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Cookbook:Cuisine of Germany

| European cuisines

German cooking is considered rather heavy. Depending on season and region it uses pork, beef, sausages, game and fish. Vegetables used include especially potatoes, peas and carrots, cauliflower, string beans, pickled cabbage (sauerkraut), kale (Grünkohl), and asparagus. There's a variety of stews. Surely not everything is Sauerkraut. Germany has various regional cuisines and specialities, and many real "haute cuisine" meals.

Bavarian recipes and dishes

 * Kalbshaxe mit Sauerkraut (veal shank with sauerkraut)
 * Semmelknödel (bread dumpling)
 * Leberknödel (liver dumplings)
 * Weißwurst mit süßem Senf (white sausage with sweet mustard)
 * Leberkäse
 * Schweinsbraten mit Kartoffelknödel (pork roast with potato dumpling)
 * Spätzle

Franconian recipes and dishes

 * Zwiebelkuchen (onion tart, served especially in autumn with new wine or "Federweisser" - fermented grape juice that has not yet turned into wine)
 * Fränkischer Sauerbraten (beef stewed in a sauce made of vinegar, gingerbread and spices)
 * Rindfleisch mit Kren (beef with horseradish mash)
 * Blaue Zipfel ("Bratwürste" - typical sausages - cooked in vinegar with onions, carrots and spices)
 * Karpfen (carp - to be eaten only September to April)

Frisian and North German recipes and dishes

 * Grünkohl mit Pinkel (curly kale with smoked sausage)
 * Grünkohl mit Kassler (curly kale with smoked pork chop)
 * Grützwurst ("grits sausage")
 * Kopfwurst ("head sausage" or head cheese)
 * Bregenwurst ("brain sausage")
 * Strammer Max (a slice of bread with butter, ham or Mettwurst and fried eggs)
 * Holsteiner Pfanne ("Holsteinian pan", roasted potatoes with tomatoes)
 * Bauernfrühstück ("farmer's breakfast", roasted potatoes with scrambled eggs)
 * Bohnen, Birnen und Speck (stew of string "beans, pears and bacon" and potatoes)
 * Niedersächsische Hochzeitssuppe ("Lower Saxon wedding soup", usually beef broth with meat balls, royale, asparagus, parsley etc.)
 * Pfannkuchen, Eierkuchen, Plinsen ("pan cake" traditionally eaten with sugar and apple butter or gooseberry compote, it can be prepared sweet or savory)
 * Kartoffelpuffer (fried rasped potatoes)
 * Spargel (aspargus with ham, potatoes and either molten butter or Sauce Hollandaise)
 * Labskaus (corned beef, potatoes, onions, red beets, marinated herring, fried egg and pickle)
 * Rote Grütze (dessert of a variety of red berries with vanilla sauce)
 * Königsberger Klopse (veal meatballs with an anchovies-caper sauce)

 * Beverages
 * East Frisian Tea

Hesse recipes and dishes

 * Ahle Worscht ("Old Sausage", dried sausage)
 * Grüne Soße ("Green Sauce", potatoes + herb/sour cream sauce)

Rhineland recipes and dishes

 * De halve Hahn (rye bap (Röggelchen) with butter, 2 slices of Gouda, 1 small sour cucumber and mustard)
 * Sauerbraten (sour roast - red meat marinated in vinegar for several days)

Swabian recipes and dishes

 * Spätzle (noodles made with a liquid dough)
 * Maultaschen
 * Nockerln
 * "Schäufele" (Pigs shoulder, scapula)
 * Linsen Eintopf (English: Lentil stew) - part of a traditional Swabian meal.

Poultry

 * Weihnachtsgans ("Christmas goose")
 * Broiler (roasted chicken)

Fish and seafood

 * Forelle blau ("trout blue")
 * Forelle Müllerin Art ("trout à la wife of the miller" or "trout meunière")
 * Hummer (Helgoland, North Sea lobster)
 * Neujahrs- or Silvesterkarpfen ("New Year's Eve carp")
 * Heringssalat ("herring salad", pieces of herring and beetroot)
 * Rollmops ("rollmop", pickled herring)
 * Matjes (marinated herring)
 * Räucheraal (smoked eel)

Game
often with Preiselbeeren (mountain/lowbush cranberry)
 * Rehrücken (roast saddles of venison)
 * Wildschweinbraten (wild boar)
 * Wildleber (fried liver with mashed potatoes, onions and apples)

Sausages

 * Wiener Würstchen
 * Frankfurter Würstchen
 * Bratwurst
 * Currywurst
 * Thüringer Rostbratwurst
 * Schinkenwurst
 * Mettwurst
 * Leberwurst
 * Blutwurst

