Physics/The Doppler Effect

The Doppler Effect

You have probably heard how the pitch of a train horn changes as it passes you. When the train is approaching, the pitch or frequency is higher than when it is moving away from you. This is called the Doppler effect. A similar, but distinct effect occurs if you are moving past a source of sound. If a stationary whistle is blowing, the pitch perceived from a moving car is higher while moving toward the source than when moving away. The first case thus has a moving source, while the second case has a moving observer.

In this section we will compute the Doppler effect as it applies to light moving through a vacuum. The figure below shows the geometry for computing the time between wave fronts of light for a stationary and a moving reference frame.

 * The time between wavefronts for the stationary observer, in the stationary frame, is T.

 * The time between wavefronts for the moving observer, in the stationary frame, is T′.

 * The time between wavefronts for the moving observer, in the stationary frame, is τ

Since the world lines of the wave fronts have a slope of unity, the sides of the shaded triangle both have the same value, C. If the observer is moving at speed U, the slope of the observer's world line is c/U, i.e

 * $$\frac{c}{U}=\frac{cT+X}{X}$$

Solving this for X and substituting in give cT′=cT+X gives

 * $$T'=\frac{T}{1-\frac{U}{c}} \quad (1)$$

In classical physics T′ and τ are the same so this formula, as it stands, leads directly to the classical Doppler shift for a moving observer.

However, in relativity T′ and τ are different. We can use the Lorentz transformation to correct for this.

The second wavefront passes the moving observer at (UT′,cT′) in the stationary observers frame, but at (0,cτ) in its own frame. The Lorentz transform tells us that.

 * $$c\tau = \gamma \left( cT'-\frac{U}{c}UT' \right)

= cT' \gamma \left( 1 - \frac{U^2}{c^2} \right) = cT'/ \gamma $$

Substituting in equation (1) gives

 * $$\begin{matrix}

\tau & = & T \frac{ \sqrt{ 1 - U^2/c^2 }}{1-U/c} \\ & = & T \sqrt{ \frac{ (1-U/c)(1+U/c)}{(1-U/c)^2}} \\ & = & T \sqrt{ \frac{c+U}{c-U} } \end{matrix}$$

From this we infer the relativistic Doppler shift formula for light in a vacuum:
 * $$\omega^{\prime}= \omega \sqrt{ \frac{c-U}{c+U} } $$

since frequency is inversely proportional to time.

We could go on to determine the Doppler shift resulting from a moving source. However, by the principle of relativity, the laws of physics should be the same in the reference frame in which the observer is stationary and the source is moving. Furthermore, the speed of light is still c in that frame. Therefore, the problem of a stationary observer and a moving source is conceptually the same as the problem of a moving observer and a stationary source when the wave is moving at speed c.

This is unlike the case for, say, sound waves, where the stationary observer and the stationary source yield different formulas for the Doppler shift.

title: Modern Physics/The Doppler Effect


text:
Modern Physics/Addition of Velocities

In classical physics, velocities simply add. If an object moves with speed u in one reference frame, which is itself moving at v with respect to a second frame, the object moves at speed u+v in that second frame.

This is inconsistant with relativity because it predicts that if the speed of light is c in the first frame it will be v+c in the second.

We need to find an alternative formula for combining velocities. We can do this with the Lorentz transform.

Because the factor v/c will keep recurring we shall call that ratio β.

We are considering three frames; frame O, frame O' which moves at speed u with respect to frame O, and frame O" which moves at speed v with respect to frame O'.

We want to know the speed of O" with respect to frame O,U which would classically be u+v.

The transforms from O to O' and O' to O" can be written as matrix equations,

\begin{pmatrix} x' \\ ct' \end{pmatrix} = \gamma \begin{pmatrix} 1 & - \beta \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} x \\ ct \end{pmatrix} \quad \begin{pmatrix} x \\ t \end{pmatrix} = \gamma' \begin{pmatrix} 1 & - \beta' \\ -\beta' & 1 \end{pmatrix} \begin{pmatrix} x' \\ ct' \end{pmatrix} $$

where we are defining the β's and γ's as

 * $$\begin{matrix}

\beta = \frac{u}{c} & \gamma = \frac{1}{\sqrt{1-\beta^2 }} \\ \beta^\prime = \frac{v}{c} & \gamma^\prime = \frac{1}{\sqrt{1-{\beta^\prime}^2 }} \end{matrix} $$

We can combine these to get the relationship between the O and O" coordinates simply by multiplying the matrices, giving

\begin{pmatrix} x \\ ct \end{pmatrix} = \gamma \gamma^\prime \begin{pmatrix} 1+\beta \beta' & - (\beta + \beta') \\ - (\beta + \beta') & 1+\beta \beta'  \end{pmatrix} \begin{pmatrix} x \\ ct \end{pmatrix} \quad (1)$$

This should be the same as the Lorentz transform between the two frames,

\begin{pmatrix} x \\ ct \end{pmatrix} = \gamma \begin{pmatrix} 1 & - \beta \\ -\beta'' & 1 \end{pmatrix} \begin{pmatrix} x \\ ct \end{pmatrix} \quad (2) \mbox{ where } \begin{matrix} \beta'' & = & \frac{U}{c} \\ \gamma & = & \frac{1}{\sqrt{1-{\beta}^2 }} \end{matrix} $$

These two sets of equations do look similar. We can make them look more similar still by taking a factor of 1+ββ' out of the matrix in (1) giving#

\begin{pmatrix} x \\ ct \end{pmatrix} = \gamma \gamma' (1+\beta \beta') \begin{pmatrix} 1 & - \frac{\beta + \beta'}{1+\beta \beta'} \\ - \frac{\beta + \beta'}{1+\beta \beta'} & 1+  \end{pmatrix} \begin{pmatrix} x \\ ct \end{pmatrix} $$

This will be identical with equation 2 if

 * $$\beta''=\frac{\beta + \beta'}{1+\beta \beta'} \mbox{ (3a)  and }

\gamma'' = \gamma \gamma' (1+\beta \beta') \mbox{ (3b)}$$

Since the two equations must give identical results, we know these conditions must be true.

Writing the β's in terms of the velocities equation 3a becomes

 * $$\frac{U}{c}=\frac{\frac{u}{c} + \frac{v}{c}}{1+\frac{uv}{c^2}}$$

which tells us U in terms of u and v.

A little algebra shows that this implies equation 3b is also true

Multiplying by c we can finally write.

$$U = \frac{u+v}{1+\frac{uv}{c^2}} $$

Notice that if u or v is much smaller than c the denominator is approximately 1, and the velocities approximately add but if either u or v is c then so is U, just as we expected.

title: Modern Physics/Addition of Velocities


text:
Russian/Appendix/Verbs

Verbs

 * to blossom, to flower -- цвести (a flower -- цветок)
 * я цвету
 * ты цветёшь
 * он цветёт
 * мы цветём
 * вы цветёте
 * они цветут

 * to live -- жить
 * я живу
 * ты живёшь
 * он живёт
 * мы живём
 * вы живёте
 * они живут

 * to speak -- говорить
 * я говорю
 * ты говоришь
 * он/она/оно говорит
 * мы говорим
 * вы говорите
 * они говорят

 * to sing -- петь
 * я пою
 * ты поёшь
 * он/она/оно поёт
 * мы поём
 * вы поёте
 * они поют

 * to drink -- пить
 * я пью
 * ты пьёшь
 * он/она пьёт
 * мы пьём
 * вы пьёте
 * они пьют

 * to be located (somewhere) -- находиться (где-либо)
 * я нахожусь
 * ты находишься
 * он/она/оно находится
 * мы находимся
 * вы находитесь
 * они находятся

 * знать -- to know

title: Russian/Appendix/Verbs


text:
Arabic/Edited Work

This is an attempt to make a long text to organize my thoughts.

