text:
Semiconductors/PN Junctions

Simple Models
These simple models are small signal approximations to how a fairly generic diode works. Zener and tunnel diodes are ignored.

At a first approximation, a diode is a device that will let current flow in one direction with zero resistance and will block current in the other direction with infinite resistance. Basically, it operates like a check valve for current.

A second approximation includes a constant voltage drop for current flow in the forward direction. The voltage drop varies with the type of diode, but is typically around 0.7 volts for fairly generic diodes and well over 1 volt for LEDs.

To get a third approximation, add a little resistance to the diode. Now things get a little more complicated. Since the v-i curve is non-linear, the resistance is not constant. However, if your signal is small and carried on a large offset, you can do a fairly accurate analysis by properly selecting the voltage drop and resistance.

Finally the fourth approximation accounts for the small leakage current that exists when the diode is reverse biased. This is typically in the range of nano-amps to a few micro-amps for most generic diodes.

If you need more accuracy than this, you will need to look at the physics that happens inside a diode.

Analysis of Diode Physics
Diodes junctions may be formed with N type and P type semi-conductor or semi-conductor and Metal junction.

as $$\Phi$$.

When the Metal and Semi-conductor come together, excess electrons diffuse from the semi-conductor to the metal until the Fermi levels are equal. As electrons diffuse from the semi-conductor to the metal, they leave behind their donor atom. This depletes the region near the interface of charge carriers and hence this region does not conduct well. Thus for charges to diffuse across this region, they require additional energy. Since the extent of the depletion of charges is some function of distance, the conduction band is bent to represent the energy required for diffusion in that region.

There will now exist 2 currents. Electrons diffuse from metal to semi-conductor over the barrier induced by the depletion region $$I_O$$. Electrons diffuse from the semi-conductor over this barrier in the opposite direction $$I_F$$. When there is no external applied voltage, there current will be equal and opposite.

Electrons will diffuse from semi-conductor to metal if they have enough energy to overcome the barrier. The energy of an electron is statistical, and is given by the Maxwell-Boltzmann distribution [1]. Thus as the temperature increases, the probability of the number of electrons occupying higher energy states increases. For low doping densities, the Maxwell-Boltzmann distribution may also be applied. However for non-zero temperature most excess electrons from the dopant will exist in the conduction band.

Maxwell-Boltzmann distribution says: given an energy W, the number of electrons with energy greater than W is given by:

$$ n_W=n\cdot e^\frac{-W}{kT} $$

where n is the number density of electrons, k Boltzmann's constant and T the temperature.

The Maxwell-Boltzmann distribution does not apply to metals. The reason may be found in the referenced [1]. For small voltages one may assume $$I_O$$ to be constant.

$$ I_F=nqe^\frac{-q\Psi}{kT} $$

$$I_O=I_F$$ for no external potential. This equilibrium exists for a diffusion potential we will call $$\Psi$$, where $$\Psi=\phi_m-\phi_s$$. Thus the energy of an electron to cross this barrier is given by $$q\Psi$$.

$$ I_O=I_F=nq\cdot e^\frac{-q\Psi}{kT} $$ $$ nq=I_O\cdot e^\frac{q\Psi}{kT} $$

The fermi level of the n-type semi-conductor may be increased by an external potential. This will increase the number of electrons with energy to diffuse across the barrier. Thus the energy for electron to cross the barrier will become $$q(\Psi-V)$$.

$$ I_F=nq\cdot e^\frac{-q(\Psi-V)}{kT} $$ $$ I_F=I_O\cdot e^\frac{q\Psi}{kT}\cdot e^\frac{-q(\Psi-V)}{kT} $$ $$ I_F=I_O\cdot e^\frac{qV}{kT} $$ Thus: $$ I_D=I_F-I_O $$ Let: $$ V_T=\frac{kT}{q} $$

Then:

$$ I_D=I_O\bigg(e^\frac{V_D}{V_T}\bigg)-I_O $$ $$ I_D=I_O\bigg(e^\frac{V_D}{V_T}-1\bigg) $$

Depletion Width
The depletion width is given by the required amount of charge to be displaced to realize the diffusion potential. Thus if one knows the doping density and relative permittivity of the material, then one may calculate the depletion width.

The depletion width is easiest realized by Poisson's Equation [1]. Assume constant doping density.

By Poisson's equation: $$ \frac{\partial^2}{\partial x^2}V(x)=\frac{\rho}{\varepsilon} $$

where $$\rho=qN_d$$ the charge density, $$\varepsilon$$ the permittivity and x the distance from the junction.

$$ V=\frac{qN_dx^2}{2\varepsilon_0\varepsilon_r} $$

The depletion width is some function of barrier height, $$(\Psi-V)$$. Therefore solve for $$x=W_d$$

$$ \Psi-V=\frac{qN_dW_d^2}{2\varepsilon_0\varepsilon_r} $$ $$ W_d=\sqrt{\frac{2\varepsilon_0\varepsilon_r(\Psi-V)}{qN_d}} $$

Depletion capacitance
Capacitance is the change in charge for change in voltage over the depletion width.

Given: $$ C=\frac{\partial Q}{\partial V} $$ $$ Q=qN_dW_dYZ $$

Then:

$$ C=\frac{\partial(qN_dW_dYZ)}{\partial(\Psi-V)} $$ $$ C=qN_dYZ\frac{\partial W_d}{\partial(\Psi-V)} $$ $$ C=qN_dYZ\cdot\frac{2\varepsilon_0\varepsilon_r}{qN_d}\cdot -\frac{1}{2}\bigg(\frac{2\varepsilon_0\varepsilon_r(\Psi-V)}{qN_d}\bigg)^{-\frac{1}{2}} $$ $$ C=-\frac{1}{2}qN_dYZ\cdot\sqrt{\frac{2\varepsilon_0\varepsilon_r}{qN_d}}\sqrt{\frac{2\varepsilon_0\varepsilon_r}{qN_d}}\cdot \sqrt{\frac{qN_d}{2\varepsilon_0\varepsilon_r(\Psi-V)}}\cdot \frac{\sqrt{\Psi}}{\sqrt{\Psi}} $$ $$ C=-\frac{1}{2}qN_dYZ\cdot\sqrt{\frac{2\varepsilon_0\varepsilon_r}{qN_d}}\cdot \frac{1}{\sqrt{\Psi}}\cdot\sqrt{\frac{\Psi}{\Psi-V}} $$ $$ C=-\frac{1}{2}qN_dYZ\cdot\sqrt{\frac{2\varepsilon_0\varepsilon_r}{qN_d\Psi}}\cdot \frac{1}{\sqrt{1-V/\Psi}} $$

Let: $$ C_0=-\frac{1}{2}qN_dYZ\cdot\sqrt{\frac{2\varepsilon_0\varepsilon_r}{qN_d\Psi}} $$ Therefore: $$C=\frac{C_0}{\sqrt{1-V/\Psi}}$$

title: Semiconductors/PN Junctions


text:
Semiconductors/MOSFET Transistors

Insulated Gate Field Effect Transistors, or IGFETs are are very important part of modern electronics. IGFETs control the flow of current using an electrical field applied at a contact, called the gate, which is electrically isolated from the current-carrying medium.

In early FETs, the gate was made using metal, with an oxide (Silicon Dioxide (SiO2)), serving as the insulator, leading to the name Metal Oxide Semiconductor FETs, or MOSFETs. This name has stuck even though gates are now commonly manufactured using polysilicon. The name has, however, stuck, and MOSFET is generally synonymous with IGFET, with a few exceptions.

MOSFET Basics
MOSFETs have two regions, called the source and drain which are heavily doped. These are embedded in a substrate, which is doped the other way. The gap between the source and drain regions, which spans the substrate, is where the current will eventually flow. A layer of insulating oxide is placed over this gap (the channel), and on top of that, a gate contact, usually made of polysilicon.

The way a MOSFET works is to modify a thin layer of this gap, or channel, using an electric field that propagates through the insulation. This modification could be either increasing the current carrying capacities of the channel, or reducing it. Thus, we have two kinds of devices - enhancement mode MOSFETs and depletion mode MOSFETs. Depending on what kind of silicon is in the channel, a MOSFET can be p-channel or n-channel. A p-channel MOSFET has p-type silicon in the channel when 'on.' This will be explained more clearly soon.

MOSFET operation

The operation of a MOSFET can be separated into three different modes, depending on the voltages at the terminals. For the NMOSFET the modes are:

Cut-off or sub-threshold mode
When

 * $$V_{GS} < V_{T}$$

where

 * $$V_{T}$$ is the threshold voltage of the device.

Here the switch is turned off, and there is no conduction between drain and source. While the current between drain and source should ideally be zero since the switch is turned off, there is a weak-inversion current, or subthreshold leakage. With MOSFET scaling subthreshold leakage composes a large percentage of total power consumption.

Triode or linear region
When

 * $$V_{GS} > V_{T}$$

and

 * $$V_{DS} < V_{GS} - V_{T}$$

The switch is turned on, and a channel has been created which allows current to flow between the drain and source. The MOSFET operates like a resistor, controlled by the gate voltage. The current from drain to source is:

 * $$I_D \approx \frac{\mu_n C_{ox}}{2}\frac{W}{L}\bigg[2(V_{GS}-V_{T})V_{DS}-{V_{DS}}^2\bigg]$$

Saturation
When

 * $$V_{GS} > V_{T}$$

and

 * $$V_{DS} > V_{GS} - V_{T}$$

The switch is turned on, and a channel has been created which allows current to flow between the drain and source. Since the drain voltage is higher than the gate voltage, a portion of the channel is turned off. The onset of this region is also known as pinch-off. In first approximation the drain current is now independent of the drain voltage, and the current is only controlled by the gate voltage:

 * $$I_D \approx \frac{\mu_n C_{ox}}{2}\frac{W}{L}(V_{GS}-V_{T})^2$$

In digital circuits the transistors are only operated in cut-off and saturation mode. The triode mode is mainly relevant for analog applications.

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

title: Semiconductors/MOSFET Transistors


text:
Semiconductors/MESFET Transistors

MESFET Operation
Assume an N channel MESFET with uniform doping and sharp depletion region shown in figure 1.

The depletion region $$W_n$$ is given by the depletion width for a diode. Where the voltage is the voltage from the gate to the channel, where the channel voltage is given for a position x along the channel as $$V_{gc}(x)$$.