Stews

 * Erbseneintopf ("pea stew" usually with sausages)
 * Linseneintopf
 * Gulasch

Side dishes

 * Salzkartoffeln (potatoes boiled in saltwater)
 * Petersilienkartoffeln (unpeeled potatoes are boiled and then fried in butter with parsley)
 * Pellkartoffeln (potatoes that are peeled after boiling)
 * Kartoffelbrei (mashed potatoes)
 * Kartoffelsalat (potato salad)
 * Leipziger Allerlei (cauliflower, peas, carrots, asparagus)

Bakery

 * Berliner, Pfannkuchen, Krapfen (jam-filled donut glazed or dusted with powdered sugar)
 * Schwarzbrot ("black bread", rye, brown bread)
 * Pumpernickel
 * Vollkornbrot ("whole grain bread", whole meal, whole wheat, bread)
 * Graubrot ("gray bread", rye, brown bread)
 * Brötchen, Semmel ("little bread", dinner rolls)
 * Brezeln (pretzels)
 * Zwieback
 * Windbeutel
 * Butterkuchen
 * Schwarzwälder Kirschtorte ("Black Forest Cake")
 * Bienenstich
 * Donauwellen Cake, also known as Schneewittchen (Snow White) cake
 * Lebkuchen (gingerbread, served at Christmas time)
 * Marzipan
 * Spekulatius (spicy biscuits, available at Christmas)
 * Stollen (a type of fruitcake with many variations)
 * Streuselkuchen
 * Spritzgeback

title: Cookbook:Cuisine of Germany


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Cookbook:Kovaim (Hungarian Cheese Dumplings)

| Cuisine of Hungary | Vegetarian cuisine

Kovaim are Hungarian cheese dumplings. They go well with cheddar cheese and ketchup.

Dough

 * 300 g flour
 * 3 large eggs
 * ½ cup water
 * Salt to taste

Filling

 * Dry ricotta cheese
 * Salt to taste

Procedure

 * 1) Mix the flour, salt, eggs and water into a dough. Knead it until it is very smooth.
 * 2) Cover the dough in a plastic bowl, and let it rest for at least half an hour.
 * 3) Combine the ricotta and salt to make the filling.
 * 4) Roll the dough out into a sheet as thin as possible (about 3 mm thick). Cut into squares, and fill each square with some filling. Fold the dough over the filling diagonally to make a triangular dumpling.
 * 5) Boil the dumplings in water for about 5 minutes.
 * 6) Serve hot.

Notes, tips, and variations

 * You can also fill these dumplings with cooked potatoes and onions, or ricotta cheese with sugar for a sweet dumpling.
 * You could add some spicier cheese to the filling like cheddar, or even Parmesan, but make sure this is not more than a third of the filling.

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


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Cookbook:Paprika Chicken

| Cuisine of Hungary | Chicken

Paprika chicken is a dish of Hungarian origin traditionally made with chicken, paprika, and lard, served over pasta. There is no official recipe. Preparations of this dish often vary between families. In this Americanized version, sour cream is used in place of lard. Vegetables are optional—the traditional Hungarian version contains none.

Ingredients

 * 1 ea. (2-3 pounds) chicken
 * 2 medium onions, chopped
 * 2–4 tablespoons sweet (not hot) paprika
 * 5–7 tablespoons olive oil
 * 1 cup sour cream
 * ¼–½ cup heavy cream
 * Salt
 * Fresh green parsley
 * 6–7 whole black peppercorns
 * 1 green bell pepper (optional)
 * ½ red bell pepper (optional)
 * 1 carrot (optional), thinly sliced
 * 1 onion (optional), thinly sliced

Procedure

 * 1) Cut the chicken into eight pieces.
 * 2) Add the oil to a medium sauté pan. Sauté the onions for about 3–4 minutes or until they turn yellow.
 * 3) Add the chicken, peppercorns, salt, and paprika, and stir it rapidly. It is a lot of paprika, but it is not hot and this is a Paprikash. Add water immediately to cover the meat.
 * 4) Simmer over low heat. If using, add peppers and carrots after 10 minutes of simmering. Stir occasionally until the chicken is tender. If necessary add more water, but very sparingly. Do not boil the chicken—it should be almost fried in the end. This takes about 45–50 minutes. During the last 15 minutes, stop adding water and wait until the liquid is almost boiled off. Keep about a half finger's width of gravy on the bottom of the pan
 * 5) Add the sour cream and heavy cream to the gravy and stir. Add fresh chopped parsley on the top.
 * 6) Serve with dumplings, rice or mashed potatoes and salad.