The book is composed of the following parts:

 * 1) Learning to Read Arabic
 * With Tashkeel
 * without tashkeel
 * 1) Phrases
 * Tourist Phrases
 * Airport
 * Taxi
 * Greeting
 * Parting
 * Question and Answer phrases
 * Conversation Changing phrases
 * Agreeing and disagreeing
 * Apologizing, and whose fault
 * Congratulating
 * 1) Basic Grammar

 * 1) Advanced Grammar
 * 2) Reader / Vocab
 * 3) Verb Reference   (with intro to how to learn the verbs)
 * 4) Reference of particles أحراف   (not verbs, and not nouns) (tells which moods go with each particle, and the meaning of each)

title: Arabic/Edited Work


text:
Talk:Russian/Alphabet

Untitled
The handwritten letters should also be shown...

This works in Firefox, but not in Konqueror: тест водка

This doesn't work in Firefox or Konqueror: тест водка

This was one of the notes:

The letter Ё/ё is "optional": it is formally correct to consistently use E/e to represent both /je/ and /jo/.
 * Please don't. The pronounciation of the letter "ё" is *always* different, and ommission of the dots to me is a sign of inaccuracy/laziness.  Soviet bureaucrats often replaced "ё" with "е", thus making the language bleak.  See also http://www.gramota.ru/mag_new.html?id=127 --Ilgiz 01:53, 10 Mar 2004 (UTC)
 * Still, ё can be and is represented by е, both by 'Soviet bearocrats' and in most public writings. Though I personally always render it as ё, it is better to include this information than to exclude it for personal reasons. (Sevens years too late? Ah, well...) DD 8630 (discuss • contribs) 07:58, 27 June 2011 (UTC)

title: Talk:Russian/Alphabet


text:
Probability

Welcome to the Wikibook of PROBABILITY

This book is an introduction to the mathematical theory of probability.

Contents

 * /Local Manual of Style/

Preliminaries

 * 1) /Introduction/
 * 2) /Set Theory/

General probability

 * 1) /Combinatorics/
 * 2) /Probability Spaces/
 * 3) /Conditional Probability/

Random variables

 * 1) /Random Variables/
 * 2) /Important Distributions/
 * 3) /Joint Distributions and Independence/
 * 4) /Properties of Distributions/
 * 5) /Conditional Distributions/
 * 6) /Transformation of Random Variables/

title: Probability


text:
Cookbook:Veal

Veal is meat from calves, the young of cattle. It is much softer, tender, pale and less strongly flavored than beef. This is because the calves are milk-fed and do not have much iron in their bodies. (Iron is used to make haemoglobin and myoglobin, respectively the red oxygen-carrying proteins in blood and muscle tissue.) They also have not developed much muscle.

You may find that some guests reject veal, believing that the calves are restricted to a small dark crate to cut costs or to keep the flesh pale and tender. (Of course, many people would not be bothered by this.) To varying degrees, such practices are illegal now.

Recipes featuring veal

 * Wiejska
 * Rosół
 * Boerewors
 * Veal with lima beans
 * Giblet Soup
 * Sarmi (Cabbage or Grape Leaf Rolls in Yogurt Sauce)
 * Frikadeller
 * Ossobuco Alla Milanese
 * Italian Meatballs

title: Cookbook:Veal


text:
Cookbook talk:Veal

The text of this module, or a previous version recorded in the page history, was copied from the Wikipedia article Veal. The page history at the time the text was coppied is included below:

 * (cur) (last) . . M 20:43, 9 Oct 2002 . . Zoe
 * (cur) (last) . . 20:26, 9 Oct 2002 . . Wesley
 * (cur) (last) . . 12:36, 1 Oct 2002 . . Gianfranco
 * (cur) (last) . . M 15:43, 25 Feb 2002 . . Conversion script (Automated conversion)
 * (cur) (last) . . 16:11, 20 May 2001 . . Neeklamy (a bit more honesty)

I don't think we want to gloss over the anti-veal feeling of many people. If someone is going to cook veal, they should know that some of their guests might find it barbaric. Gentgeen 06:26, 3 Mar 2004 (UTC)

title: Cookbook talk:Veal


text:
Cookbook:Brussels Sprout

Brussels sprouts are a variety of brassica resembling a miniature cabbage. It's believed that they were first cultivated in Belgium, and they are therefore named after its capital, Brussels.

Characteristics
Brussels sprouts grow on long thick stalks, from which they must be picked, usually by hand. The individual sprouts resemble small cabbages, with a round to slightly ovoid shape and tightly packed leaves. They are usually no more than about 2 inches (5 cm) in diameter, and smaller sprouts tend to be sweeter than larger ones. The leaves are a darker green on the outside and get more tender and yellow toward the inside. The flavor is slightly bitter and cabbagey when raw, but when cooked and browned they get sweeter and more complex in flavor.

As brassicas, brussels sprouts harbor sulfur-containing compounds, which can result in unpleasant odors and flavors when prepared improperly. Overcooking them, especially by boiling, tends to amplify such characteristics. When suitably prepared, brussels sprouts have a sweet and nutty flavor.

Seasonality
Brussels sprouts are a typical winter vegetable. Frost is generally considered to improve their flavor—so, though they are available from October or earlier, the prime season starts in November.

Selection and storage
Brussels sprouts may be sold attached to or separated from their stalks—if feasible, purchasing on-the-stalk sprouts will help prolong their storage life. Select brussels sprouts that are green and firm, whose leaves are tightly clustered and not wilted. They will last about a week in the fridge after purchasing, and 3–5 weeks at 32°F (0°C). If you want to freeze them, it's best to blanch and dry them first.

Preparation
To prepare brussels sprouts, slice off the base of the sprout and remove the outer layer or two of leaves, which are tougher and can become battered during transport and storage. Wash the buds well before cooking. Because they are a small vegetable, sprouts can generally be cooked whole, but they may also be halved or sliced to encourage even cooking. When cooking larger sprouts whole, you can cut a small cross in the base to help the sprout cook evenly. When eating them raw, they should be shaved or thinly sliced.

Use
Overcooking is a common problem with brussels sprouts, especially when boiling or steaming. This makes them soggy and discoloured, and it releases the strong 'cabbagey' odour of all brassicas. When boiling sprouts, keep an eye on their color, flavor, and texture. Whole sprouts require 4–10 minutes of boiling time, depending on how cooked you want them. Steaming time ranges from 8–15 minutes. It's best to taste test the sprouts during cooking to see whether they have the desired tenderness. When done, they should be bright green.

Brussels sprouts can also be roasted or fried with oil and seasonings. Dry cooking in this way allows the sprouts to caramelize, which produces a nutty sweetness, and avoids overall sogginess. Sprouts may also be shaved and eaten raw, although this tends to be less common than cooking them.