 * $$W_n(x)=\sqrt{\frac{2\varepsilon_0\varepsilon_r(\Psi-V_{gc}(x))}{qN_d}}$$

 * $$W_n(x)^2=\frac{2\varepsilon_0\varepsilon_r(\Psi-V_{gc}(x))}{qN_d}$$

 * $$\frac{W_n(x)^2qN_d}{2\varepsilon_0\varepsilon_r}=\Psi-V_{gc}(x)$$

 * $$V_{gc}(x)=\Psi-\frac{W_n(x)^2qN_d}{2\varepsilon_0\varepsilon_r}$$

 * $$\frac{dV_{gc}(x)}{dW_n(x)}=-\frac{2W_n(x)qN_d}{2\varepsilon_0\varepsilon_r}$$            (1)

The current density in the channel is given by:

 * $$J_n=\sigma\xi$$

 * $$I_n(x)=\sigma \xi\cdot W\cdot b(x)$$

 * $$I_n(x)=-\sigma \frac{dV_{gc}(x)}{dx}W(a-W_n(x))$$

where:

 * $$\xi=-\frac{dV_{gc}(x)}{dx}$$

Therefore,

 * $$I_n(x)=-\sigma aW\bigg(1-\frac{W_n(x)}{a}\bigg)\frac{dV_{gc}(x)}{dWn(x)}\frac{dWn(x)}{dx}

$$

 * $$\int_0^L I_n(x)\, dx=\int_0^L-\sigma aW\bigg(1-\frac{W_n(x)}{a}\bigg)\frac{dV_{gc}(x)}{dW_n(x)}\frac{dW_n(x)}{dx}\, dx$$

 * $$I_n\cdot L= -\sigma aW\int_{Wn(0)}^{W_n(L)}

\bigg(1-\frac{W_n(x)}{a}\bigg)\frac{dV_{gc}(x)}{dW_n(x)}\, dW_n(x)$$

Substituting from equation 1:

 * $$I_n= \frac{-\sigma aW}{L}\int_{W_n(0)}^{W_n(L)}

\bigg(1-\frac{W_n(x)}{a}\bigg)\bigg(-\frac{2W_n(x)qN_d}{2\varepsilon_0\varepsilon_r}\bigg)\, dWn(x)$$

 * $$I_n= \frac{\sigma aW2qN_d}{2\varepsilon_0\varepsilon_rL}\int_{W_n(0)}^{W_n(L)}

\bigg(W_n(x)-\frac{W_n(x)^2}{a}\bigg)\, dWn(x)$$

 * $$I_n= \frac{2\sigma aWqN_d}{2\varepsilon_0\varepsilon_rL}

\bigg[\frac{W_n^2(x)}{2}-\frac{W_n^3(x)}{3a}\bigg]_{W_n(0)}^{W_n(L)} $$

 * $$I_n= \frac{2\sigma aWqN_d}{2\varepsilon_0\varepsilon_rL}

\bigg[\frac{W_n^2(L)-W_n^2(0)}{2}-\frac{W_n^3(L)-W_n^3(0)}{3a}\bigg] $$

 * $$I_n= \frac{2\sigma aWqN_da^2}{6L\cdot

2\varepsilon_0\varepsilon_r} \bigg[\frac{3(W_n^2(L)-W_n^2(0))}{a^2}-\frac{2(W_n^3(L)-W_n^3(0))}{a^3}\bigg] $$

One defines constant Β as the channel conductance with no depletion. And the work function to deplete the channel W00 [1]:

 * $$W_{00}=\Psi-V_{to}=\frac{qN_da^2}{2\varepsilon_0\varepsilon_r}$$

 * $$\beta = \frac{\sigma a}{3LW_{00}}$$

We now define Vto, the voltage such that the channel is pinched off. d is the ratio of channel depletion to maximum depletion for the drain. s the ratio of channel depletion to maximum depletion for the source.

 * $$d=\frac{W_n(L)}{a}=\frac{\sqrt{\frac{2\varepsilon_0\varepsilon_r(\Psi-V_{gd})}{qN_d}}}{\sqrt{\frac{2\varepsilon_0\varepsilon_r(\Psi-V_{to})}{qN_d}}}=\sqrt{\frac{\Psi-V_{gd}}{W_{00}}}

$$

 * $$s=\frac{W_n(0)}{a}=\frac{\sqrt{\frac{2\varepsilon_0\varepsilon_r(\Psi-V_{gs})}{qN_d}}}{\sqrt{\frac{2\varepsilon_0\varepsilon_r(\Psi-V_{to})}{qN_d}}}=\sqrt{\frac{\Psi-V_{gs}}{W_{00}}}

$$

Substituting:

 * $$I_n= W\cdot \frac{\sigma a\cdot W_{00}}{3L}

\big[3(d^2-s^2)-2(d^3-s^3)\big]$$

 * $$I_n= W \cdot\beta W_{00}^2 \big[3(d^2-s^2)-2(d^3-s^3)\big]$$ (2)

Equation 2 is Shockley's expression [2] for drain current in the linear region. When the device enters saturation, one end is pinched off(normally the drain). Thus $d=1$ and one may derive the equation for the saturation region:

 * $$I_{sat}=\beta W_{00}^2(1-3s^2+2s^3)$$

 * $$g_m=3\beta W_{00}(s-1)$$

 * $$G_{DS}=3\beta W_{00}(1-d)$$

Simpler Model

 * $$I_{ds}=\frac{3}{2}\beta

W_{00}^2\bigg[\frac{(V_{gs}-v_{to})^2}{W_{00}^2}-\frac{(V_{gd}-v_{to})^2}{W_{00}^2}\bigg] $$

 * $$g_m=3\beta W_{00}(V_{gs}-V_{to})$$

 * $$G_{ds}=3\beta W_{00}(V_{gd}-V_{to})$$

General power law:
It was found that a general power law provided a better fit for real devices [3].

 * $$I_{ds}=\beta\big[(V_{gs}-V_{to})^Q-(V_{gd}-V_{to})^Q\big]$$

Where Q is dependent on the doping profile and a good fit is usually obtained for Q between 1.5 and 3. A general power law is approximately equal to Shockley's equation for Q = 2.4. Β is also empirically chosen and is proportion to the previous Β

 * $$\beta \mbox{ proportial to } \frac{\sigma aW}{3LW_{00}}$$

Modelling the various regions is done though model binning. This however infers that a sharp transition exists from one region to another, which may not be accurate.

 * $$I_{ds}=\left\{

\begin{matrix} 0&V_{gs}V_{gd} \end{matrix}\right. $$

title: Semiconductors/MESFET Transistors


text:
User talk:Dexter73~enwikibooks

Semiconductor Device Physics and Fabrication Technology
I am merging your book, Semiconductor Device Physics and Fabrication Technology, into the new book on Semiconductors. My hope is that by combining all the different resources we have on the subject into a single book, we will have a higher-quality resource, and more people will be able to contribute to it. let me know if you have any objections to this, and I can undelete your book. --Whiteknight (talk) (projects) 18:10, 30 May 2006 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Dexter73. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Dexter73~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:09, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 04:53, 19 April 2015 (UTC)

title: User talk:Dexter73~enwikibooks


text:
Semiconductors/Doping

When examining crystals we make a few assumptions about them for simplicity. Such as:
 * They are Periodic
 * They are symmetrical
 * They are infinitely large
 * They compose a lattice

Although the infinite claim is a bit sketchy, it is a pretty good approximation as these crystals are often huge compared to distances in the crystal.

Since crystals exist in 3D space, and are periodic, if we move the size of the crystal in one of its axis we are at a corresponding spot. Mathematically we say that we have three linearly independent vectors and can move integer values of them to corresponding places in the lattice.

$$ \mathbf{r} = u\mathbf{a} + v\mathbf{b}+ w\mathbf{c} $$

Unit Cells
If we have a lattice it must be made up of smaller components, the unit cell is the smallest undivisible part of the lattice. And by placing many unit cells together, you can create the whole lattice again. Imagine a large stack of dice, you could say that one die was the unit cell for this crystalline lattice.

There is an infinite number of possible unit cells, but only 14 types of lattice. This can be generalised to 7 crystal classes.

Also there is a minimum size of the unit cell, with its volume being 1 lattice point (atom/molecule/whatever).

Cubic
Imagine a cubic unit cell with one point at each corner. So 8 points in total, but each of these is shared between it and 7 other cubes.

So the amount of lattice points per cell is 8 * 1/8 = 1. We call this the Primitive.

Now if we imagine each point as a sphere which touches its neighbours, then we can define the Packing Fraction as the volume taken up by the points compared to the volume of the cell.

In Cubic cells each sphere will have a radius half the size of the crystal. So we say r = a/2. $$V_{point} = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi \left(\frac{a}{2}\right)^3$$

And there is one per cell, ie the primitive = 1. So:

$$Packing Fraction = \frac{V_{point}}{V_{cell}} = \frac{\frac{4}{3}\pi \left(\frac{a}{2}\right)^3}{a^3} = \frac{\pi}{6}$$ or 52%

Body Centre Cubic
Like a cubic unit cell except has another lattice point directly in its centre.

Primitive = 8 * 1/8 + 1 = 2.

It has a Packing Fraction of 68%.

Face Centre Cubic
This is a close packed lattice, it's like a cubic except all 6 faces have another point at their centre.

Primitive = 8 * 1/8 + 6 * 1/2 = 4

Packing Fraction = 74%

title: Semiconductors/Doping


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Semiconductors/Transistors as Switches

A transistor is the most basic of digital devices. It has three terminals: the base, the collector, and the emitter. These terminals are connected to a piece of silicon that has been treated to have certain properties that allow it to behave as an amplifier, a switch, or other electronic devices.

Transistors as a Digital Device
At the most basic level, a transistor can be thought of as a switch, allowing current to flow if the base of the transistor is fed a reasonably high voltage compared to the emitter. This means that in a five volt system (+5v), voltages between 0v and 1v are considered 'low' and voltages between 4v and 5v are considered 'high'. Voltages between 1v and 4v are considered undefined and should not be applied to the input of a transistor acting as a logic device. Because transistors do not act like an ideal switch, they are arranged in ways that allow them to more closely emulate an ideal logic gate. More information on the different types of transistor families and associated logic will come later.

Transistor Anatomy
There are two main types of transistors: Bipolar Junction (BJT) and Metal-Oxide Semiconductor (MOS). These transistors are fabricated differently from slightly different materials, however they can be arranged to perform similar functions.

Inside a BJT
A Bipolar Junction Transistor is made up of a piece of silicon that has been treated to create a 'channel' in the silicon. First, an undoped (untreated) piece of silicon is selected (undoped silicon acts as an insulator, preventing the flow of current). Next, a process is used to create an N-type channel in the silicon. This is silicon that has an overabundance of negative charge carriers (electrons). A second channel is created in the first, this time with P-type silicon. P-type silicon carries current because of the overabundance of positive charge carriers ('holes', they're called). The process has created three regions in the silicon, an N-type region, a P-type region, and another N-type region. The base terminal is connected to the P-type region, while the collector and emitter are connected to the opposing N-type regions.

Effectively, for the average transistor, when the base of the transistor rises above 0.7V (compared to the emitter), the impedance between the collector and the emitter begins to fall, until at about 1.2V it has virtually no resistance.

BJTs are current controlled devices. The following assumes that the BJT is NPN construction. It is possible to construct BJTs as PNP but these are normally slower and thus not popular. The base emitter junction is a diode. If the device is based on silicon (as opposed to germanium or 3-5 compounds like gallium/arsenides) then the forward volt drop of the diode is assumed to be 0.7V. In practice the transistor starts to change state at 0.65V and by 0.75V is fully saturated. This means that the electric current flowing from collector to emitter is near zero whilst the base emitter voltage is below 0.65V (and hence there is no electric current flowing from base to emitter) then there is no current flow from collector to emitter. As the voltage from base to emitter is raised to 0.75 the current flow from base to emitter will increase by at least a factor of 100. This will increase the flow of current from collector to base by at least the same amount. By the time the base to emitter voltage is greater than 0.75V the current flowing from collector to emitter will typically be 100 times greater (this is amplification) and the collector to emitter voltage will have dropped to 0.3V and cannot be further reduced hence the transistor will be said to be in saturation. Technically the transistor is saturated when the collector voltage is below the base voltage.