Notes, tips, and variations

 * It is important not to use hot paprika, as this would make the soup too spicy and of the wrong consistency. In the United States, paprika is typically mild and sweet unless otherwise indicated.
 * If desired, for every carrot used add ⅓ parsnip, thinly sliced. This takes away the slight sweetness of the carrot.
 * If you want to avoid the heavy cream, use more sour cream.

title: Cookbook:Paprika Chicken


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Cookbook:Potato Casserole (Rakott Krumpli)

| Cuisine of Hungary

Rakott krumpli is a popular Hungarian potato casserole.

Ingredients

 * 6 medium potatoes
 * 4 hard boiled eggs
 * Salt
 * 1 kolbász (Hungarian sausage) or 1 cup diced ham (optional)
 * 4 tablespoons olive oil
 * 350 grams sour cream around 40% fat (called tejföl or smetana)

Procedure

 * 1) Cook the potatoes for about 15–20 minutes.
 * 2) Peel the potatoes and the eggs.
 * 3) Slice the potatoes, eggs, and sausage.
 * 4) Put a little olive oil in a baking dish. Arrange alternating layers of potatoes, eggs, and salt or sausage in the dish.
 * 5) Drizzle olive oil over the filling, and top with sour cream.
 * 6) Bake in a preheated oven at 200°C for about 35 minutes.

Notes, tips, and variations

 * Traditionally most Hungarian cooking used lard (preferably home-made) but olive oil works just as well as a substitution.


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Cookbook:Silvash Gombotz (Hungarian Plum Dumplings)

| Desserts | Cuisine of Hungary

Silvash gombotz (Hungarian spelling: Szilvás gombóc) is a traditional dessert recipe for plum dumplings from Hungary. During the freezing cold winters, warm filling desserts like these are very popular.

Ingredients

 * 2 pounds peeled potatoes
 * 1 egg
 * 1 pinch salt
 * 3 tbsp oil
 * Plain flour
 * Fresh plums, plum jam or plum compote

Procedure

 * 1) Cook potatoes until very soft (at least an hour). Peel and mash with a fork.
 * 2) Add the egg to the mixture, along with the little salt and oil. Mix well.
 * 3) Mix in enough flour to make a smooth dough.
 * 4) Knead the potato dough, then roll it out into a thin sheet (about 5 mm thick), and cut into squares.
 * 5) Put half a plum or 1 tsp of plum jam on each square. Wrap the dough around the filling and seal the edges.
 * 6) Cook in boiling water for 10–15 minutes.
 * 7) Serve with bread crumbs fried in butter.

Notes, tips, and variations

 * When filled with apricot, the dumplings are called "barackos gombóc" or "Baratzkosh Gombotz".


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Cookbook:Dutch Oven

A Dutch oven is a heavy, cast-iron or aluminium pot with a tight-fitting lid. Once heated, it retains heat exceptionally well, making Dutch ovens a good choice for slow-cooked stews. Dutch ovens are one of the best ways to cook over an open fire, and can be used to make a wide range of dishes. While camping, it can be useful to line a dutch oven with aluminium foil, thus saving the job of cleaning it afterwards. Some Dutch ovens come with a heavy wire stand, instead of cast-in legs, and this has the advantage of other uses around the camp, but if an oven has legs, there is the possibility that a long drop may break off a leg, or worse, drive it up through the bottom, destroying a valuable treasure.

You can put plain potatoes in a Dutch oven and really enjoy the results. Or maybe try some chicken.

Care and feeding
A Dutch oven (actually developed in many cultures, but many were imported by the Dutch to the Americas as a trade item) must be seasoned before use, by putting grease or oil (animal or vegetable-anything edible) over the entire inside including the lid and heating to cooking temperature for an hour or so, replenishing the oil as needed, then allow to cool. To clean the oven, wipe the food remnants out of the inside, while still warm, and re-oil. Washing should be done with plain water and a brush. Soap or detergent can remove the seasoning. This is true for any iron cooking surface. An alternate method of seasoning (the bottom) is to deep fry in it. Any Dutch oven worth its weight, iron or aluminium, will have a turned up edge on the lid, to keep coals, either wood or charcoal, from dropping into the food, when the lid is removed. An aluminium pot should be carefully heated: overheating may warp the lid! Most dishes will do much better, especially breads, when the pot is preheated before adding the food.

Use
Heat can be introduced in one of two ways, charcoal briquettes or coals form the fire. Each has their own benefits and drawbacks. Charcoal briquettes provide a consistent heat source that can easily be adjusted for temperature. Charcoal is fairly easy to light and quickly reaches the desired temperature. Each briquette contributes 10-20 degrees to your cooking temperature. Doing some simple multiplication will get you to consistent cooking temps easily. For example a 12 inch oven requires 16 bricks on the top and 8 on the bottom to bake at 350 degrees. The drawback is that you have to carry a bag with you every time you want to cook. Hardwood coals from a camp fire will also work, but its is much harder to control temperature. When baking for example you should put 1/3 of your coals on the bottom and 2/3 on the top. In addition, it takes a large amount of coals to keep several ovens going. It may take an hour or two of a roaring fire to get the desired amount of coals. Also placing the ovens too close to the fire may cause uneven cooking or burning on the sides if not careful.