In the UK, brussels sprouts are a traditional winter vegetable, and are often eaten boiled with a roast dinner. They can also be stir fried or made into soup. Simple serving suggestions include sprinkling the sprouts with lemon juice, salt, fresh nutmeg, or grated cheese, possibly with a light coating of butter or olive oil.

title: Cookbook:Brussels Sprout


text:
Cookbook:Broccoflower

Broccoflower refers to two cultivated varieties of Brassica oleracea.

Characteristics
The first variety of broccoflower resembles a traditional cauliflower, but with light green flesh. The second variety is known as romanesco broccoli, and it has a bright yellow-green appearance with spiky florets growing in fractal patterns. Both taste more like cauliflower than broccoli.

Selection and storage
When selecting broccoflower, go for heads that are bright, firm and not soft or wilted, and dense. Avoid those with discoloration or slimy spots. Do not wash before storing in the fridge —they should keep for at least a week, and perhaps longer.

Use
Broccoflower is prepared much the same as cauliflower and broccoli, served raw, roasted, steamed, and boiled. Like them, overcooking broccoflower can quickly make it turn dull and mushy.

title: Cookbook:Broccoflower


text:
Cookbook:Venison

| Game

Venison is the meat of the deer and related species. Deer meat is an excellent source of lean animal protein and a healthy alternative to processed foods. As with the meat of other large animals, it is eaten as steaks, roasts, sausages and ground meat. Organ meats are sometimes eaten but would not be called venison. Venison is lower in calories, cholesterol and fat than common cuts of beef, pork or lamb.

Types of venison
Deer live in woodlands all over Europe, Asia, northern Africa and America. There are many deer species of various sizes but all the males grow antlers. The meat is lean and has a gamey flavor that can be made milder if soaked overnight.

Elk meat tastes like mild (almost sweet) beef, with only a very faint venison flavor. Elk can be substituted equally for venison in most standard venison recipes. Elk are from North America, Europe, and Asia.

Moose is the largest member of the venison family standing about 6 1/2 feet at the shoulder. It’s native from North America. The meat is similar to elk.

Caribou (reindeer) are slightly larger than white-tailed deer. Both males and females have antlers. The meat is somewhat sweeter than other venison. They live primarily in North America and Siberia.

Antelope are currently farmed in Texas, where black buck and nilgai antelope, native to Africa, are allowed to roam on huge preserves. Males are called bucks, bulls, or stags; females, does or cows; and unweaned young are fawns or calves. Antelope meat is leaner, but similar in taste, to that of deer.

Pronghorn (once classified as "antelope") are native to North America and the only surviving species in the Antilocapridae ("goat-antelope") family. The largest populations are in Wyoming and Montana. Pronghorn meat is leaner, but similar in taste, to that of deer.

Venison as "red" meat
All venison is red meat. The coloring comes from myoglobin, a protein in muscle tissue that stores oxygen for later use, and not from blood, which is red due to hemoglobin, the protein that transports oxygen throughout the body.

In extremely fresh meat, the myoglobin is a dark purple color, but it quickly brightens to cherry red when exposed to oxygen in the air, usually within 15 minutes. Over several days, the meat will darken to a brown color as the oxidation of the meat continues.

Cooking guidelines
After shooting the animal, the hunter should promptly remove the vital organs from the stomach cavity to preserve the quality of the meat. Remaining carcass can either be self-processed or butchered professionally. The tenderness of a particular cut of venison is similar to the corresponding cut of domestically-raised meat. All game tends to be leaner than that of domesticated animals, which have been bred for tenderness and fat marbling. Overcooking can toughen venison. You can use moist heat, basting, and larding or barding (inserting slivers of fat or wrapping in bacon) to help keep the meat tender during cooking. Fast searing over high heat can also work for smaller cuts, such as tenderloin medallions or rib chops.

Venison recipes

 * Roast European Bison (Poland)
 * Saddle of Venison Roast (Poland)
 * Venison Sausages
 * Bambi Bolognaise
 * Chili con Bambi

title: Cookbook:Venison


text:
Cookbook:Broccoli

Broccoli is a member of the brassica family, like cauliflower and cabbage.

Characteristics
The part of the plant we eat is actually an undeveloped flower head with many tiny flower buds crowded onto it. The most common type of broccoli—calabrese broccoli—is green. Purple sprouting broccoli has a purple tinge to the heads and is finer and leafier than the smaller-headed calabrese. As its name suggests, it is harvested closer to its sprouting stage, making it softer and bendier. A less common variety is the white sprouting broccoli, which behaves much like its purple brother. When left to grow, broccoli heads turn a bright yellow.

Seasonality
The calabrese broccoli season starts slowly in the summer, and peaks in the autumn.

Purple and white sprouting broccoli have their seasons in the late winter, making them ideal for bridging the gap between the winter and spring vegetables.

Out of season broccoli can be imported from countries near the equator. It is sometimes available frozen.

Selection and storage
When buying broccoli, look for clean stems that are not dry or split. When buying sprouting broccoli, the stalks should bend flexibly without breaking. Broccoli with yellow buds should be avoided, as this is a clear indication that the broccoli is old. Broccoli is best when kept in the fridge no longer than a few days.

Preparation
The thick broccoli stem is usually cut off, but it is edible and tastes the same as the florets. However, its outside is rather fibrous and must be trimmed or peeled to get to the tender core. Slicing crosswise into thin disks is another option to shorten tougher fibers.

Use
Broccoli can be prepared in a variety of ways, ranging from raw to lightly cooked, roasted, or slow-cooked. Many people prefer it lightly blanched or cooked more thoroughly. When overcooked, it takes on a yellow-brown cast and becomes mushy. Properly cooked broccoli is very bright green, having lost the bluish cast of raw broccoli but without any hint of a yellow-brown cast.

Because it only needs light cooking, broccoli is usually added to recipes near the end of the cooking time, and it is ideal for stir-frying or steaming. When roasting, a thin coating of oil is useful to prevent drying out and burning, and cooking at high temperature (425°F / 220°C) yields a crisped exterior. Alternatively, broccoli can be braised or stewed slowly for a long time.

The stem can be chopped or finely sliced and used in casseroles because it withstands more cooking than the florets. It also goes well in stir-fry as if it were a different vegetable from the flower buds. Stems can also commonly be seen in grocery stores in matchstick or julienne form, packaged as broccoli slaw.

Unlike the calabrese, purple sprouting broccoli can cooked whole, with leaves and stem attached.

title: Cookbook:Broccoli


text:
Geometry/Angles

An angle is the union of two rays with a common endpoint, called the vertex. The angles formed by vertical and horizontal lines are called right angles; lines, segments, or rays that intersect in right angles are said to be perpendicular.

Angles, for our purposes, can be measured in either degrees (from 0 to 360) or radians (from 0 to $$2\pi$$). Angles length can be determined by measuring along the arc they map out on a circle. In radians we consider the length of the arc of the circle mapped out by the angle. Since the circumference of a circle is $$2\pi$$, a right angle is $$\frac{\pi}{2}$$ radians. In degrees, the circle is 360 degrees, and so a right angle would be 90 degrees.