An example
Reference: "Transistor Manual" published in 1964 by General Electric Company

Page 44 has two graphs related to the NPN Transistor 2N1613, whose tabulated data are shown on page 548.
 * a) The Volt-Current (VI) characteristic (Collector current versus collector-to-base voltage) for different base currents), and
 * b) The base current versus base-to-emitter voltage.

The VI graph shows that
 * negligible collector current flows if the base current is zero,
 * about 9 mA flows if the base current is roughly 0.1 mA,
 * about 45 mA flows if the base current is roughly 0.4 mA,
 * about 75 mA flows if the base current is roughly 0.7 mA.

 * A change of base current from 0.1 to 0.4, of 0.3 mA, results in a collector current change from 9 to 45, of 36 mA, so that the approximate current gain is 36 divided by 0.3 = 120.
 * A change of base current from 0.4 to 0.7 mA, of 0.3 mA, results in a collector current change from 45 to 75, of 30 mA, so that the approximate current gain (current amplification) is (30 divided by 0.3 =) 100.
 * MAXIMUM collector to base voltage, per data sheet, is 60 V, but the VI graph only shows voltage up to 50, which is below the permitted maximum.
 * The graph for the base VI relationship shows that there is negligible base current if the base to emitter voltage is below 0.5 V.
 * It also shows that for a base current of 5 mA to flow only about 0.9 V from base to emitter is needed. From above, only up to about 0.7 mA base current are needed.
 * From the data sheet we see that the rated maximum power for this transistor is only 0.8 Watt if the case temperature in free air cooling is no more than 25 degrees Celsius, but that a maximum of 1.7 Watts would result in a very hot case temperature of about 100 degrees C. Let us use only up to 0.8 Watts; this power curve is not shown on the collector current graph, but we can add it there: For 0.8 Watts and a collector current of 100 mA the collector-emitter voltage is (0.8 W divided by 0.1 A =) 8V, 90 mA and 8.9V, 80 mA and 10V, ...... 45V and 18 mA, 50V and 16 mA.
 * Assume we have two resistors in series connected to the power supply, and that one of these resistors is variable, the other one is fixed. As we vary the variable resistor from zero Ohms to its maximum, the voltage across it varies too. Obviously if it is zero resistance then the voltage across it is also zero, and the current is a maximum, being the voltage of the power supply divided by only the fixed resistor. If we set the variable resistor to its maximum value then the current is the power supply voltage divided by the sum of the two resistor values, and the voltage across the variable resistor is its resistance times the power supply voltage divided by the sum of the values in Ohms of these two resistors. Think of the variable resistor being the transistor, it being connected to the power supply via a fixed resistor. We can show on the collector current versus voltage graph a straight line for any such resistor, and we choose a line that is still within the permitted power area, possibly approximately a tangent to the maximum power curve. The graph in the book shows such a line, called a load line, connecting the 0V, 80 mA point with the 45V, 0 mA point. It indicates that the power supply voltage is 45V (zero current), and that the load resistor passes 80 mA if full power supply voltage is across it. Therefore the value of that load resistance is (45V divided by 0.08 A =) 562.5 Ohms. The nearest standard size resistor is 560 Ohms, to be connected between the power supply and the transistor.
 * The collector graph shows also that as the base current is varied as above, there is a corresponding voltage change across the transistor, with the voltage change across the resistor being the difference between the voltage of the power supply and the voltage at the collector. Note that this voltage change is much higher than the voltage change in the base circuit; voltage amplification results.

title: Semiconductors/Transistors as Switches


text:
Practical Electronics/7400 Series/Labelling

74 and 54 Series ICs all have a code on the top of the carrier body. This can be printed (usually in white) or etched. It consists of a "core" code which identifies the IC's function, and the type of electronics used within it. This is often supplemented by manufacturer's prefixes and suffixes which give information about the manufacturer and the carrier's physical form.

Device Family

 * 74 means a "commercial grade" device, suitable for general use.
 * 54 means a "military grade" device, for use in more hostile environments.

Device Function
This code tells you what the IC does. There are hundreds of these codes. Many common ones are given on the 74 Series page.

Prefix
This tells you which company made the IC. There a many hundred of different prefixes, and dozens of manufacturers, some sharing the same prefix. As these prefixes are not unique to the 74 Series of ICs, these prefixes are listed separately and can be found here.

Suffix
This tells you the type of package that the IC comes in, for example a DIP. It can also give information about the material (e.g. ceramic or plastic). These suffixes are not unique to the 74 Series of ICs, and as such, they are listed on a separate page, here.


text:
User:Freeshot

Greetings, i am Freeshot aka ThePieSwindler. Im 17 years old, and interested in learning about Computer Science and Economics this summer.

title: User:Freeshot


text:
User:Carle49

INDEPENDENT TELEVISION in the UK 1955-2005 Bold text

title: User:Carle49


text:
User:Imp Wit

Starting to learn wikibooks

WikiBooks are for TextBook...

Need to borrow my space for a second...

Moved Forth book to correct location so far in accordiance with Naming_policy

Forth

title: User:Imp Wit


text:
User talk:Imp Wit

Just in case someone wants to talk to me.

Hello and welcome to Wikibooks!

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 * Eventually, you might want to read the Manual of Style and Policies and Guidelines.
 * Take a look at Naming policy
 * See list of templates which can save your time
 * Remember to maintain a Neutral point of view.
 * Explore, be bold in editing pages, and have fun!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page.

Good luck! -- LV (Dark Mark) 03:11, 31 May 2006 (UTC)

title: User talk:Imp Wit


text:
Talk:Learning the vi Editor/Advanced tasks

Where did the motion 1G to move the beginning of the file come from? I can't find that in the documentation. I use gg.

-- SefTarbell@gmail.com


text:
Topology/Manifolds

The surface of a sphere and a 2-dimensional plane, both existing in some 3-dimensional space, are examples of what one would call surfaces. A topological manifold is the generalisation of this concept of a surface. If every point in a topological space has a neighbourhood which is homeomorphic to an open subset of $$\mathbb{R}^n$$, for some non-negative integer $$n$$, then the space is locally Euclidean. This formalises the idea that, while a surface might be unusually connected, patches of the surface can still resemble a euclidean space. For instance, the surface of a Klein bottle cannot be immersed in three dimensions without intersecting itself, and there is no distinction between the inside and the outside of a Klein bottle, but small patches of it still look like a euclidean space. A small creature living on the surface of a Klein bottle may not be aware of how it is connected overall or how it curves, but it could still make a rectangular or square map of its immediate area and use the map to measure lengths and directions and so forth.

A topological manifold is a locally-Euclidean Hausdorff space. Other properties are usually included in the definition of a topological space, such as being second-countable (having a countable base), which is included in the definition below. A topological manifold being Hausdorff excludes some pathological examples, such as the line with two origins, which is created by replacing the origin of the real line with two points. Any neighbourhood of either of the two origin points will contain all points in some open interval around zero, and thus will contain the other origin point, so their neighbourhoods will always intersect, so the space is not Hausdorff. See Non-Hausdorff manifold for other examples.

Note: As a convention, the ball $$B^0$$ is a single point. Any space with the discrete topology is a 0-dimensional manifold.

Note also that all topological manifolds are clearly locally connected.

To emphasize that a given manifold $$M$$ is $$n$$-dimensional, we will use the shorthand $$M^n$$. This is not to be confused with an $$n$$-ary cartesian product $$M\times...\times M$$. However, we will prove later that such a construction does exist as well.

The alert reader may wonder why we require the manifold to be Hausdorff and second-countable. The reason for this is to exclude some pathological examples. Two such examples are the long line, which is not second-countable, and the line with two origins, which is not Hausdorff.

Proof: Since all topological manifolds are clearly locally connected, the theorem immediately follows.

Given a pair $$(M,A)$$, where $$M$$ is an $$n$$-manifold and $$A$$ is an atlas on $$M$$, properties that $$M$$ may satisfy are often expressed as properties of the transition functions between charts in $$A$$. This is how we will define our notion of a differentiable manifold.

Note that in a smooth atlas, all transition functions are diffeomorphisms.

A chart $$(V,\psi)$$ with the property described above is said to be compatible with $$A$$.

Proof: We have to show that the transition functions between any pair of charts in $$A_{\mathrm{max}}$$ are smooth. This is obvious if one of then is in$$A$$, so let $$(V_1,\psi_1)$$ and $$(V_2,\psi_2)$$ be charts in $$A_{\mathrm{max}}$$ that are not in $$A$$, such that $$V_1\cap V_2\neq \emptyset$$. Let $$(U,\phi)\in A$$ be a chart such that $$U\cap V_1 \cap V_2 = W\neq \emptyset$$. Then $$\phi\circ \psi_1^{-1}|_W$$ and $$\psi_2\circ \phi^{-1}|_W$$ are both smooth, since, both $$(V_1,\psi_1)$$ and $$(V_2,\psi_2)$$ are compatible with $$A$$. Then, $$\psi_2\circ \psi_1^{-1}|_W=\psi_2\circ (\phi^{-1}\circ\phi )\circ \psi_1^{-1}|_W=(\psi_2\circ \phi^{-1}|_W) \circ (\phi\circ \psi_1^{-1}|_W)$$ is smooth since it is a composition of smooth maps. An identical argument for $$\psi_1\circ \psi_2^{-1}$$ completes the proof.

It should be obvious that if $$A^\prime$$ is a smooth atlas containing a smooth atlas $$A$$, then $$A_{\mathrm{max}}^\prime=A_{\mathrm{max}}$$.

Smooth maps
Proof: $$f$$ is continuous since $$\psi\circ f\circ\phi^{-1}$$ is smooth and thus continuous and $$\phi$$ and $$\psi$$ are homeomorphisms. Let $$(U^\prime,\phi^\prime)$$ and $$(V^\prime,\psi^\prime)$$ be two other charts at $$p$$ and $$f(p)$$. Then $$(\psi^\prime\circ f\circ{\phi^\prime}^{-1})=(\psi^\prime\circ\psi^{-1})\circ (\psi\circ f \circ \phi^{-1})\circ (\phi\circ {\phi^\prime}^{-1})$$ which is a composition of smooth functions since the atlases on $$M$$ and $$N$$ are smooth, and is therefore smooth.

Proof: Let $$(U,\phi)$$, $$(V,\psi)$$ and $$(W,\xi)$$ be charts on $$M,N,P$$ at $$p,f(p),g\circ f(p)$$ respectively. Then $$\xi \circ g\circ f\circ \phi^{-1}(\phi(p))=(\xi\circ g\circ\psi^{-1})\circ (\psi\circ f\circ\phi^{-1})(\phi(p))$$ which is a composition of smooth maps of euclidean space and is hence smooth.

title: Topology/Manifolds


text:
Algorithm Implementation/Date and time

 * /Conversion of a Gregorian date to an International Fixed Calendar date/

title: Algorithm Implementation/Date and time


text:
Algorithm Implementation/Date and time/Conversion of a Gregorian date to an International Fixed Calendar date

This Source code converts a Gregorian Calendar date into a International Fixed Calendar date. The Gregorian date can be abbreviated, or

nothing entered for today's date. The program is in 2 classes: IFCConverter and TestIFCConverter.


text:
Bodybuilding and Weight Training/Finding a Gym

Finding A Gym
Finding a gym can be a difficult task. You can do a few things. For example there's probably a recreation center or fitness club in your area. Try looking in the telephone book to find numbers of local workout centers. Many high schools also have designated days where you can come and use the gym for a small fee. Ask friends where they work out and find out others opinions on places you might want to go. Or, if you want to work out close to home, build your own gym. You can get weights, a yoga mat, a treadmill (or take walks), workout bicycle (or bike around), and jump ropes. You can do all your working out in your home at a very low cost. Research what option is best for you, and go for it. This is the easy part!