Cooking times may vary with either method depending on wind or elevation. If you begin to smell the cake it may be time to check the oven.

All the coals should be on the bottom for simple boiling or using the bottom for frying. The majority of the coals should be on the bottom for simmering a soup or cooking a stew, only about 20% on top. To roast meats, put equal amounts of coals on top and bottom. Baking breads and casseroles should have most of the coals on top, about 2/3.

title: Cookbook:Dutch Oven


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Electronics/RCL

RLC Series
An RLC series circuit consists of a resistor, inductor, and capacitor connected in series:
 * [[Image:RLC series circuit v1.svg|left|100px]]

By Kirchhoff's voltage law the differential equation for the circuit is:
 * $$L \frac{dI}{dt} + IR + \frac{1}{C} \int I dt = V(t)$$

or
 * $$L \frac{d^2I}{dt^2} + R \frac{dI}{dt} + \frac{I}{C} =\frac{dV}{dt} $$

Leading to:
 * $$s^2 + \frac{R}{L} s + \frac{1}{LC} = 0$$

 * $$s = -\alpha$$ ± $$ \sqrt{\alpha^2 - \beta^2}$$

with
 * $$\alpha = \frac{R}{2L}$$ and $$\beta = \sqrt{\frac{1}{LC}}$$

There are three cases to consider, each giving different circuit behavior, $$ \alpha^2 = \beta^2, \alpha^2 > \beta^2, or \alpha^2 < \beta^2 $$.

 * $$\alpha^2 = \beta^2$$.
 * $$\frac{R}{2L}$$ = $$\sqrt{ \frac{1}{LC}}$$
 * $$R = 2 \sqrt{\frac{L}{C}}$$

Equation above has only one real root
 * s = -α = $$\frac{R}{2L}$$
 * $$I = A e^{(-\frac{R}{2L}) t}$$

 * $$\alpha^2 > \beta^2$$ ,
 * $$\frac{R}{2L}$$ > $$\sqrt{\frac{1}{LC}}$$
 * $$R > 2\sqrt{\frac{L}{C}}$$

Equation above has only two real roots
 * $$s = -\alpha$$ ± $$\sqrt{\alpha^2 - \beta^2}$$
 * $$I = e^(-\alpha + \sqrt{\alpha^2 - \beta^2}) t + e^-(\alpha + \sqrt{\alpha^2 - \beta^2}) t$$
 * $$I = e^(-\alpha) e(\sqrt{\alpha^2 - \beta^2}) t - e^-(\sqrt{\alpha^2 - \beta^2}) t$$

 * $$\alpha^2 < \beta^2$$.
 * $$R < 2\sqrt{\frac{L}{C}}$$

Equation above has only two complex roots
 * $$s = -\alpha$$ + j$$\sqrt{\beta^2 - \alpha^2}$$
 * $$s = -\alpha$$ - j$$\sqrt{\beta^2 - \alpha^2}$$
 * $$I = e^j(-\alpha + \sqrt{\beta^2 - \alpha^2}) t + e^j(-\alpha + \sqrt{\beta^2 -\alpha^2}) t$$

R = 0
If R = 0 then the RLC circuit will reduce to LC series circuit. LC circuit will generate a standing wave when it operates in resonance; At Resonance the conditions rapidly convey in a steady functional method.
 * $$Z_L = Z_C$$
 * $$\omega L = \frac{1}{\omega C}$$
 * $$\omega = \sqrt{\frac{1}{LC}}$$

R = 0 ZL = ZC
If R = 0 and circuit above operates in resonance then the total impedance of the circuit is Z = R and the current is V / R

At Resonance
 * $$Z_L + Z_C = 0 $$ Or $$Z_L = Z_C$$
 * $$\omega L = \frac{1}{\omega C} $$
 * $$\omega = \sqrt{\frac{1}{LC}}$$
 * $$Z = Z_R + Z_L + Z_C = R + 0 = R$$
 * $$I = \frac{V}{R}$$

At Frequency
 * I = 0 . Capacitor opens circuit . I = 0
 * I = 0 Inductor opens circuit . I = 0