Naming Conventions
Angles are named in several ways.
 * By naming the vertex of the angle (only if there is only one angle formed at that vertex; the name must be non-ambiguous) $$\angle B$$
 * By naming a point on each side of the angle with the vertex in between. $$\angle ABC$$
 * By placing a small number on the interior of the angle near the vertex.

Classification of Angles by Degree Measure
Acute Angle
 * an angle is said to be acute if it measures between 0 and 90 degrees, exclusive.

Right Angle
 * an angle is said to be right if it measures 90 degrees.
 * notice the small box placed in the corner of a right angle, unless the box is present it is not assumed the angle is 90 degrees.
 * all right angles are congruent

Obtuse Angle
 * an angle is said to be obtuse if it measures between 90 and 180 degrees, exclusive.

Special Pairs of Angles

 * adjacent angles
 * adjacent angles are angles with a common vertex and a common side.
 * adjacent angles have no interior points in common.

 * complementary angles
 * complementary angles are two angles whose sum is 90 degrees.
 * complementary angles may or may not be adjacent.
 * if two complementary angles are adjacent, then their exterior sides are perpendicular.
 * supplementary angles
 * two angles are said to be supplementary if their sum is 180 degrees.
 * supplementary angles need not be adjacent.
 * if supplementary angles are adjacent, then the sides they do not share form a line.
 * linear pair
 * if a pair of angles is both adjacent and supplementary, they are said to form a linear pair.
 * vertical angles
 * angles with a common vertex whose sides form opposite rays are called vertical angles.
 * vertical angles are congruent.

title: Geometry/Angles


text:
Probability/Introduction

Overview
Probability theory provides a mathematical model for the study of randomness and uncertainty. Many important decisions, whether from business, government, science, recreation or even one's personal life must be made with incomplete information or some degree of uncertainty. Hence, a formalized study of uncertain or random outcomes occupies an important role in modern society. In situations where one of any number of possible outcomes may occur, the mathematical model of probability theory offers methods for quantifying the likelihoods associated with those outcomes. Probability also provides tools which allow us to move beyond simply describing the information contained within a set of data (descriptive statistics) to actually inferring further information from that data (inferential statistics). Many of the early attempts to model likelihood arose from games of chance. For a brief history of probability see this Wikipedia article.

Although probability theory is now a very formal branch of mathematics, the language of probability is often used informally in everyday speech. We express our beliefs about likelihoods of outcomes in situations involving uncertainty using intuition guided by our experiences and in some cases statistics. Consider the following examples:

 * Bill says "Don't buy the avocados here; about half the time, they're rotten". Bill is expressing his belief about the probability of an event — that an avocado will be rotten — based on his personal experience.
 * Lisa says "I am 95% certain the capital of Spain is Barcelona". Here, the belief Lisa is expressing is only a probability from her point of view, because only she does not know that the capital of Spain is Madrid (from our point of view, the probability is 100%). However, we can still view this as a subjective probability because it expresses a measure of uncertainty. It is as though Lisa is saying "in 95% of cases where I feel as sure as I do about this, I turn out to be right".
 * Susan says "There is a lower chance of being shot in Omaha than in Detroit". Susan is expressing a belief based (presumably) on statistics.
 * Dr. Smith says to Christina, "There is a 75% chance that you will live." Dr. Smith is basing this off of his research.
 * Nicolas says "It will probably rain tomorrow." In this case the likelihood that it will rain is expressed in vague terms and is subjective, but implies that the speaker believes it is greater than $$\frac12$$ (or 50%). Subjective probabilities have been extensively studied, especially with regards to gambling and securities markets. While this type of probability is important, it is not the subject of this book. A good reference is "Degrees of Belief" By Steven Vick (2002).

Notice that in the previous examples the likelihood of any particular outcome is expressed as a percentage (between 0% and 100%), as is common in everyday language. However, probabilities in formal probability theory are always expressed as real numbers in the interval $$[0,1]$$ (e.g. a probability of .25 may be expressed as 25%, or a probability of $$\frac{1}{\pi}$$ may be expressed as approximately 31.83%). Other differences exist between common expressions of probabilities and formal probability theory. For example, a probability of 0% is typically taken to mean that the event to which that probability is assigned is impossible. However, in probability theory (usually in cases where there are infinitely many possible outcomes) an event ascribed a probability of zero may actually occur. In some situations, it is certain that such an event will occur (e.g. in selecting a real number between 0 and 1, the probability of selecting any given number is zero, but it is certain that one such number will be selected).

Another way to express the probability of an outcome is by its odds: the ratio of the probability of "success" (event occurs) to the probability of "failure" (event does not occur). In gambling odds are expressed as the ratio of the stakes risked by each participant in a wager. For instance: a bookmaker offering odds of 3 to 1 "against" a horse will pay a punter three times their stake (if the horse wins). In fact, the bookmaker (ignoring factors such as his potential need to "lay off" bets which are exposing him to the possibility of an unacceptable overall loss) is announcing that he thinks the horse has a $$\frac14$$ chance of winning. If we express odds as "chance of winning": "chance of not winning", then 3 to 1 against would be represented as $$1:3=\frac14:\frac34$$ or $$\frac13$$. So an event with a probability of $$\frac14$$ or 25% has odds of 33%. This disparity is even more clear where an event has a probability of 50% (e.g., the odds of a coin showing heads is 50%:50% = 1:1 or $$\frac{1}{2}$$).

Types of probability
As mentioned earlier, probability can be expressed informally in a variety of different ways, but even formal definitions and approaches vary. The most general and rigorous approach is known as axiomatic probability theory, which will be the focus of later chapters. Here we briefly discuss a few other approaches, their uses and limitations. All of these approaches rely in one way or another on the concept of an experiment. Recall that probability provides means to study randomness and uncertainty.

An experiment is any action or process whose outcome is subject to uncertainty or randomness.

Here the term experiment is used in a wider sense than its usual connotation with controlled laboratory situations. Further clarification on experiments will be given later, but for now the following examples of experiments will suffice:
 * observing whether or not a commercial product is defective.
 * tossing a coin one or more times or selecting a card from a card deck.
 * conducting a survey.
 * measuring the wind speed or rainfall in a particular area.

Assuming that an experiment can be repeated under identical conditions, then each repetition of an experiment is called a trial.

Basic Concepts
There are two standard approaches to conceptually interpreting probabilities: the relative frequency approach and the subjective belief (or confidence approach). In the Frequency Theory of Probability, probability is the limit of the relative frequency with which certain outcomes occur in repeated trials (note that the outcome of any single trial cannot depend on the outcome of other trials). The relative frequency approach requires that experiments be random and that all possible outcomes be known before execution of the experiment. The probability of any set of outcomes is expressed as the relative frequency with which those outcomes will occur among many repeated trials.

Physical probabilities fall within the category of objective or frequency probabilities, and are associated with random physical systems such as roulette wheels, rolling dice and radioactive atoms. In such systems, a given outcome (such as a die yielding a six) tends to occur at a persistent rate, or 'relative frequency', in a long run of trials. Physical probabilities either explain, or are invoked to explain these stable frequencies.