Created By: writerhawk 23:21, 30 May 2006 (UTC)


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MediaWiki talk:Pagemovedtext

Broken code
The [ current code] was broken by recent changes to the way MediaWiki parses interface messages. The code below corrects the problem, links directly to the page (ignoring the redirect), and adds links to allow for easy whatlinkshere access and deletion. This version is currently used on Meta-Wiki and the English Wikisource.

 Page " [ $1]" (links • [ delete]) moved to "$2".

Please check if this move has created any double redirects, and fix them as necessary.

Alternately, simply removing the <nowiki> tags will correct the display problem with the current version. // Pathoschild 23:38, 30 May 2006 (UTC)

title: MediaWiki talk:Pagemovedtext


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Talk:Blender 3D: Noob to Pro/Every Material Known to Man/Light Effects

Could you possibly provide a .blend for us, it seems to be difficult to reproduce.

--I got good results by lowering the alpha value of the material to about 0,2-0,3


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User:Patrickward

Patrick Ward

How my comments work
For Bad Leadership I have put in post it notes. They are noted in my comments on the main page with the page numbers.

Nice personal page patrick :) Aburda 23:03, 2 June 2006 (UTC)

title: User:Patrickward


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User:(chubbstar)

my "real" page is at Wikipedia here.

title: User:(chubbstar)


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User talk:(chubbstar)

Hiya, I have accounts on several wikimedia site and thusly do not check them all as frequently as possible. So, you you wanna leave a general chat with me about nothin particlar you should probly just leave that at my wikipedia talk page, here. Otherwise, for instance if you are a bot or I forgot to add copyright details or its an issue directly related to this wikimedia project or somethin' you should just leave your message here. I'm sure ill get to it eventually. Ta ta.

title: User talk:(chubbstar)


text:
Suomen kieli ulkomaalaisille/Sisältö/us-uus-y-yys nominals

This page is in development

Partitive formation varies for example

lupaus (nom.) > lupauksen (gen.) lupausta (ptv.)
 * abstract noun lupaus permission is derived from luvata (to permit)
 * verb stem lupaa- + -us (abstract noun formation) > laupaus (with standard dropping of the extra vowel)

kaipaus (nom.) > kaipauksen (gen.) kaipausta (ptv.)
 * abstract noun kaipaus longing or yearning is derived from kaivata (to yearn

for, long for, to miss)
 * verb stem kaipaa- +-us (abstract noun formation) > kaipaus (with standard dropping of the extra vowel)

lisäys (nom.) > lisäystä (ptv.)
 * abstract noun lisäys addition or increase is derived from lisätä ('to increase. add to)
 * verb stem lisää- + -us (abstract nounn formation) > lisäys (with standard dropping of the extra vowel)

lykkäys (nom.) > lykkäystä (ptv.)
 * abstract noun lykkäys postponement is derived from lykätä to postpone
 * stem lykkää- become (abstract noun, postponement)

The group that forms differently are usually formed from nouns and adjectives. These forms have genitive forms where the s transforms to d but partitive forms where the s transforms to t.

vapaus (nom.) freedom > vapautta

rakkaus becomes rakkautta

onnettomuus becomes onnettomuutta

ystävyys becomes ystävyyttä

kateus becomes kateuta

pimeys becomes pimeyttä

The rule is that if the word is derived from a nominal form, then the partitive will take the standard -ta or -tä ending which usually attached to nominals ending in a consonant. BUT, if the word is derived from an adjective (which may itself be derived from a verb) the s usually mutates to to a t before the -ta or -tä ending ius added.


text:
Learning the vi Editor/Print version

= Learning the vi Editor=

= Getting acquainted = = Basic tasks = = Making your work easier = = Advanced tasks = = Details =

= Vi clones =

= Vim =

= vile =

= BusyBox vi =

= vi Reference =

= Authors =

title: Learning the vi Editor/Print version


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Talk:European History/Europe: 1918 to 1945

Problems in Weimar Republic
Very impressed with the wikibook in general, but I thought this bit was vague: "The Spartacists, or communists, challenged the authority. The Freikorps, a band of paramilitary soldiers that were left demobilized and unemployed after the war, caused additional problems for the Republic. Finally, the Brown Shirts, the reactionary conservative group, also created problems for the government." Just saying that these groups 'challenged authority' or 'caused problems' for the Weimar government doesn't seem very clear. If you like I could have a go at changing it (I was thinking of just mentioning the Spartacist's attempted uprising c.early 1919 and the Freikorps' involvement in the Kapp Putsch) but I thought I'd check here before I went about altering your lovely book. Also, nitpicking, but shouldn't we say 'Sturmabteilung' instead of 'Brown shirts'? It would seem more consistent with the use of the German names for the Freikorps and such. Anyway, those are my thoughts. If nobody has anything to say on the matter I'll go ahead and change it myself.--AsaRoast 13:32, 31 May 2006 (UTC)

Holocoust
I believe that mentioning of WWII without mentioning the Holocoust or the extermination of 6 million Jews is a real problem. Dror Kronstein 01:02, 24 July 2006 (UTC)

Too true, there should be something about the Holocaust here. I think there should be mentions about how many lives were lost (Jews, non- Jews) in total because of the WWII (both in battle and in camps: Nazi camps and in Soviet camps). That's one of the things that have to be remembered about WWII, the scale of it and how much it changed the whole Europe.

Russian Losses
Well, 6 million Jews were lost, but I don't hear anybody willing to talk about 20 million Russians who were lost...
 * That's a good point. Why not be bold and add this information to the article. I look forward to seeing what you have to say about the topic :) Reece   (Talk)   (Contributions)  18:39, 25 September 2008 (UTC)

Bias
The part about Wilson's suggestions being perfect and the Treaty of Versailles being evil is ludicrously biased.--219.78.47.79 06:58, 24 December 2006 (UTC)

Expansion
I think Expansion is needed on WWII. If you look at WWI, it is much, much more detailed, and WWII seems to be just a brief overview. Red4tribe (talk) 14:06, 21 June 2008 (UTC)

World War II
I think world war II should have its own page due to its signifigance.

Turkish War of Independence
I think it was a very significant event in the history of Europe. Though the "borders" of Europe come into question for something like this, I believe the wars of 1918-1923 greatly influenced Europe at large. The Megali Idea died in Greece, the Turkish Republic became the first ally of the Soviet Union and a great power was lost to history. I'm not sure if the Caucasus counts as a region of Europe, but the Turkish War of Independence (particularly regarding Armenia) and the Russian Civil War caused sweeping changes in terms of ethnicities in the region.

I will write a brief overview of the war if people think it is appropriate for the topic of European history, or appropriately influential. What are your view on this? Johnanth (discuss • contribs) 14:38, 27 May 2012 (UTC)


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Talk:Spanish/Infinitive

Untitled
I was wondering.... Should I use the imperfect subjunctive for the verb on that last example?
 * Yo pensaba que nadar era divertido. User:Jack-o-jason 31 May 2006 (UTC)

Are there anymore different examples of infinitives that I should put? User:Jack-o-jason 1 May 2006 (UTC)

title: Talk:Spanish/Infinitive


text:
Talk:German/Level I/Freizeit/Test

Untitled
Wenn John Carrie nicht um 7.00 Uhr trifft, werden sie nicht in Kino gehen.

title: Talk:German/Level I/Freizeit/Test


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A-level Biology/Biology Foundation/enzymes

Enzymes are protein molecules that act as biological catalysts on reactions. This means that they speed up said reactions, but remain unchanged at the end. A vast majority of metabolic reactions are catalysed by enzymes. Enzymes are globular molecules just as we learnt about in the previous chapter, and so have their hydrophilic r R groups on the outside, making them soluble, but enzymes also have a speciality in that they possess an 'active site'. This active site allows other molecules to bind to the enzyme, and these molecules are known as the 'substrate', and the shape of the active site allows them to fit perfectly. Temporary bonds form between the substrate and the R groups on the enzymes amino acids, forming the enzyme-substrate complex. The substrate then unbinds in the form of its product.

Enzymes are said to be specific in their nature, as the shape of the active site will only allow one molecule to fit. It may either catalyse the breaking up or joining of molecules, forming one or more products. When the reaction is complete, these products leave the active site, and as said before - the enzyme is unchanged and can thus receive another substrate molecule. Most enzymes end in 'ase', for example lipase, amylase

=Activation Energy=

All reactions require energy - and the threshold of the reaction happening is known as the activation energy - the point at which the reaction begins to occur. Enzymes lower this activation energy, and sometimes this not only speeds up the process, but makes it happen in the first place as most of the reactions in living cells would be so slow that they would (for practical purposes), not happen at all.

One way of providing energy to reactions is to heat them, since this gives kinetic energy to the molecules and increases the rate of collisions between enzymes and substrates (more in the temperature section). Humans do this, maintaining a constant 37c within the body, but this is not enough for the activation energy required for many reactions - and thus enzymes solve this problem by reducing the activation energy. Enzyme-catalysed reactions take place faster and at a much lower temperature than standard reactions.

A reaction's path
We have said that when an enzyme catalyses a reaction, substrates combine with enzymes to form enzyme-substrate complexes, and a product is created. Take a look at this graph. Time is along the bottom, amount of product produced on the y-axis. It is a generic enzyme catalysed reaction - as you can see there is a large amount (the majority) of product collected quickly, but then the line levels off.

This is because when there is a large amount of both enzyme and substrate, all enzyme molecules have a substrate in its active site, and collisions are constantly occurring, but as the reaction goes on and more substrate is used up, the longer it takes for a substrate to hit an enzyme (as the substrate becomes more scarce), and thus the more time enzymes are simply waiting. This initial period is known as the initial rate of reaction.

=Reaction Rate=

Rate of reaction can be measured in a couple of ways - the amount of product produced, the amount of substrate left (as subtracted from the initial total). For example, the rate at which amylase breaks down starch - both the product and substrate remain colourless in the mixture. Last chapter, the food test section showed that we test for the presence of starch with iodine - so to see the rate of reaction we can take a sample from the reaction mixture, mix it with iodine and test the intensity of the change with a colourimeter. This will give us results that can be plotted into a graph - the amount of starch remaining against time.

Various things affect a reaction rate, enzyme concentration, substrate concentration, temperature, pH and presence of enzyme inhibitors. These are discussed next.

Enzyme Concentration
Take a look at this graph. It shows how increasing enzyme concentration increases the rate of the reaction, and this is in line with enzyme theory as so long as there is sufficient substrate, there is more enzyme-substrate complexes being formed per second and thus, more product being formed every second - faster rate of reaction. However, something to note is that left to their own devices, these reactions would all eventually produce the same amount of product (assuming each has the same amount of substrate). The rate of reaction increases linearly with enzyme concentration.