Plot the three value of I at three I above we have a graph I - 0 At Resonance frequency $$\omega = \sqrt{\frac{1}{LC}}$$ the value of current is at its maximum $$I = \frac{V}{R}$$. If the value of current is half then circuit has a stable current $$I = \frac{V}{2R}$$does not change with frequency over a Bandwidth of frequencies É1 - É2. When increase current above $$I = \frac{V}{2R}$$ circuit has stable current over a Narrow Bandwidth. When decrease current below $$I = \frac{V}{2R}$$ circuit has stable current over a Wide Bandwidth

Thus the circuit has the capability to select bandwidth that the circuit has a stable current when circuit operates in resonance therefore the circuit can be used as a Resonance Tuned Selected Bandwidth Filter

title: Electronics/RCL


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Electronics/RCL time domain

Figure 1: RCL circuit

When the switch is closed, a voltage step is applied to the RCL circuit. Take the time the switch was closed to be 0s such that the voltage before the switch was closed was 0 volts and the voltage after the switch was closed is a voltage V. This is a step function given by $$V\cdot u(t)$$ where V is the magnitude of the step and $$u(t)=1$$ for $$t\geq0$$ and zero otherwise.

To analyse the circuit response using transient analysis, a differential equation which describes the system is formulated. The voltage around the loop is given by:

$$ Vu(t)=v_c(t)+\frac{di(t)}{dt}L+Ri(t) \mbox{ (1)} $$

where $$v_c(t)$$ is the voltage across the capacitor, $$\frac{di(t)}{dt}L$$ is the voltage across the inductor and $$Ri(t)$$ the voltage across the resistor.

Substituting $$i(t)=\frac{dv_c(t)}{dt}$$ into equation 1:

$$ Vu(t)=v_c(t)+\frac{d^2v_c(t)}{dt^2}LC+R\frac{dv_c(t)}{dt}C $$ $$ \frac{d^2v_c(t)}{dt^2}+\frac{R}{L}\frac{dv_c(t)}{dt}+\frac{1}{LC}v_c(t)=\frac{Vu(t)}{LC} \mbox{ (2)} $$

The voltage $$v_c(t)$$ has two components, a natural response $$v_n(t)$$ and a forced response $$v_f(t)$$ such that:

$$ v_c(t)=v_f(t)+v_n(t)\mbox{ (3)} $$

substituting equation 3 into equation 2.

$$ \bigg[\frac{d^2v_n(t)}{dt^2}+\frac{R}{L}\frac{dv_n(t)}{dt}+\frac{1}{LC}v_n(t)\bigg]+\bigg[\frac{d^2v_f(t)}{dt^2}+\frac{R}{L}\frac{dv_f(t)}{dt}+\frac{1}{LC}v_f(t)\bigg]=0+\frac{Vu(t)}{LC} $$

when $$t>0s$$ then $$u(t)=1$$:

$$ \bigg[\frac{d^2v_n(t)}{dt^2}+\frac{R}{L}\frac{dv_n(t)}{dt}+\frac{1}{LC}v_n(t)\bigg]=0 \mbox{ (4)} $$ $$ \bigg[\frac{d^2v_f(t)}{dt^2}+\frac{R}{L}\frac{dv_f(t)}{dt}+\frac{1}{LC}v_f(t)\bigg]=\frac{V}{LC} \mbox{ (5)} $$

The natural response and forced solution are solved separately.

Solve for $$v_f(t):$$

Since $$\frac{V}{LC}$$ is a polynomial of degree 0, the solution $$v_f(t)$$ must be a constant such that:

$$ v_f(t)=K $$ $$ \frac{dv_f(t)}{dt}=0 $$ $$ \frac{d^2v_f(t)}{dt}=0 $$

Substituting into equation 5:

$$ \frac{1}{LC}K=\frac{V}{LC} $$ $$ K=V $$ $$ v_f=V \mbox{ (6)} $$

Solve for $$v_n(t)$$:

Let: $$ \frac{R}{L}=2\alpha $$ $$ \frac{1}{LC}=\omega_n^2 $$ $$ v_n(t)=Ae^{st} $$

Substituting into equation 4 gives:

$$ \frac{d^2Ae^{st}}{dt^2}+2\alpha\frac{dAe^{st}}{dt}+\omega_n^2Ae^{st}=0 $$ $$ s^2Ae^{st}+2\alpha Ae^{st}+\omega_2^2Ae^{st}=0 $$ $$ s^2+2\alpha s+\omega_n^2=0 $$

$$ s=\frac{-2\alpha\pm\sqrt{4\alpha^2-4\omega_n^2}}{2}=-\alpha\pm\sqrt{\alpha^2-\omega_n^2} $$

Therefore $$v_n(t)$$ has two solutions $$Ae^{s_1t}$$ and $$Ae^{s_2t}$$

where $$s_1$$ and $$s_2$$ are given by:

$$ s_1=-\alpha+\sqrt{\alpha^2-\omega_n^2} $$ $$ s_2=-\alpha-\sqrt{\alpha^2-\omega_n^2} $$

The general solution is then given by:

$$ v_n(t)=A_1e^{s_1t}+A_2e^{s_2t} $$

Depending on the values of the Resistor, inductor or capacitor the solution has three posibilies.