Relative frequency probabilities are always expressed as a figure between 0% (the outcome essentially never happens) and 100% (the outcome essentially always happens), or similarly as a figure between 0 and 1. According to the Frequency Theory of Probability, saying that "the probability that A occurs is p%" means that if you repeat the experiment many times under essentially identical conditions, the percentage of time for which A occurs will converge to p. For example, a 50% chance that a coin lands "heads up" means that if you toss the coin over and over again, then the ratio of times the coin lands heads to the total number of tosses approaches a limiting value of 50% as the number of tosses grows. Notice that the outcome of one toss never depends on another toss, and that the ratio of heads to total number of tosses is always between 0% and 100%.

In the Subjective Theory of Probability, probability measures the speaker's "degree of belief" that a set of outcomes will result, on a scale of 0% (complete disbelief that the event will happen) to 100% (certainty that the event will happen). According to the Subjective Theory, saying that "the probability that A occurs is $$\frac23$$ " means that I believe that A will happen twice as strongly as I believe that A will not happen. The Subjective Theory is particularly useful in assigning meaning to the probability of outcomes that in principle can occur only once. For example, how might one assign meaning to the following statement: "there is a 25% chance of an earthquake on the San Andreas fault with magnitude 8 or larger before 2050"? It would be very hard to qualify this measure in terms of relative frequency.

One way to represent an individual's degree of belief in a statement, given available evidence, is with the Bayesian approach. Evidential probability, also called Bayesian probability, can be assigned to any statement whatsoever, even when no random process is involved. On most accounts evidential probabilities are considered degrees of belief, defined in terms of dispositions to gamble at certain odds. The primary evidential interpretations include the classical interpretation, the subjective interpretation, the epistemic or inductive interpretation, and the logical interpretation.

The next several sections discuss the principal theories within the relative frequency approach to probability.

Classical theory of probability
The classical approach to probability expresses probability as a ratio of the number of favorable outcomes in a series of successive trials to the number of total possible outcomes. Note the immediate implication that the number of total possible outcomes be known. Furthermore, all possible outcomes are assumed to be equally probably and no two possible outcomes can both result from the same trial. Here, the term "favorable" is not subjective, but rather indicates that an outcome belongs to a group of outcomes of interest. This group of outcomes is called an event, which will be formalized with the introduction of axiomatic probability theory.

For example, a standard deck of cards (without jokers) has 52 cards. If we randomly draw a card from the deck, we can think of each card as a possible outcome. Therefore, there are 52 total outcomes. We can now look at various events and calculate their probabilities:
 * Out of the 52 cards, there are 13 clubs. Therefore, if the event of interest is drawing a club, there are 13 favorable outcomes, and the probability of this event is $$\frac{13}{52}=\frac14$$.
 * There are 4 kings (one of each suit). The probability of drawing a king is $$\frac{4}{52}=\frac{1}{13}$$.
 * What is the probability of drawing a king OR a club? This example is slightly more complicated. We cannot simply add together the number of outcomes for each event separately ($$4+13=17$$) as this inadvertently counts one of the outcomes twice (the king of clubs). The correct answer is $$\frac{16}{52}$$ from $$\frac{13}{52}+\frac{4}{52}-\frac{1}{52}$$ where this is essentially $$p(\text{club})+p(\text{king})-p(\text{king of clubs})$$.

Classical probability suffers from a serious limitation. The definition of probability implicitly defines all outcomes to be equiprobable. While this might be useful for drawing cards, rolling dice, or pulling balls from urns, it offers no method for dealing with outcomes with unequal probabilities.

This limitation can even lead to mistaken statements about probabilities. An often given example goes like this:
 * I could be hit by a meteor tomorrow. There are two possible outcomes: I will be hit, or I will not be hit. Therefore, the probability I will be hit by a meteor tomorrow is $$\frac12=50%$$.

Of course, the problem here is not with the classical theory, merely the attempted application of the theory to a situation to which it is not well adapted.

This limitation does not, however, mean that the classical theory of probability is useless. At many points in the development of the axiomatic approach to probability, classical theory is an important guiding factor.

Empirical or Statistical Probability or Frequency of occurrence
This approach to probability is well-suited to a wide range of scientific disciplines. It is based on the idea that the underlying probability of an event can be measured by repeated trials.

It is of course impossible to conduct an infinite number of trials. However, it usually suffices to conduct a large number of trials, where the standard of large depends on the probability being measured and how accurate a measurement we need.

A note on this definition of probability: How do we know the sequence $$\frac{n_A}{n}$$ in the limit will converge to the same result every time, or that it will converge at all? The unfortunate answer is that we don't. To see this, consider an experiment consisting of flipping a coin an infinite number of times. We are interested in the probability of heads coming up. Imagine the result is the following sequence:
 * HTHHTTHHHHTTTTHHHHHHHHTTTTTTTTHHHHHHHHHHHHHHHHTTTTTTTTTTTTTTTT...

with each run of $$k$$ heads and $$k$$ tails being followed by another run twice as long. For this example, the sequence $$\frac{n_A}{n}$$ oscillates between roughly $$\frac13$$ and $$\frac23$$ and doesn't converge.

We might expect such sequences to be unlikely, and we would be right. It will be shown later that the probability of such a run is 0, as is a sequence that converges to anything other than the underlying probability of the event. However, such examples make it clear that the limit in the definition above does not express convergence in the more familiar sense, but rather some kind of convergence in probability. The problem of formulating exactly what this means belongs to axiomatic probability theory.

Axiomatic probability theory
Although axiomatic probability theory is often frightening to beginners, it is the most general approach to probability and has been employed in tackling some of the more difficult problems in probability. It begins with a set of axioms which, although not immediately intuitive, are guided by the more familiar classical probability theory. These axioms are discussed in the (as yet unwritten) following chapter.

About This Book
This book is going to discuss the topic of mathematical probability using Calculus and Linear Algebra. Readers of this book should have a good understanding of both those topics before attempting to read and understand this book completely.

title: Probability/Introduction


text:
Cookbook:Roux

| Sauces

Roux is a basic cooked mixture of varying ratios of flour and fat (usually butter), useful for making sauces, and for thickening soups or gravies. The benefits of using a roux include: It does not have to be cooked very long to remove a floury taste, clumps of flour are removed, and it creates unique flavors. It can be cooked to different degrees (from white to brown) depending upon the intended use, and a darker roux (one that has been cooked longer) will also be thicker and have more flavor, but will have less thickening power.

Technique
To make a basic roux, butter (solid or clarified) is melted in a saucepan. An equal weight of wheat flour is then whisked in. The mixture is then cooked for a variable amount of time to remove the raw flour taste and, sometimes, to add color. For a white roux, cook for a few minutes to remove the raw flavor. For a blond roux, cooking must cease as soon as the color of the roux begins to change, and before the appearance of any progressive coloring whatsoever. Brown roux is ready when it has acquired a fine, light brown color and nutty odor, characteristic of baked flour. A good roux will have a slight shine to it, and neither the texture nor the taste of the flour will be apparent.

Depending upon how you plan to use your roux, you may need to add the sauce's other ingredients before the roux is fully cooked.

When making a dark roux, switching from butter to an oil with a high smoke point (such as soybean oil or Canola oil) will allow for a higher cooking temperature, decreasing cooking time. Keep in mind that different fats will give the roux a somewhat different taste.