Substrate Concentration
This graph shows how increasing the substrate concentration increases the rate of reaction, to a certain point. This is because, as substrate concentration increases, more enzyme-substrate complexes can be formed per second, but point x on the graph is known as the saturation point, which is where every enzymes active site is working continuously, and the enzymes cannot work faster so substrate simply ends up 'queuing up' and waiting for an active site.

Temperature
The speed at which molecules move around freely is determined by the temperature giving them more or less energy to do so - at high temperatures the molecules, the enzymes and the substrates, move around fastest, and thus there are the most collisions per second, causing enzyme-substrate complexes to be formed more frequently. So why does the graph show the rate of reaction peak at 40c? After 40c, the bonds in the enzyme molecule begin to shake so violently that they begin to break, which causes the enzyme to lose it's shape, which changes the shape of the active site (so the substrate will no longer fit) and the enzyme is said to be denatured. The reason the rate of reaction does not immediately drop to 0 is that the enzyme slowly loses shape as the temperature increases, and so the substrate fits less well and eventually not at all, and so the catalysis does not occur. The process of denaturing is usually irreversible.

In humans 40c is the optimum temperature for enzyme reactions- the temperature at which an enzyme catalyses a reaction at the maximum rate. We keep our bodies at 37c to make sure we never go above 40c as enzymes would start to denature at even the slightest upward variation - which would be extremely dangerous as nearly all reactions within the body rely on enzymes.

Effects of pH
This graph shows the effect of pH on enzyme activity. pH is a measure of hydrogen ions in a solution (think: percentage Hydrogen), and these affect enzyme activity since the hydrogen ions can react with the enzyme and change the enzymes shape, deforming the active site. As with temperature, too high or low pH for the enzyme will denature the enzyme. The graph is just an example, since different enzymes have different optimum pH

=Inhibition=

There are two types of enzyme inhibition - that is, things that inhibit the enzymes function. They can be harmful in that they can stop a reaction happening, or helpful in stopping a reaction from running wild - perhaps the end-product of a chain of reactions will be an enzyme inhibitor to prevent the reaction continuing indefinitely.

Competitive

Competitive enzyme inhibitors are named as such because they compete with the substrate for the enzyme molecule's active site. They achieve this by being a similar shape and fitting into the active site, temporarily blocking substrate from entering. This makes the reaction slower, since there is less chance that genuine substrate will collide with an enzyme and form product. Competitive inhibitors affect on the enzyme is always reversible. They can be overcome by increasing the substrate concentration. This then increases the chances of for substrate to bond with the enzyme active site, and decreases the chances for the inhibitor to have a successful collision.

Non-Competitive

Non-competitive inhibitors are ones that do not compete for the active site. Instead they bind to another region of the enzyme called it’s allosteric site. This is turn causes the enzymes active site to change shape and denature because the inhibitor that has bind to the enzyme changes the tertiary structure of the enzyme and thus also the active site stopping any substrate from bonding/fitting. This is usually permanent. Unlike, competitive inhibitors, non-competitive inhibitors cannot be over come by increasing substrate concentration. This is because the substrate and non-competitive inhibitor do not compete for the active site, thus allowing a non-competitive inhibitor to bind to an enzyme’s allosteric site, without a substrate preventing/competing with it at that site.

End product
Some products will inhibit the enzyme that created them, in a process known as end product inhibition. This acts as a negative feedback loop for the enzyme, as this will stop the enzyme from creating an excess of product, as well as ensuring that valuable resources are not wasted by the enzyme.


text:
GCSE Mathematics

GCSE Mathematics is offered by all five examination boards: AQA, CCEA, Edexcel, OCR and WJEC. This wikibook should be suitable for all exam boards, but check with your exam board to see what topics you need to know. For the exam structure used by AQA that is similar to that of most exam boards see the relevant chapter.

Numbers

 * /Factorisation/
 * /Sequences/
 * /Percentages/
 * /Probability/
 * /Money/
 * /Matrices/

Algebra

 * /Linear Equations/
 * /Simultaneous Equations/
 * /Quadratic Equations/

Graphs

 * /Travel Graphs/
 * /Graphing Functions/
 * /Linear Programming/

Shapes

 * /Area and Volume/
 * /Circles/
 * /Trigonometry/
 * /Similar Shapes/
 * /Transformations/
 * /Constructions and Loci/

Other

 * /Statistics/
 * /Sets/
 * /Functions/

title: GCSE Mathematics


text:
GCSE Mathematics/AQA Specification

For the latest specification (syllabus) see this page of the AQA's website.

The overall subject content is the same as for 2007, but there are changing to deal with the changes to tiers.

Assessment Objectives
These are the four assessment objectives:

AO1: Using and applying mathematics AO2: Number and algebra AO3: Shape, space and measures AO4: Handling data

AO1 is assessed in contexts provided by other Assessment Objectives.

Internal assessment
This is compulsory. It comprises half the marks for AO1 and AO4 (the other halves being from written papers). A04 has a handling data task, A01 has an investigative task, which assesses AO1 in the context of AO2 and/or AO3.

Mark Scheme
The written papers are structured so that 50% of marks are focused on the lowest two grades available, with 25-30% being focused on the top two grades.

Paper 1
Written, no calculator. 40% of total marks. Foundation Tier - 1.5 hours, Higher Tier - 2 hours. All questions compulsory, question and answer book provided. Marked out of 100.

Paper 2
Written, calculator. 40% of total marks. Foundation Tier - 1.3 hours, Higher Tier - 2 hours. All questions compulsory, question and answer book provided. Marked out of 100.

Coursework
20% of total marks. Either Centre/AQA-set tasks, Centre marked or AQA-set tasks, AQA marked. Marked out of 48.

title: GCSE Mathematics/AQA Specification


text:
General Astronomy/Luminosity

Luminosity is the total amount of light emitted by a particular star. It is the star's intrinsic brightness, or the amount of energy it emits each second. In many cases, this energy comes from thermonuclear reactions inside the star. In general, a star's luminosity depends on its current stage in its evolutionary sequence. Stars radiate various forms of electromagnetic radiation, such as ultraviolet radiation, visible light, infrared, and radio waves.

The luminosity of a star depends on its temperature and size. If two stars of different sizes both have the same temperature, the larger of the two will have a greater luminosity due to its larger surface area.

The formula for calculating luminosity is

$$L=4 \pi r^2 * \delta T^4 $$

where T is the star's temperature in Kelvins.

Another way to measure a star's luminosity is by calculating its magnitude. There are two types of magnitude: apparent and absolute. Apparent is the luminosity of an object when viewed from Earth; absolute is the brightness of that object if it were 10 parsecs away from Earth.

The Sun has an apparent magnitude of -26.7, and an absolute magnitude of 4.8, while a full moon has an apparent magnitude of -12. Most visible stars range in apparent magnitude from -1.5 (Sirius) to 4 (Alcor). The dimmest ones just visible to the naked eye are around 6.5 (4 in the city with bright lights).

Stars are classified by orders of magnitude: alpha, beta, gamma, delta, epsilon, and zeta. These are called Bayer letters.

Cepheid variables are pulsating stars. They have a longer period and higher magnitude than other stars. By calculating the star's period and magnitude, the distance can be determined. Astronomers discovered from the Cepheid variables in Andromeda that the nebula was 2 million light-years away.

The stars which have been measured by astronomers up to date range from $$\frac {1}{500}$$ to 500,000 times the luminosity of our Sun.

title: General Astronomy/Luminosity


text:
Introduction to Astrophysics/Galaxies:Shapes and Evolution

The three main types of galaxies are elliptical (E), irregular (I), and spiral (S).

Galaxies are typically found in clusters. These clusters are categorized as regular or irregular. Regular clusters contain mainly E galaxies, and are large, dense, and spherical. These clusters have formed from many collisions, causing the S galaxies to become E galaxies. Irregular clusters are thin and have no symmetry. They contain mostly S galaxies.


text:
Introduction to Astrophysics/Albert Einstein

Albert Einstein, a rare genius, has something that changes the whole way of looking to the heavens. He developed the theory of relativity, one of the most important discoveries of the 20th century. This deals with very large objects, such as planets and galaxies. The Special Theory pertains to motion without a gravitational field, and the General Theory pertains to motion with a gravitational field.


text:
Apache Ant/Background

What is Apache Ant?

 * An operating system and language neutral XML based "Build Tool"
 * A scripting language for executing any complex file-system processes that can be run from a command line interface
 * A.N.T. – Another Neat Tool
 * Used for Building the Project

History of Ant

 * Built by James Duncan Davidson
 * Frustrated with the UNIX "make"
 * Invented while turning a product from Sun into open source
 * Make used "tab" as a record separator
 * Tabs frequently got converted to spaces during copy/paste operations

Why is Ant Strategic?
Ant is important because it helps organizations create repeatable build processes.

Repeatability is critical to organizations reaching the next level of CMU's Capability Maturity Model :

 * 1) Initial
 * 2) Repeatable
 * 3) Defined
 * 4) Managed
 * 5) Optimized

Ant helps you get from the Initial to the Repeatable level.

Ant is a Process Discipline

 * Process discipline helps ensure that existing practices are retained during times of stress
 * When these practices are in place, projects are performed and managed according to their documented plans
 * Answers the Question: How did the prior developer compile, test and install their system?
 * Excellent aid for software archeologists

Software Project Lifecycle

 * Version 1 and version 2 of a software package are frequently done by different groups
 * Sometimes version 1 and 2 are done years apart by different teams in different countries
 * Contractors and internal staff need to use the same tools
 * Shared development processes, like those used in the Open Source community, would be almost impossible without tools like Ant

Ant is Operating System and Language Neutral

 * Builds run on both Windows and UNIX/Linux systems
 * Should run anywhere a Java VM runs
 * Ant files "know" about file separators "/" vs. "\"
 * Build targets can still do OS-specific tasks
 * Works with anything that can be done from the command line
 * Easy to extend (with Java)
 * It is possible to extend with other languages as long as they have Java bindings but in practice, most people use Java to extend Ant

Ant and XML

 * If you are familiar with XML (or even HTML) you will probably learn Ant quickly
 * If you are not already familiar with XML you will need to learn some XML before you use Ant
 * One of the best ways of doing this is to read many small Ant sample tasks
 * This book should help you do this

Next Chapter
../Adoption/

title: Apache Ant/Background


text:
Apache Ant/Adoption

Organizational Ant Adoption
We have found that most people new to Ant go through several stages of learning Ant.

The three stages of learning core Ant functionality

 * 1) Learn the grammar and syntax of XML and a build file- this goes quickly if you know any XML or HTML (about 15 minutes)
 * 2) Learn the Ant concepts of properties, dependencies and task (about two hours)
 * 3) Build your vocabulary: Learn the basic Ant tasks you need to get your job done (duration depends on what you are doing)

Transform Your Organization!

 * Integrating it into your development/QA process
 * Making it a requirement of all projects
 * Specifying that all vendor deliverables MUST include a reproducible build process
 * Problematic for Microsoft developers that are not familiar with Ant
 * Problematic for "visual only" development environments (Microsoft Visual Studio, Microsoft Analysis Services)

Next Chapter
../XML Summary/

title: Apache Ant/Adoption


text:
Apache Ant/XML Summary

Brief Overview of XML
This chapter has a summary of the things you need to know about XML to use Apache Ant. It is not a lot of information and users that are already familiar with HTML will pick up the concepts very quickly.