1. If $$\alpha > \omega_n$$ the system is said to be overdamped

2. If $$\alpha = \omega_n$$ the system is said to be critically damped

3. If $$\alpha < \omega_n$$ the system is said to be underdamped

Example
Given the general solution

$$ \alpha=\frac{R}{2L}=1000 $$ $$ \omega_n=\frac{1}{\sqrt{LC}}\approx 4472 $$

$$ s_1=-1000-4359j $$ $$ s_2=-1000+4359j $$

$$ v_n(t)=A_1e^{(-1000-4359j)t}+A_2e^{(-1000+4359j)t} $$

Thus by Euler's formula ($$e^{j\phi}=\cos{\phi}+j\sin{\phi}$$):

$$ v_n(t)=e^{-1000}\big[A_1\big(\cos(-4359t)+j\sin(-4359t)\big)+A_2\big(\cos(4359t)+j\sin(4359t)\big)\big] $$ $$ v_n(t)=e^{-1000t}\big[(A_1+A_2)\cos(4359t)+j(-A_1+A_2)\sin(4359t)\big] $$

Let $$B_1=A_1+A_2$$ and $$B_2=j(-A_1+A_2)$$

$$ v_n(t)=e^{-1000t}\big[B_1\cos(4359t)+B_2\sin(4359t)\big] $$

Solve for $$B_1$$ and $$B_2$$:

From equation \ref{eq:vf}, $$v_f=1$$ for a unit step of magnitude 1V. Therefore substitution of $$v_f$$ and $$v_n(t)$$ into equation \ref{eq:nonhomogeneous} gives:

$$ v_c(t)=1+e^{-1000t}\big[B_1\cos(4359t)+B_2\sin(4359t)\big] $$

for $$t=0$$ the voltage across the capacitor is zero, $$v_c(t)=0$$

$$ 0=1+B_1\cos(0)+B_2\sin(0) $$ $$ B_1=-1\mbox{ (7)} $$

for $$t=0$$, the current in the inductor must be zero, $$i(0)=0$$

$$ i(t)=\frac{dv_c(t)}{dt}C $$ $$ i(0)=100\cdot10^{-9}\big[e^{-1000t}\big(-4359B_1\sin(4359t)+4359B_2\cos(4359t)\big)-1000e^{-1000t}\big(B_1\cos(4359t)+B_2\sin(4359t)\big)\big] $$ $$ 0=100\cdot10^{-9}\big[4359B_2-1000B_1 \big] $$

substituting $$B_1$$ from equation \ref{eq:B1} gives

$$ B_2\approx-0.229 $$

For $$t>0$$, $$v_c(t)$$ is given by:

$$ v_c(t)=1-e^{-1000t}\big[\cos(4359t)+0.229\sin(4359t)\big] $$

$$v_{out}$$ is given by:

$$ v_{out}=V_{in}-v_c(t) $$ $$ v_{out}=Vu(t)-v_c(t) $$

For $$t>0$$, $$v_{out}$$ is given by:

$$ v_{out}=e^{-1000t}\big[\cos(4359t)+0.229\sin(4359t)\big] $$

title: Electronics/RCL time domain


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File:Bioeconomics fig3 2.png

Efficient production

title: File:Bioeconomics fig3 2.png


text:
Talk:Fluid Mechanics

Book structure and level
Engineering fluid mechanics is a subject requiring several semesters of university study to properly understand - is it practical to cover this topic in this context? What level of complexity should this book be? Is it possible to split it into two books, one simple and one advanced? I think it should start where a introductory fluid mechanic course does.

Chapter 1: key concepts, Density, Viscosity, ect Chapter 2: Statics, Hydrostatics, ect Chapter 3: Fluid Dynamics, Bernouli, ect Just to start off... (unknown contributor)

- first, what do you mean by mechanics? Math or Physics? a Fluid behavior book for everyone? or a book with strongMath-basis for engineers and scientists? if the latter, I have worked eagerly 2 years to find an excellent structure for a fluid mechanics book. here is what I reached.