It is very important that brown roux should not be cooked too rapidly. When cooking takes place with a very high heat in the beginning, the starch gets burned within its shrivelled cells. The binding principle is thus destroyed, and double or triple the quantity of roux becomes necessary in order to obtain the required consistency. But this excess of roux in the sauce chokes it up without binding it, and prevents it from clearing. At the same time, the cellulose and the burnt starch lend a bitterness to the sauce of which no subsequent treatment can rid it.

Use
A roux is typically used as a thickener, often for soups and sauces. In these cases, liquid is gradually whisked into the roux. Don't go the other way, adding the roux to the liquid, as you will get lumps. Once enough liquid has been added to the roux, you can safely add it back into another liquid.

title: Cookbook:Roux


text:
Talk:Organic Chemistry/Book outline

Can I ask something of the individuals working on wikibooks, that there should be a printable form that would in fact create a real honest to goodness paper textbook? This would be a wonderful thing for all us po' boys who can;t afford to buy a library but would love to have one.
 * Yes, the plan is to print up the books when they are ready. (This one is far from complete, agreed ?) I'm a po' boy myself and can't wait to see what this project does to reign in textbook prices. --Karl Wick

I'm excited to see how this book takes shape. One of my great hopes, though, is that the book breaks out of the traditional mold. I have a recommendation -- tie spectroscopy in to structure & bonding and introduce it early. Most lab courses these days have students using IR within the first semester, often long before the topic is covered in the lecture part of the course. I've had good luck teaching IR "out of order," and I'd like to expand that to UV, MS, and especially NMR in the next few years. Especially with spectroscopy having gained greater importance than we might have predicted 20-30 years ago, this idea (and this text as a whole) might be a great showcase for an important philosophical change in how we teach organic chemistry.

title: Talk:Organic Chemistry/Book outline


text:
Cookbook:Bell Pepper

| Chili Pepper The bell pepper, sometimes ambiguously known simply as "pepper" or "capsicum", is a type of large mild chili pepper used as a vegetable.

Characteristics
Bell peppers have multiple lobes and a hollow center with a clump of small white seeds and a bitter white membrane.

The peppers are green while developing and then ripen to a variety of colors—red and yellow are the most common but purple and orange are also found, along with white and almost black. They are edible while still green, where they have a somewhat grassy and bitter flavor. When mature, the peppers still retain a fairly vegetal taste, with no spiciness despite their close relatedness to hot chiles.

Seasonality
Green bell peppers are available all year, with their peak season falling between July and November. The ripe varieties are more sensitive to season and will reach their peak towards the end of the green pepper's season.

Selection and storage
Like many fresh vegetables, bell peppers should have few blemishes, wrinkles, or soft spots. The skin should be shiny and well-colored around the stem (if ripeness is desired), and the peppers should feel solid.

Store bell peppers in the fridge—avoid using an airtight bag, the resulting moisture of which can cause mold growth.

Use
For general usage, cut bell peppers open, then remove the stem, core, seeds, and white veins. For some applications, the peppers are charred over a fire, such as a gas range, to loosen the skins and add a distinctive smoky flavor.

Bell peppers are commonly added to pizza, stir-fry, and sweet and sour. They can also be deseeded, stuffed, and baked.

title: Cookbook:Bell Pepper


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Cookbook:Chiles

| Herbs and Spices | Chilli The chile or chile pepper is the spicy fruit of the plant capsicum. To avoid confusion, this cookbook—like several other sources  —uses "chile" with an "e" to refer to the pure chile pepper (either fresh or processed), and "chili" with an "i" to refer to the stew and its associates such as the spice blend chili powder. However, do know that the spellings can be ambiguous in other sources, and make sure to use the correct ingredient required in a given recipe.

Characteristics
With many varieties, chiles come in a wide array of shapes, sizes, and colors. They are all hollow, with crisp watery flesh, tapered ends, and inner seeds. The shape can be long, rounded, narrow, wrinkled, and more, with colors ranging from greens (when unripe) to yellows, oranges, and reds of various shades.

Chiles are infamous for their spicy heat, which is caused by the substance capsaicin found in the chile's inner membrane. This compound triggers heat sensors in the body to cause pain and inflammation when consumed. The "heat" is measured in Scoville Heat Units (SHU) and varies according to chile variety and growing seasons   —bell peppers rank at 0 SHU, jalapeños at 3000–6000 SHU, and habaneros at 300,000 SHU. Heat aside, fresh chiles also have fruity and vegetal flavors that vary between varieties.

Chiles may be dried —whereupon they may be given new names —and can be processed to make smoked chiles, chile flakes, chili powder, chili pastes, and more. During these processes, they may gain and/or lose flavors according to the preparation method.

Varieties
Chile varieties are derived from cultivars of four primary species: Capsicum annuum (e.g. jalapeños); Capsicum frutescens (e.g. cayenne), Capsicum chinense (e.g. habaneros), Capsicum pubescens (e.g. rocoto), and Capsicum baccatum (e.g. ají amarillo).

Selection and storage
Select fresh chiles like you would many other fresh vegetables. The flesh and skin should be firm, crisp, and unblemished. Overly soft or wrinkled skin indicates chiles past their prime. High-quality dried chiles should still have some moisture, much like raisins, and should not feel like paper or cardboard.

Fresh chiles should be stored in the refrigerator for several days. They can also be frozen after chopping, though this will destroy their fresh texture—frozen chiles should be used puréed and/or cooked. Dried chiles should be stored in an airtight container away from light and heat for several months. Chile powders lose their complex aromas during storage and are primarily used to contribute heat to a dish.

Preparation
To prepare all chiles for use, remove the tough stem. The inner membrane can be removed to reduce the heat if desired. The seeds, while not very spicy, are bitter and may also be removed accordingly. Dried chiles may be toasted to amplify their flavors.

Safety
Be very cautious in handling chiles, and do not touch your eyes or any other sensitive body part after handling. For some especially strong chiles, it is advisable to wear gloves while handling them and to wash the hands with soap immediately after.

Use
Chiles are eaten cooked or raw for the hot flavor they contribute to dishes. Though unknown to the Old World prior to colonization of the Americas, chiles have become well integrated into many European, Asian, and African cuisines, which originally used peppercorn and other spices for heat. Well-known dishes with a strong chile flavor include salsa, New Mexican chili con carne and Indian vindaloo.

To tame their heat, chiles are often paired with foods high in fat and protein (such as dairy), which both help interrupt the capsaicin binding to the body's receptors—water will not help. Acidic ingredients make the heat more intense. People who are regularly exposed to a lot of chile become less sensitized to the heat over time, so make sure to tailor the spice level of your dishes to the specific palate of the diner.

title: Cookbook:Chiles


text:
Cookbook talk:Chiles

This article was copied from Chilli pepper in the English Wikipedia.