You do not need to know a lot about XML to use Ant. Just a few syntaxical items.

First, you must be very careful to match up the "begin" and "end" tags. Begin tags start with a "<" and end tags have the same name but start with a " Data Inside...

Train your eye to look for the

Understanding this paired nesting structure is critical to creating working Ant build files.

XML Attributes
The XML begin tag may also have attributes.

Text

Next Section
../Getting Started/

title: Apache Ant/XML Summary


text:
Talk:Apache Ant

I am in the process of restructuring to break the book into chapters and sub-chapters.

- Dan

Dan just my thought here, the "Core Concepts" should be moved up to it's own level. Not as a sub to "Getting Started". --173.25.223.151 (discuss) 23:25, 9 March 2011 (UTC)

Also IMO the last two sections are full of examples, and should be children of "Examples" section instead of main sections. Thanks for the intro to Ant! --173.25.223.151 (discuss) 00:18, 10 March 2011 (UTC)

title: Talk:Apache Ant


text:
Apache Ant/Getting Started/Hello World

Hello World in Ant
Create a directory "My Project". Use a text editor such as Kate, Gedit or Notepad to create a file called build.xml within directory "My Project":

The first line in the file is flush left (no indentation). It tells ant that this is an XML file:

The next line names the (required) project "My Project" and its default target "hello":

The central three lines name and define the only target ("hello") and task ("echo") in the file:

Hello World!

You can now open a shell and cd to the "My Project" directory you created and type "ant"

Output of Hello World
Buildfile: build.xml hello: [echo] Hello World! Build Successful Total time 0 seconds

Variations
Try changing the echo line to be the following:

What is the result? Try the following also:

Next Section

title: Apache Ant/Getting Started/Hello World


text:
Apache Ant/Getting Started

This section contains three chapters:

 * 1) /Installation/ How to download, install Apache Ant
 * 2) /Testing/ Testing Apache Ant
 * 3) /Hello World/ How to run a small ant program that prints "Hello World!"

title: Apache Ant/Getting Started


text:
Apache Ant/Getting Started/Installation

Before starting, you will need to have a running version of the Java Development Kit (JDK) 1.2 or later. For Ant 1.7, JDK 1.5 is recommended. Ant will need a JAXP-compliant XML parser installed and available in your classpath. The binary distribution of Ant includes the latest version of the Apache Xerces2 XML parser.

To install Ant on Windows, you can use WinAnt the Windows installer for Apache Ant. Download and run the latest version of WinAnt, and follow the directions in the installer. WinAnt will place the "ant" executable on your system path, which allows you to run the command "ant" from the command line at any directory in your system. Or, you can follow these directions:

The first step is to download Apache Ant.

There are two options. You can compile the program from the source or for beginners we recommend just downloading a binary file called a zip file.

You can find the download page here: http://ant.apache.org/bindownload.cgi

You will need to download a file such as

apache-ant-1.7.0-bin.zip

This is a compressed archive file. You will need to uncompress it.

If you are not familiar with the process of uncompressing a file you should consult your computer operating system manual.

After this is done, you will see a folder such as:

apache-ant-1.7.0

This folder contains another folder called "bin". Within that there is a file called ant.bat that you can run directly from the command line.

Next Chapter

title: Apache Ant/Getting Started/Installation


text:
Apache Ant/Getting Started/Testing

To test that apache ant is installed correctly type "ant -version" from the command line:

C:\ ant -version Apache Ant version 1.6.5 compiled on June 2, 2005 C:\

If you do not see this, you have to check your PATH variable to make sure that ant is in your path. You can do this by opening a prompt and typing the "set" command.

To add the path in Windows: Right click on "My Computer" and find the Environment Variables button. Find the System Variable "Path", and add the path for Ant's bin folder (C:\ant\bin or whatever it is) separating it from other paths with a semicolon.

Next Chapter

title: Apache Ant/Getting Started/Testing


text:
Apache Ant/Core Concepts/Terminology

Ant Terminology

 * Ant Task – something that ant can execute such as a compile, copy or replace. Most tasks have very convenient default values.  See the Ant manual for a complete list of tasks.
 * Ant Target – a fixed series of ant tasks in a specified order that can depend on other named targets. Targets can depend only on other targets, not on projects or tasks.  A target represents a particular item to be created, it can be a single item like a jar, or a group of items, like classes.
 * Ant Project – a collection of named targets that can run in any order depending on the time stamps of the files in the file system. Each build file contains one project.

Next Chapter

title: Apache Ant/Core Concepts/Terminology


text:
Apache Ant/References

Links

 * ant.apache.org
 * Manual
 * Wikipedia.org

Books

 * Ant: The Definitive Guide, 2nd Edition by Holzner Steve (April 14, 2005)
 * Pro Apache Ant by Matthew Moodie (Nov 16, 2005)
 * Java Development with Ant by Erik Hatcher and Steve Loughran (Aug 2002)
 * Ant Developer's Handbook by Allan Williamson, et al. (Nov 1, 2002)

Articles

 * Top 15 Ant Best Practices

title: Apache Ant/References


text:
Apache Ant/Best Practices/Standard Targets

=Standard Targets=

One of the things that you learn is that if you name things consistently between projects, it is much easier to find things you are looking for. When you work with other people, you also want to have targets that you both are familiar with.

build.xml

 * Place your main build in a file called build.xml in the main directory of your project.
 * Do not put references to local file systems (Windows C:\ etc.) in your build file. Isolate these all in a local.properties file in the main directory.

Folder standards

 * - the location of your source code
 * - the output of a build process

This target should create all temporary directories within the build folder.

This target should remove all compiled and intermediate files leaving only source files. It should remove anything that can be derived from other files. This would be run just prior to creating a zip file of the project, and in case of gremlins occurring during the build process.

This target should compile sources and perform transforms of raw data.

The install target should be used to copy files to a testing or production system.

Other Standards
Use the  element to describe what your target does.

If you have more than around 100 targets in your build file, it becomes unwieldy. You could consider calling a separate build file, but that adds other complications such as the dependency between targets.


text:
Apache Ant/Best Practices/Local Property Files

=Using a Property file=

One of the best ways to keep your build files free of local dependencies is to use a local property file

Here is a sample of a property file:

# Property file for Project X # Author # Date # Note that the format of this file adheres to the Java Property file specification # http://docs.oracle.com/javase/7/docs/api/java/util/Properties.html#load(java.io.Reader) # to use the file put the following in your ant file: #  # All file names on local hard drives should be stored in this directory # Where Ant is installed. Will not work with 1.5 due to exec/spawn calls antHome=C:/Apps/apache-ant-1.6.5 Saxon8HomeDir=C:/Apps/saxon8 saxon8jar=${Saxon8HomeDir}/saxon8.jar # used to make sure Saxon gets the right XSLT 2.0 processor processor=trax


text:
Apache Ant/Best Practices/Local File Systems

=Dealing with Local File System Issues=

 * Each developer or user has the right (or is forced by administrators) to put resource such as jar files and libraries in different locations
 * Try to avoid having ANY local file system location dependencies in your build files. Make sure you NEVER put C: in a build file.  This is just plain bad behavior
 * Separate local file system access points in an external "property file"
 * Warning: property files are read by Java tools and are not always path separator aware. You can use "\\" on java, or, knowing that Ant expands existing properties, ${path.separator}
 * Allow people to check out all the files in a project including the build.xml file, customize their local library paths and build
 * Third party projects such as Ivy and Maven2 Ant tasks try to automate the entire library management process. Consider these on a very large/complex project.


text:
Apache Ant/XML

Ant provides targets to validate and transform XML documents.

XMLwellformed - how to use Apache ant to check an XML file for well formedness

XMLvalidate - how to use Apache ant to validate an XML file against an XML Schema

XSLT - how to use Apache ant to run an XML transform

title: Apache Ant/XML


text:
User:Torax~enwikibooks

Torax
I am Torax. Torax I am. Below is a list of Wikibooks that I am interested in and I think you will find interesting too.

Interesting Wikibooks
School of Magic (Make sure you add your username to the School of Magic:List of Participants if you want to take the courses officially.)

Lucid Dreaming (For dreams beyond your wildest dreams...)

title: User:Torax~enwikibooks


text:
User:Rudeboy124

I’m a wanna-be mathematician and comp. sci enthusiast who’s recently discovered wikipedia. Praise God for wikipedia.

http://www.gegn.net

title: User:Rudeboy124


text:
User:Deli Suarez

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.

title: User:Deli Suarez


text:
Forth/Introduction

 Introduction

 Forth

Forth was first developed in the early 1970s for scientific applications by Charles Moore and Elizabeth Rather.

For computers to be useful, they must be programmed. For beginners this usually means programming in BASIC, but for professional programmers this could include:

 * BASIC (considered too slow for many applications)
 * Pascal
 * Fortran
 * C
 * C++
 * Ada
 * Assembler (machine code – Not user friendly and limited to one CPU)

Forth is easy to start to learn, is compact and has fast execution.

This book is a complete guide to Forth programming. In this book you will be introduced to the language through examples and comparisons with BASIC. Later we will explore some of the more unusual aspects of Forth, some that have no equivalent in other languages.

 * The 1994 ANS Forth standard dialect is adopted for much of this book.

How is Forth Different
There are many statements that at first glance may seem contradictory or confusing when you are talking about Forth. For example, Forth:

 * was traditionally a threaded-interpretive language, but now exists as native code systems as well
 * is commonly written in Forth, but also has been written in Assembler, C and recently Python
 * is extensible,
 * is a high-level language with direct access to primitive operators and hardware
 * is both an interpreter and a compiler (and most systems have an Assembler)
 * is an easy language for absolute beginners to program in,
 * is difficult to master for programmers experienced in other languages,
 * is a two stack machine (data stack, return stack)
 * programmer must use the data stack directly
 * uses Reverse Polish Notation (RPN),
 * commonly uses integer arithmetic,
 * uses spaces to separate program tokens
 * does not start to execute any of your typed input until after you press the enter key,
 * is an interactive language, like BASIC,
 * but can incrementally compile code one line at a time at the console,
 * and allows (even promotes) interactive Assembly language,

Cheat Sheet
You can quickly get a taste of Forth syntax in a LearnXinYminutes tutorial which is written in Forth!

Learn Forth in Five Minutes

This brief tutorial does not teach Forth programming "style" or the mind-set that is used to get the most out of Forth's collection of low level routines.

That is best learned by studying the code of experience Forth programmers and applying the methods to your own programming problem.

title: Forth/Introduction


text:
IB Mathematics (HL)/Vectors

A vector has both a magnitude and a direction. It is represented in terms of 2d or 3d entity.

In 2D representation: r = 7i+8j

In 3D representation: r = 9i+5j+6k

i, j, k denotes magnitude in the direction of x,y and z axis respectively.

title: IB Mathematics (HL)/Vectors


text:
IB Mathematics (HL)/Trigonometry

Topic 3: Circular functions and trigonometry discusses functions based on the circle, how trigonometric functions are derived, and various trigonometric identities and rules.