Theory: Applied Fluid-Tech
 * Fluid statics
 * Pressure - pascal's law, equilibrium law
 * Force - Horizontal, Vertical *
 * Fluid Kinematics
 * Scalar approach - Flow and potential functions
 * Vector approach - Closed Subject's R, V & A fields (Eulerian), Open Subject's V & A Fields (Lagrangian)
 * Tensor approach - Stress & strain tensors
 * Flow visualization lines - streamline & streamtube, pathline, streakline, timeline
 * Flow visualization techniques - Reflective techs, Surface techs
 * Fluid kinetics
 * Integral formulation - Reynold's conservation of 1.mass, 2.Force, 3.Moment, 4.Energy and 5.entropy increase
 * differential formulation - conservation of 1.mass (continuity), 2.Force (naiver-stokes), 4.Energy(Bernoulli)
 * Internal-incompressible flow *
 * Internal-compressible flow
 * Free surface flow *
 * External flow
 * in turbo-machines
 * in power/refrigeration cycles

the sections marked with * seems to talk about liquid only. but it's not true. everything is the same for vapors & gases. --Saeed.Veradi (talk) 20:15, 7 May 2010 (UTC)

 * Some questions regarding pressure:
 * What are the topics under Force (under Fluid Statics)? Aint it lowkey covered within pressure already?
 * Also, though pressure comes under fluid statics, but where/how to cover the pressure of fluid in rotating/accelerating container?? Does that come under fluid statics too or does it shift to dynamics? Yashpalgoyal1304 (discuss • contribs) 09:58, 31 March 2021 (UTC)

- anyway, you face two obstacles:
 * 1) Yunus Cengel has written Fluid Mechanics Fundamentals and Applications for engineers. it has knocked out his 5 strong rivals, together with the older books. yes, I've red all these books. no, not every word.
 * 2) Munson
 * 3) Fox
 * 4) Streeter
 * 5) Shames
 * 6) White
 * 7) even if you write for general audience, why do you write fluid mechanics? It's a highly visual subject for which we can's provide a fraction of sufficient pictures.

--Saeed.Veradi (talk) 20:15, 7 May 2010 (UTC)

Chapter 2: Vector Calculus
In this section we have some equation such as: $$Insert formula here$$A+B=c

Headline text
This formula says that the addsition of vector A & B is vector C.

Conservation of Mass
Should the derrivation of these laws be done in both Lagrangian and Eulerian reference frames? ??That comes from Mr.Reynolds theory of transport... which unites Lagran. and Eules stuff into one pakaged deal no ??? -Mechstudent Either way, it yeilds:

$$ \frac{\partial\rho}{\partial t} + \nabla\cdot \left(   \rho \mathbf{V}  \right) = 0 $$ In words we may suggest to use:

Rate of mass accumulation = Rate of mass in - Rate of mass out

new idea
There is an open content fluid mechanics textbook by Dr. Bar-Meir at www.potto.org. It seem that someone should take that book and insert it to here.

chapter order
Fluid Mechanics/Analysis Methods should come before Fluid_Mechanics/Fluid_Statics - as that would make analysis methods' chapter to be like an overview of coming chapters - in which each chapter is generally a more in depth view of an analysis method told there. The chapters (of statics only i guess) will need slight modifications to fit this flow. I can put some drafts if this sounds agreeable. Yashpalgoyal1304 (discuss • contribs) 19:33, 31 March 2021 (UTC)

title: Talk:Fluid Mechanics


text:
Latin/Lesson 8-Imperfect and Future

= Imperfect and Future constructs =

Warning: Beyond the imperfect, this page is not entirely clear. Do not use it beyond the basic imperfect if you are a first time Latin student. See discussion for my thoughts on this.

Imperfect Active Indicative
The imperfect is a construct like:
 * I was seeing.

In Latin it would look like this:
 * Vidēbam.

English has a similar construct called progressive past. Actions seem incomplete, and so the imperfect label. For example, "I was running," "We were sailing," "They were calling." Note that 'to be' is always there. Latin, however, would sometimes use imperfect like simple past; accordingly, "We were sailing" could be translated as "We sailed." Other translations of imperfect can be used to/kept such as "We used to sail/We kept sailing."

Regardless of language, the concept of an imperfect is important. Imperfect is called imperfect for a reason - in Latin, the verb "perficere" means to finish/complete, which is what perfect is from. Thus, imperfect, in the grammatical sense, means not finished - that the action could be or could not be completed. Perfect instead means it has been finished - I saw. You have already seen, and it is now completed. I was seeing implies that the action is not yet completed.

The perfect tense, which we will learn later, is a more immediate reference to the past. The name, imperfect, helps you remember its use: in situations where you can't say when an event started or ended or happened, you must use the imperfect.

In situations where you can know when an event started or ended or happened, use the perfect.

You conjugate the imperfect tense this way: verb + -bā- + personal ending

The endings for imperfect are:

Sg. Pl.
 * 1) -bam
 * 2) -bās
 * 3) -bat
 * 1) -bāmus
 * 2) -bātis
 * 3) -bant

Note that the only thing we add are ba + the personal endings (the same as in the present tense) to the infinitive stem. This gives us the imperfect conjugation.