Legend: (cur) = difference with current version, (last) = difference with preceding version, M = minor edit

* (cur) (last). . M 16:23, 15 Dec 2003. . Nikai (link cayenne pepper) * (cur) (last). . 15:30, 14 Dec 2003. . 65.50.167.13   * (cur) (last). . 15:13, 30 Nov 2003. . 12.248.194.91   * (cur) (last). . M 15:15, 28 Nov 2003. . Kowey (ms:) * (cur) (last). . M 07:38, 21 Sep 2003. . Christian (da:) * (cur) (last). . 04:14, 15 Sep 2003. . Heron (Diego Alvarez Chanca) * (cur) (last). . M 04:05, 15 Sep 2003. . Rmhermen * (cur) (last). . M 03:31, 15 Sep 2003. . Heron (copyedit) * (cur) (last). . 03:20, 15 Sep 2003. . 64.76.182.78   * (cur) (last). . M 01:31, 12 Sep 2003. . AdamRaizen (links) * (cur) (last). . M 09:36, 4 Aug 2003. . Rob Hooft (nl:) * (cur) (last). . M 19:36, 28 Mar 2003. . Setup (japanese link) * (cur) (last). . M 13:10, 13 Mar 2003. . Tarquin (fm) * (cur) (last). . 11:35, 13 Mar 2003. . Jketola (Added info about capsaicin and birds.) * (cur) (last). . M 10:30, 23 Feb 2003. . Mkweise (the nightshade (Solanaceae) family) * (cur) (last). . M 10:20, 23 Feb 2003. . Sannse (Disambiguating Indian) * (cur) (last). . M 03:40, 16 Feb 2003. . Hephaestos (sp. etc.) * (cur) (last). . M 03:23, 16 Feb 2003. . Hephaestos (paragraph structure) * (cur) (last). . 04:49, 7 Feb 2003. . Heron (report of dead link was exaggerated) * (cur) (last). . M 02:32, 16 Jan 2003. . Liftarn (sv) * (cur) (last). . 15:46, 3 Dec 2002. . Marj Tiefert (restore) * (cur) (last). . 15:32, 3 Dec 2002. . 128.114.163.191   * (cur) (last). . M 06:23, 10 Oct 2002. . 64.236.243.243   * (cur) (last). . M 16:12, 24 Sep 2002. . Lee Daniel Crocker * (cur) (last). . 16:11, 24 Sep 2002. . Lee Daniel Crocker * (cur) (last). . M 20:07, 22 Sep 2002. . 65.33.176.248   * (cur) (last). . M 20:06, 22 Sep 2002. . 65.33.176.248   * (cur) (last). . M 03:56, 13 Sep 2002. . Andre Engels (disambiguate) * (cur) (last). . M 21:04, 5 Sep 2002. . Andre Engels (typo) * (cur) (last). . 21:03, 5 Sep 2002. . Andre Engels (if Columbus brought it back in 1493, it must have been from the Caribbean) * (cur) (last). . M 05:53, 21 Aug 2002. . PierreAbbat (replace box character with a meaningful punctuation mark; sp) * (cur) (last). . M 03:30, 21 Aug 2002. . Marj Tiefert (link to capsaicin) * (cur) (last). . M 02:39, 21 Aug 2002. . Heron (note on spelling) * (cur) (last). . M 21:13, 20 Aug 2002. . Rmhermen * (cur) (last). . 20:40, 20 Aug 2002. . Rmhermen * (cur) (last). . 20:19, 20 Aug 2002. . Rmhermen * (cur) (last). . 20:06, 20 Aug 2002. . Rmhermen (add some chili names) * (cur) (last). . 19:56, 20 Aug 2002. . Rmhermen * (cur) (last). . 19:44, 20 Aug 2002. . Rmhermen * (cur) (last). . 19:17, 20 Aug 2002. . Heron

This module was the featured recipe on the main Cookbook page from May 16, 2004 - May 23, 2004. Gentgeen 12:35, 23 May 2004 (UTC)

title: Cookbook talk:Chiles


text:
Cookbook talk:Cuisine of France

I am reformatting the regions section so that I can index the recipes back to their exact place on the Cuisine of France page. This requires a heading, a bullet won't work with the template. I will do it a little at a time. Mike Hayes (talk) 07:36, 19 August 2008 (UTC)

I advise you to look at fr:Cuisine:Recettes du Nord or fr:Cuisine:Recettes du Sud-Ouest.

Some of the text of this module was taken from the Wikipedia article Cuisine of France. Below is the last 50 edits made to that page at the time the text was coppied:

 * (cur) (last) . . 11:48, 23 Feb 2004 . . Dysprosia (moving back list, i'm sure we can work something out)
 * (cur) (last) . . 11:43, 23 Feb 2004 . . Anthere (let's remove culture...sorry, recipee)
 * (cur) (last) . . M 13:40, 12 Feb 2004 . . Fabiform (shortbread)
 * (cur) (last) . . 12:22, 18 Dec 2003 . . 81.65.22.223 (added ref to haute cuisine)
 * (cur) (last) . . M 12:00, 18 Dec 2003 . . Olivier
 * (cur) (last) . . 11:53, 18 Dec 2003 . . 81.65.22.223 (a few refinements and additions)
 * (cur) (last) . . 11:56, 17 Dec 2003 . . Anthere
 * (cur) (last) . . 11:53, 17 Dec 2003 . . 195.115.82.180
 * (cur) (last) . . M 10:32, 12 Oct 2003 . . Anthere (remove crème brûlée as a title, it is not a general term)
 * (cur) (last) . . M 10:09, 12 Oct 2003 . . Minesweeper (wiki)
 * (cur) (last) . . 22:46, 2 Oct 2003 . . Wetman (a few links)
 * (cur) (last) . . M 22:37, 2 Oct 2003 . . Angela
 * (cur) (last) . . M 22:37, 2 Oct 2003 . . Angela (fix link)
 * (cur) (last) . . M 11:15, 12 Jun 2003 . . Anthere (duck !?!)
 * (cur) (last) . . 11:02, 12 Jun 2003 . . Fraise (Missed a couple words.)
 * (cur) (last) . . 11:00, 12 Jun 2003 . . Fraise (Several edits (spelling, word usage, formatting), added recipes)
 * (cur) (last) . . 11:51, 22 Apr 2003 . . Olivier
 * (cur) (last) . . M 08:11, 17 Apr 2003 . . Olivier (fr:)
 * (cur) (last) . . 14:45, 15 Apr 2003 . . Olivier
 * (cur) (last) . . M 21:52, 6 Mar 2003 . . Sannse (Disambiguating French)
 * (cur) (last) . . M 16:33, 6 Mar 2003 . . Timo Honkasalo
 * (cur) (last) . . 18:12, 4 Feb 2003 . . Olivier
 * (cur) (last) . . 18:10, 4 Feb 2003 . . Olivier
 * (cur) (last) . . 18:09, 4 Feb 2003 . . Olivier
 * (cur) (last) . . M 18:07, 4 Feb 2003 . . Olivier
 * (cur) (last) . . 18:07, 4 Feb 2003 . . Olivier
 * (cur) (last) . . 18:02, 4 Feb 2003 . . Olivier
 * (cur) (last) . . M 16:22, 4 Feb 2003 . . Olivier
 * (cur) (last) . . M 11:16, 4 Feb 2003 . . Olivier
 * (cur) (last) . . M 11:14, 4 Feb 2003 . . Olivier
 * (cur) (last) . . M 11:13, 4 Feb 2003 . . Olivier
 * (cur) (last) . . 11:11, 4 Feb 2003 . . Olivier
 * (cur) (last) . . 11:09, 4 Feb 2003 . . Olivier (started list of specialties by region/city)
 * (cur) (last) . . M 12:09, 3 Feb 2003 . . Sannse (Disambiguating German)
 * (cur) (last) . . M 11:42, 3 Feb 2003 . . Sannse (removed duplicate link)
 * (cur) (last) . . 09:26, 3 Feb 2003 . . 81.80.245.147
 * (cur) (last) . . M 10:05, 29 Oct 2002 . . Olivier
 * (cur) (last) . . M 10:04, 29 Oct 2002 . . Olivier
 * (cur) (last) . . M 10:02, 29 Oct 2002 . . Olivier
 * (cur) (last) . . 10:00, 29 Oct 2002 . . Olivier
 * (cur) (last) . . 23:45, 27 Sep 2002 . . Maveric149
 * (cur) (last) . . 09:04, 27 Sep 2002 . . Quintessent (added explanations of dishes with French names)
 * (cur) (last) . . M 15:51, 25 Feb 2002 . . Conversion script (Automated conversion)
 * (cur) (last) . . 15:47, 22 Jan 2002 . . 64.105.112.xxx
 * (cur) (last) . . 09:43, 22 Aug 2001 . . 63.192.137.xxx