3.1 The circle and radian measure
Section content: The circle: radian measure of angles; length of an arc; area of a sector

 * 1 radian is defined as the angle that is subtended by an arc of a length equal to the radius of the circle. 180° = π radians.
 * The angle θ radians subtended by an arc of length l in a circle of radius r is defined as: $$\theta = \frac{l}{r}$$.
 * The arc length l can be calculated using: $$l = \theta r$$, where θ is in radians or $$l = \frac{\theta}{360}(2\pi r)$$, where θ is in degrees. These are derived from the formula for the circumference of a circle. (The former equation is included in the formula booklet)
 * The area of a sector is calculated using: $$A = \frac{1}{2}\theta r^2$$, where θ is in radians or $$A = \frac{\theta}{360}(\pi r^2)$$, where θ is in degrees. (The former equation is included in the formula booklet)

3.2 Trigonometry as circular functions
Section content: Definition of cos θ, sin θ and tan θ in terms of the unit circle; exact values of sin, cos and tan of 0, π/6, π/4, π/3, π/2 and their multiples; definition of the reciprocal trigonometric ratios sec θ, csc θ and cot θ; Pythagorean identities

Consider a unit circle (circle of radius 1) on a coordinate plane with centre at origin O(0, 0) and a point P(x, y) on its circumference. The angle θ is the angle made by OP from the positive x-axis and the angle α is the angle made by OP from the x-axis (either positive or negative). OP = 1 unit. Tip: These relationships are not found in the formula booklet. However, you can use ASTC (all, sine, tangent, cosine) to make a phrase to make this relationship with the quadrants easier to remember; for example: "Add sugar to coffee".
 * In the first quadrant (0 < θ < π/2): $$\sin{\theta} = \frac{y}{OP} = y$$, $$\cos{\theta} = \frac{x}{OP} = x$$, and $$\tan{\theta} = \frac{y}{x}$$. All ratios are positive.
 * In the second quadrant (π/2 < θ < π): $$\sin{\theta} = \sin{(\pi - \alpha)} = \frac{y}{1} = y = \sin{\alpha}$$, $$\cos{\theta} = \cos{(\pi - \alpha)} = \frac{-x}{1} = -x = -\cos{\alpha}$$, and $$\tan{\theta} = \tan{(\pi - \alpha)} = \frac{y}{-x} = -\frac{y}{x} = -\tan{\alpha}$$. Only the sine ratio is positive.
 * In the third quadrant (π < θ < 3π/2): $$\sin{\theta} = \sin{(\pi + \alpha)} = \frac{-y}{1} = -y = -\sin{\alpha}$$, $$\cos{\theta} = \cos{(\pi + \alpha)} = \frac{-x}{1} = -x = -\cos{\alpha}$$, and $$\tan{\theta} = \tan{(\pi + \alpha)} = \frac{-y}{-x} = \frac{y}{x} = \tan{\alpha}$$. Only the tangent ratio is positive.
 * In the fourth quadrant (3π/2 < θ < 2π): $$\sin{\theta} = \sin{(2\pi - \alpha)} = \frac{-y}{1} = -y = -\sin{\alpha}$$, $$\cos{\theta} = \cos{(2\pi - \alpha)} = \frac{x}{1} = x = \cos{\alpha}$$, and $$\tan{\theta} = \tan{(2\pi - \alpha)} = \frac{-y}{x} = -\frac{y}{x} = -\tan{\alpha}$$. Only the cosine ratio is positive.

A straight line passing through the origin O(0, 0) has the equation $$y = \tan{\theta}$$.

Exact values for acute angles are summarised in the following table. Paper 1 uses this knowledge and you may also be asked to use this in Paper 2. Tip: The exact values of sine and cosine ratios are the reverse of each other, so you only have to remember one set of values. Also tan θ is just (sin θ)/(cos θ).

The exact values for angles greater than π/2 radians can be derived by using the quadrant relationships above.

The reciprocal trigonometric functions are: The first two of these are included in the formula booklet.
 * $$\sec{\theta} = \frac{1}{\cos{\theta}}$$
 * $$\csc{\theta} = \frac{1}{\sin{\theta}}$$
 * $$\cot{\theta} = \frac{1}{\tan{\theta}}$$

The Pythagorean identities are: All three of these are included in the formula booklet.
 * $$\cos^2 {\theta} + \sin^2 {\theta} = 1$$
 * $$1 + \tan^2 {\theta} = \sec^2 {\theta}$$
 * $$1 + \cot^2 {\theta} = \csc^2 {\theta}$$

3.3 Compound and double angle identities
Section content: Compound angle identities; double angle identities

The compound angle identities expresses trigonometric ratios of the sum of two angles in terms of ratios of separate angles. These are: The proofs of these have been explicitly excluded from the syllabus.
 * $$\sin{(A \pm B)} = \sin{A}\cos{B} \pm \cos{A}\sin{B}$$
 * $$\cos{(A \pm B)} = \cos{A}\cos{B} \mp \sin{A}\sin{B}$$ (note that the signs between the terms are opposite of that in the original function)
 * $$\tan{(A \pm B)} = \frac{\tan{A} \pm \tan{B}}{1 \mp \tan{A}\tan{B}}$$

The double angle identities can be derived from the compound angle identities.

All identities in this section are included in the formula booklet.

3.4 Composite trigonometric functions
Section content: Composite functions of the form f(x) = a sin(b(x + c)) + d Rewind: 2.3 - Transformation of graphs: translations; stretches; reflections in the axes

Consider a composite trigonometric function f where $$f(x) = a\sin{(b(x + c))} + d$$ Tip: The phase shift c is sometimes confused with bc due to the common perception that b = 1. Examination questions may trick you by expanding the expression in the trigonometric function. For example $$f(x) = \sin{(2x + 8)}$$, c is not 8 but instead 8 divided by 2, which is 4.
 * a| is the amplitude, half the vertical distance between the maximum and minimum height. When a is negative the graph is also reflected along y = d.
 * b is the number of cycles of the wave in the normal period (π radians for tangent, 2π radians for sine and cosine). The new period is hence the normal period divided by b i.e. in this case period = 2π/b.
 * c is the phase shift, or horizontal shift (to the left or right). When c is positive the graph moves to the left; when c is negative the graph moves to the right.
 * d is the vertical shift (up or down). When d is positive the graph moves up; when d is negative the graph moves down.

3.5 Inverse trigonometric functions
Section content: The inverse functions arcsin x, arccos x, arctan x; their domains and ranges; their graphs Rewind: 2.1 - Inverse function, including domain restriction; 2.3 - The graph of the inverse function as a reflection in y = x

 * The inverse trigonometric functions (arcsin x, arccos x and arctan x) finds the angle given the trigonometric ratio (sin θ, cos θ and tan θ respectively); it effectively is the reverse of the standard trigonometric functions.
 * From topic 2.1 you would have known that inverse functions will only exist for one-to-one functions and not for many-to-one functions, hence in technicality the inverse of a trigonometric function is not a function. However the domain of f(x) can be restricted so that it becomes a one-to-one function and hence has an inverse.
 * The domain of $$f(x)$$ is the range of $$f^{-1}(x)$$, and the range of $$f(x)$$ is the domain of $$f^{-1}(x)$$. Hence the domain of arcsin x and arccos x is -1 ≤ x ≤ 1, while arctan x has the domain x: x exist for all real numbers.
 * Alternative notation (especially on calculators; will also be used throughout the guide): $$\arcsin{x} = \sin^{-1} {x} ; \arccos{x} = \cos^{-1} {x} ; \arctan{x} = \tan^{-1} {x}$$
 * The graphs of the inverse functions can be obtained by first restricting the domains to -π/2 ≤ x ≤ π/2, then reflecting the graph in y = x.
 * You should know the identity for sum of arctan functions
 * And that arcsin(x) + arccos(x) = π/2

3.6 Solutions to trigonometric equations
Section content: Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorisation Rewind: 2.6 - Solving polynomial equations both graphically and algebraically; use of technology to solve a variety of equations, including those where there is no appropriate analytic approach

Trigonometric equations can be solved through algebraic and graphical means. To solve such an equation algebraically:
 * The key idea is to have all trigonometric ratios reduced to the same basic one: either sine or cosine (or tangent if algebra forces it to sin θ/cos θ). Use the trigonometric identities to help you do so. If this is not possible it may be easier to use the graphical method of solutions.
 * Usually this will reduce to a single linear equation. However it can also be reduced to a polynomial whereby x is replaced with a trigonometric function. In this case solve as you would with a standard polynomial equation e.g. by factorisation, quadratic formula or the binomial theorem. You could also use the graphing function of your GDC to solve the equation when it is too difficult to solve manually.
 * After reducing to the form sin θ = x, check the domain - what do they want your answers to be in between? Use the quadrant relationships in 3.2 to help you solve the equation for that domain. For example, if the domain is π/2 ≤ θ ≤ 3π/2, then your solutions would be $$\theta = \pi - \sin^{-1}{x}, \pi + \sin^{-1}{x}$$.

To solve graphically, plot the left hand side of the equation as one graph and right hand side of the equation as another graph on the same pair of axes. Make sure that your GDC is on the correct angle mode (degrees or radians) - assume radians unless stated by the exam paper. The intersection points will be the solutions to the equation.

3.7 Sine and cosine rules
Section content: The cosine rule; the sine rule including the ambiguous case; area of a triangle as 1/2 ab sin C

The sine and cosine rules relate the sides and angles in any triangle.

Consider a triangle ABC where a = BC, b = AC and c = AB.
 * If you are given the lengths of any two sides a and b and the angle C between those two sides (the included angle), then you can use the cosine rule to calculate the length of the remaining side c: $$c^2 = a^2 + b^2 - 2ab\cos{C}$$. A specific case of this is the Pythagoras' theorem for right-angled triangles, where C = π/2 radians or 90 degrees, then $$c^2 = a^2 + b^2$$.
 * If you are given the lengths of all three sides a, b and c, then you can use the cosine rule to calculate any angle within the triangle; for angle C this is: $$\cos{C} = \frac{a^2 + b^2 - c^2}{2ab}$$.
 * If you are given any two angles A and B and the length a side not in between these two angles - either a or b, then you can use the sine rule to calculate one of the missing sides: $$\frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}}$$.
 * If you are given the lengths of any two sides a and b, and the angle A or B that is not in between the two sides, then this is known as the ambiguous case as the sine rule: $$\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}$$ will result in two possible sizes for one of the missing angles: $$A = \sin^{-1}{\frac{a\sin{B}}{b}}, \pi - \sin^{-1}{\frac{a\sin{B}}{b}}$$. In this case, determine which one of the angles is not possible from the information given in the question.

The area of any triangle, given the lengths of any two sides a and b and the included angle C, is: $$Area = \frac{1}{2}ab\sin{C}$$

These rules are given in the formula booklet.

title: IB Mathematics (HL)/Trigonometry


text:
A-level Mathematics/OCR/C2/Integration/Problems

Example problems
1. Evaluate these indefinite integrals:

(a) $$ \int 2x^5 \, dx $$

(b)$$ \int 7x^6 + 2x^3 - x^2 \, dx $$

2. Find the equation of the curve in each of the following cases:

(a) Given $$ \frac{dy}{dx} = x + 5 $$ and that the point $$(0, 3)$$ lies on the curve.

(b) Given $$ \frac{dy}{dx} = 3x^2 + 7x +0.1 $$ and that the point $$(2, 2)$$ lies on the curve.

3. Evaluate these definite integrals

(a)$$ \int_{0}^{2} x + 1 \,dx $$

(b)$$ \int_{-3}^{4.7} \frac{1}{7}x^{\frac{1}{3}} + 1 \,dx $$

4. Find the area between the curve of $$y=-x^4-\frac{1}{2}x^3+3x^2$$ and the x-axis from $$x=-2$$ to $$x=0$$.

Worked solutions can be found here: Integration problem solutions

Integration/Problems


text:
User talk:Andrew Watt

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.

Deleted
I've deleted material from my talk page that is more than 2 years old.

Re: Ancient History
Great to see you here! I haven't seen a proposal to move the book into becoming part of the broader book on World History, but I have thought about it. Reading the talk pages in Ancient History, I see there were disputes over what constitutes "ancient" history. Moving it into an-all inclusive history book might be better, and would resolve the issue of whether or not peoples like the Aztecs were "ancient". What do you think? Νεοπτόλεμος ( talk | email | contribs ) 23:34, 24 February 2008 (UTC)
 * It isn't deleted, just hidden. I'll make it visable again if you'd like. Νεοπτόλεμος ( talk | email | contribs ) 03:40, 25 February 2008 (UTC)

History of Greece
Just thought I'd let you know I added you to the list of contributors in History of Greece, because the first five chapters of the Greek history book are the exact same pages that exist in Ancient History/Greece, since it would be silly to have two separate pages on the same subject. Νεοπτόλεμος ( talk | email | contribs ) 03:43, 8 March 2008 (UTC)

Saylor.org's Ancient Civilizations of the World
Hey there, I noticed that you've made significant contributions the Ancient History textbook. I am currently working with the Saylor Foundation to create a similar textbook to host and use in our course on ancient civilizations on our website (which we offer free to anyone online). We would love for you to check out Saylor.org's Ancient Civilizations of the World and give us some feedback or make some contributions! --Azin (discuss • contribs) 19:13, 6 June 2012 (UTC)

title: User talk:Andrew Watt


text: Wikibooks:Book of the month/June 2006

title: Wikibooks:Book of the month/June 2006


text:
User:Andrew Watt

About Me
My name is Andrew, and I teach history and English composition at a middle school. I hold two master's degrees, in Christian theology and ancient history. I'm thinking about getting a third in English literature. I'm a fencing coach, a Boy Scout, and a freelance writer.

What I'm Working On
I tried out Wikipedia, to work on the free encyclopedia. I liked it. It's cool, and interesting. Yet I discovered that I didn't have the heart to go in and do lots of editing of existing pages. I'm a history teacher, and I do a lot of editing on student papers, which I find deeply frustrating. However, I discovered this Ancient History wikibook project was nearly empty. The notion that this is something I can work on directly, in collaboration with peers and colleagues and talented amateurs, to bring the book into existence... That is a whole lot more exciting to me than constantly editing and revising, which is what I constantly do to my students' work.

After a long absence, I'm back. I'm still working on Ancient History as my principal project, especially since I think it's particularly important given the OLPC project from MIT and Nicholas Negroponte. Otherwise, this part of the study of history will likely vanish from the scene. At the moment I'm planning on working on the Mesopotamia and Rome sections.

My Goal
My goal is to provide some basic text and stubs right now for this wikibook project, Ancient History, so as to draw attention to the project and get others to contribute. I hope this is acceptable, and I look forward to working with you all.

Also, I want to finish, and get into PDF format, How to write a research paper in History, and maybe talk Fastfission back into collaborating on that project. The book is 75%-80% complete, and I'd like to get it done across the board.

I'd also like to begin, organize, and get into PDF format, How to Study because this and How to write a research paper in History would be good books for me to use in class. The existing copyright versions of such books are usually quite expensive, and having ones that I could photocopy and distribute would be useful. I wouldn't mind doing a High School Guide to English Grammar, either, but I'm going to finish my first three projects, or at least get some more people contributing to them, first.

My Contributions

 * 1) Ancient History
 * 2) Introduction
 * 3) Ancient History/Egypt/Pharaohs
 * 4) Ancient History/Egypt
 * 5) Geography
 * 6) Human Origins
 * 7) Paleolithic Age
 * 8) Wikijunior:Ancient Civilizations/Minoans
 * 9) Wikijunior:Ancient Civilizations/Sumerians
 * 10) How to write a research paper in History
 * 11) Organization
 * 12) Writing the Paper

My Projects

 * How to Study
 * High School Guide to English Grammar
 * How to Study
 * High School Guide to English Grammar

title: User:Andrew Watt


text: Wikibooks:Collaboration of the Month/June 2006


text:
Wikibooks:Collaboration of the Month/July 2006 voting

Voting Rules
The following are the rules and conditions for voting for the Collaboration of the Month:

 * 1) Only registered users with 20 or more edits to their name may vote.
 * 2) You may vote under as many nominations as you wish, however, you may only vote for each book itself once.
 * 3) Sign your vote with a number sign and four tildes, #~, after the most previous vote under the book's nomination. Unsigned votes will be removed.
 * 4) Any nomination which receives 3 or more votes in one month will automatically be renominated the next month if it is not chosen.
 * 5) If you wish to include a short comment for your vote, then you may do so. Make sure that if you leave such a comment, that you are also voting for that page.
 * 6) Do not post objection votes as these will have no effect on the final tally.
 * 7) Voting will end at 00:00 UTC on the first day of each month. In essence, when a new month begins voting ends. Any votes added after this time will be discarded.
 * 8) The nomination with most votes will be chosen as the Collaboration of the Month. In case of a tie, the older Wikibook will be chosen.

Shaving
Votes:

 * 1) Easyas12c 20:07, 14 June 2006 (UTC)
 * 2) Nathan8225 02:04, 23 June 2006 (UTC)
 * 3) Klingoncowboy4 19:33, 24 June 2006 (UTC)
 * 4) Master  mind   007  17:51, 27 June 2006 (UTC)

Comments:
 * The book is a guide to a simple everyday task. This makes the book both usefull and editable to many. It has a clear structure which makes appending easy. The book contains already the most basic information regarding the subject, but has been lacking attention for a while. --Easyas12c 20:07, 14 June 2006 (UTC)

Brazilian History
Votes:

 * 1) Brandon Johnsonn 15:47, 14 June 2006 (UTC)

Comments:
 * This book is off to a good start, but needs more information. #Brandon Johnsonn 15:47, 14 June 2006 (UTC)

Guide to Success & Fulfillment
Votes:

 * 1) Rancor
 * 2) Klingoncowboy4 06:53, 8 June 2006 (UTC)
 * 3) Regre7 19:47, 14 June 2006 (UTC)
 * 4) Master  mind   007  17:51, 27 June 2006 (UTC)

Comments: This book is a nove idea that has the potential to become something of great benefit to everybody, but especially the youth

William Shakespeare's Works
Votes:

 * 1) Brandon Johnsonn 20:04, 1 June 2006 (UTC)
 * 2) Andrew Watt 19:57, 6 June 2006 (UTC)
 * 3) Klingoncowboy4 06:53, 8 June 2006 (UTC)
 * 4) Dilaudid 19:16, 10 June 2006 (UTC)
 * 5) Master  mind   007  17:51, 27 June 2006 (UTC)

Comments: I think this book is very useful for any person who either enjoys the bard or needs a copy of one of his plays and the play's history for a homework assignment. Needs much help on adding plays.Brandon Johnsonn 20:04, 1 June 2006 (UTC)

The plays are complex, but we'd be much more likely to get them annotated as a collaboration than with casual efforts. Andrew Watt 19:57, 6 June 2006 (UTC)

Muggles' Guide to Harry Potter
(renominated with 4 votes for June)

Votes: Comments: This book is lacking a lot of important information, many characters' pages have not been created, and Wikibooks need to get going on this. --Fbv65 e del (discuss)>del 16:58, 4 June 2006 (UTC)
 * 1) Fbv65 e del (discuss) 16:58, 4 June 2006 (UTC)
 * 2) Matt 17:39, 6 June 2006 (UTC)
 * 3) Klingoncowboy4 06:53, 8 June 2006 (UTC)
 * 4) Lag 15:14, 11 June 2006 (UTC)
 * 5) Regre7 18:27, 20 June 2006 (UTC)
 * 6) Whiteknight (talk) (projects) 00:42, 27 June 2006 (UTC)
 * 7) Hedwig0407
 * 8) Dolive35 18:24, 27 June 2006 (UTC)
 * 9) Rob Horning 15:20, 28 June 2006 (UTC)

I agree, I think that we should get a good example of an annotated text/study guide up on the main page, as a template for future books. --Whiteknight (talk) (projects) 00:42, 27 June 2006 (UTC)

I tried to fix some of the pages up etc. a few months ago, this definatelly needs it.Dolive35 18:24, 27 June 2006 (UTC)

Italian
(renominated with 3 votes for June)

Votes:
 * 1)  Master  mind   007  16:43, 4 June 2006 (UTC)
 * 2) Geoking66 23:12, 18 June 2006 (UTC)
 * 3) Klingoncowboy4 19:33, 24 June 2006 (UTC)

Comments: This book is developing a lot. It's making progress and I hope it'll come to the standards of other books like French & German. Master mind  007  16:43, 4 June 2006 (UTC)

Korean
Comments: This book requires much work. There are good books for both Chinese and Japanese, but Korean has been neglected. For English-speakers trying to learn an oriental language, Korean is the most preferale because the Hangul writing system is phonetic, and words are divided up into syllabic blocks.

C Sharp Programming
(renominated with 3 votes for June)

Votes:
 * 1) Sblive 21:57, 1 June 2006 (UTC)
 * 2) Klingoncowboy4 06:53, 8 June 2006 (UTC)
 * sv:User:Per9000 20060622 1518 CET.

Comments:
 * I think this book can rapidly become a valuable source of information, if only we make some effort in it. It has a solid base... but it's still unfinished. Sblive 21:57, 1 June 2006 (UTC)

 * I am a messed up programmer: I use C, C++, C#, Fortran, Python, JavaScript, php, etc. I know what you can program, but all languages have their own special syntax - I'd be really happy to see a structured quickguide to C#. This book has my vote (although I'm not registered in wikibooks I have a english/swedish/common wikipedia useraccount) /sv:User:Per9000 20060622 1518 CET

Special relativity
Votes:
 * 1)  Master  mind   007  16:43, 4 June 2006 (UTC)
 * 2) RobinH 19:42, 4 June 2006 (UTC)
 * 3) Klingoncowboy4 06:53, 8 June 2006 (UTC)
 * 4) Rob Horning 17:17, 18 June 2006 (UTC)
 * 5) hagindaz 01:23, 22 June 2006 (UTC)
 * 6) Whiteknight (talk) (projects) 00:39, 27 June 2006 (UTC)

Comments: I see that it's a nice book related to Physics with good illustrations and diagrams. Nice! Master mind   007  16:43, 4 June 2006 (UTC)

So much to do and so little time... The mathematical approach in particular needs a lot of beefing up and elaboration in this book. The more helpers the better.

Reverse Engineering
Votes:

 * 1) Whiteknight (talk) (projects) 17:03, 12 June 2006 (UTC)
 * 2) --Easyas12c 18:27, 15 June 2006 (UTC)
 * 3) Klingoncowboy4 19:33, 24 June 2006 (UTC)

Comments: This book has alot of potential, and could attract alot of contributors if we put it up on the main page. I think it's in a good condition, even if there are a few stub pages and sections here or there. --Whiteknight (talk) (projects) 17:03, 12 June 2006 (UTC)