title: Cookbook talk:Cuisine of France


text:
General Mechanics

Newton's Laws: First principles

 * 1) /Fundamental Principles of Kinematics/
 * 2) /Fundamental Principles of Dynamics/
 * 3) /Work and Power/
 * 4) /Partial Derivatives/
 * 5) /Motion in Two and Three Dimensions/
 * 6) /Force and Motion/
 * 7) /Momentum/
 * 8) /Statics/

Rotational Dynamics

 * 1) /Cross Product/
 * 2) /Torque and Angular Momentum/
 * 3) /The Uneven Dumbbell/
 * 4) /Index Notation/
 * 5) /Many Particles/
 * 6) /Rigid Bodies/

Newton's Laws: A second look

 * 1) /Motion Under Constraint/
 * 2) /Alternate Formulations of Newton's Laws/
 * 3) /Velocity Dependent Forces/
 * 4) /Mach's Principle/

Harmonic Oscillators

 * 1) /Energy Analysis/
 * 2) /Analysis Using Newton's Laws/
 * 3) /Forced Oscillator/
 * 4) /Coupled Oscillators/
 * 5) /The Continuum Limit/

title: General Mechanics


text:
Quantum Mechanics

Matter Waves

 * 1) /Introduction to QM/
 * 2) /Waves and Modes/
 * 3) /Blackbody Radiation/
 * 4) /Bragg's Law/
 * 5) /X-Ray Diffraction Techniques/
 * 6) /Operators and Commutators/
 * 7) /Heisenberg Uncertainty Principle/
 * 8) /Meaning of Quantum Wave Function/
 * 9) /Time Independent Schrödinger/
 * 10) /Mass, Momentum and Energy/
 * 11) /Perturbation Theory/
 * 12) /Quantum Scattering/

 * Problems

Symmetry and Bound States

 * 1) /Complex Waves/
 * 2) /Symmetry and Quantum Mechanics/
 * 3) /Confined Matter Waves/

 * Problems

Measuring the Very Small

 * 1) /Continuous Matter or Atoms?/
 * 2) /The Ring Around the Moon/
 * 3) /The Geiger-Marsden Experiment/
 * 4) /Cosmic Rays and Accelerator Experiments/
 * 5) /Commentary/

 * Problems

Atoms

 * 1) /Fermions and Boson/
 * 2) /The Hydrogen Atom/
 * 3) /The Periodic Table of the Elements/
 * 4) /Atomic Spectra/

 * Problems

The Standard Model

 * 1) /Quarks and Leptons/
 * 2) /Quantum Chromodynamics/
 * 3) /The Electroweak Theory/
 * 4) /Grand Unification?/

 * Problems

Further Reading and Resources

 * This Quantum World - a complete introductory text.
 * Quantum theory of observation
 * Quantum Field Theory
 * Quantum Chemistry

量子力学

title: Quantum Mechanics


text:
Cookbook:Garlic

| Herbs and Spices

Garlic is a pungently odoriferous member of the allium family (like onions and leeks). Garlic is generally used as a spice or a seasoning rather than as a vegetable due to its extremely strong flavour.

Harvesting
Garlic grows in large papery clusters. These clusters are called 'heads', 'bulbs' or 'knobs'. Each small, individual segment of a garlic head is a clove.

Freshly-harvested garlic bulbs must be cured prior to long-term storage. This curing dries out the exterior of the head, protecting the inner cloves from moisture and rot. Uncured garlic must be used quickly as it is more prone to spoilage.

Storing
Cured garlic heads will keep for a long time as long as they are stored in a cool, dark place. Do not keep them in the fridge or they will start to sprout and become bitter. They can be frozen without ill-effect, or simply stored in a dark cupboard away from moisture. Garlic braids should be hung to prevent crushing any of the cloves.

If you want to store the cloves individually and ready for use, the garlic must be either dried or processed. One good way to freeze prepared garlic is to crush or mince it in a food processor and mix it with a little water, then freeze it in ice cube trays so you can get cubes out as needed. Otherwise it can be frozen in olive oil, or frozen whole.

Never store garlic in olive oil at room temperature or leave garlic in oil to sit on the counter. Because garlic is grown in the ground it is frequently contaminated with botulism spores, which are almost impossible to remove. These are harmless in their normal state, but because they are an anaerobic bacteria they will grow at room temperature if submersed in oil. The spores can not grow in the cold, so freeze or refrigerate it, or better yet, store the garlic in vodka, wine or vinegar rather than oil.

Peeling
Before using cured garlic, the cloves must be separated from their papery casing. One method of extracting the clove from the bulb is to hold it in your hand and then use a paring knife to gently stab at the center of a clove, pulling it out from the skin. One can also grasp the top (or point) of the clove, and push the coarse base into a hard surface (such as a cutting board) until the paper "cracks", allowing it to be easily peeled from the clove.

Processing
As a rule, the finer you chop or especially crush fresh garlic, the stronger the flavour will be. This is because the flavour compounds are released by breaking cell walls. It is usually crushed with the side of a knife (which also aids in peeling) or finely minced. You can also use a garlic press, or crush it in a pestle and mortar with a bit of salt added to help reduce the garlic to a smooth paste. But it can also be used in slivers or as whole cloves, with a much milder result. Whole cloves are often roasted, and as they cook their flavour changes dramatically to become sweeter and less pungent.

Garlic can be purchased pre-crushed or pre-chopped in a jar. This type of pre-prepared garlic keeps for a long time in the fridge or freezer, and the garlic odour cannot penetrate the glass. It is milder and often sweeter than fresh garlic because the garlic flavor declines slightly with time.

Measurement
For cooking purposes, garlic is often measured in cloves. However, depending on the head of garlic, an average clove can vary from 1 g to 6 g, and personal judgment must be used to determine the amount to use. North American manufacturers of pre-packaged minced garlic consider one clove as 2.5 g. One teaspoon of minced garlic would contain the equivalent of 2 cloves.

Recipes featuring garlic
Many recipes use garlic, but those listed below feature garlic as a major contributor to the recipe.
 * Le Tourin
 * Pesto
 * Tahini Goddess Dressing
 * Garlic Bread
 * Aioli

Kookboek/Knoflook

title: Cookbook:Garlic