Note that in third and fourth conjugations, you will have to form it differently. There is *no* rule to explain this, it just is, although there are memorization techniques that can help.

venire is 4th conjugation and is formed like:
 * veniēbam
 * veniēbās
 * veniēbat
 * veniēbāmus
 * veniēbātis
 * veniēbant

For third conjugation -iō stem verbs, the imperfect is like so: capere (to capture or seize)
 * capiēbam
 * capiēbās
 * capiēbat
 * capiēbāmus
 * capiēbātis
 * capiēbant

Note that it is easiest to think of what the endings -ere and ire lack. The imperfect -bā- + the personal ending, which we can call the imperfect conjugation, must be prefixed by -iē-.

A few examples:
 * amābam - I was loving (A-conjugation--1st)
 * monēbātis - You were warning [object/personage] (of something negative) (Pl.) (2nd Conjugation)
 * vincēbāmus - 'We were defeating (3rd conjugation)
 * capiēbant - They were catching (short I-conjugation--3rd conjugation)
 * pellēbat - She/he/it was propelling (drive something (not a vehicle), propel something) (consonantic conjugation)

(Wiki-reading tips: See discussion. Some of the above may be unclear, however the clarifying '--' and '/' indicate verification. We may not know what the original author intended, but we know what conjugations the examples are.)

Future I, Active
Future active is a tense which, unsurprisingly, refers to something which has not yet happened. The endings are fairly basic, and follow fairly regular rules - however, the future endings used in 1st and 2nd conjugation differ from the endings of 3rd, 3rd-iō, and 4th.

For example - "amō, amāre" (1st conjugation) would be
 * Amābō - I will love
 * Amābis - You will love
 * Amābit - He/She/It will love
 * Amaābimus - We will love
 * Amābitis - Y'all will love
 * Amābunt - They will love

NOTE: 1st person singular and 3rd person plural use -bō and -bunt, not -bi-.

NOTE: the B and the BIs - the distinguishing feature of future tense in Latin.

With "veniō, venīre" (4th conjugation), however, the endings are different. In future, this is what they look like:

 * Veniam - I will come
 * Veniēs - You will come
 * Veniet - He/She/It will come
 * Veniēmus - We will come
 * Veniētis - Y'all will come
 * Venient - They will come

[deleted paragraphs go here. deleted to maintain rigorous accuracy, which we will go back to striving for.)

To clarify: venīre, veniō.. we know it is 4th conjugation verb and if we look at its first person singular conjugation, we see that it is an -iō verb, because the conjugation of the first person singular is "venio". (an io category exists within 3rd and fourth conjugations and is a more general concept which we will briefly introduce here by using venire, venio as an example).

Let's first identify what we know.

We know it is 4th conjugation -io because it ends in īre, which tells us that it is 4th conjugation, and io because its first person singular ends in io (venio). Because it is -iō, we leave the -i- in. So, when we are asked (as all textbooks should phrase these new questions):

What are the steps to form the future 2nd person conjugation?

We say:

 * 1) It is better to know more than you need: check the infinitive nominative singular, we now know that it is 4th conjugation io.
 * 2) We now know that we can form the stem: the stem is veni and can then add a personal ending--leaving in the i. We leave in the i because it is io. Because it looks weird, we never leave the i in the future perfect.

What is the form for venīre, in the future tense, in the 2nd person?

The answer is veniēs.

Future conjugation
Example: I will love:
 * amābō

The table at the end of this page tries to summarize the future tense, with both sets of personal endings. As the warning notes, this summary may confu panda

As an aid to your understanding, this table only applies to the future tense. Do not assume the table is displaying a pattern that is somehow applicable to all of Latin.

(Wiki-reading-tip: This is why they are in the future section, and were not discussed before.)

The A- and the E- conjugation are (relatively) straight-forward. The others are more advanced, and as the warning notes, could confuse a first-time student. Commercial textbooks probably explain it better at this point, although laying their explanation in a table like the one below is well-advised. Leave items marked with a ? in until issues are resolved.

Take a look at the following table:

The vocabulary mostly consists of verbs, and can easily be looked up in a dictionary. We will give a limited translation below, and the rest, for those who are particularly adept at language learning, can be learned through immersion.

 * capere (3rd conjugation--short ere): to seize, metaphorically or literally [see dictionary for full explanation]
 * amare (first conjugation -are): to love

title: Latin/Lesson 8-Imperfect and Future


text:
C Programming/Operators and type casting

Operators and Assignments
C has a wide range of operators that make simple math easy to handle. The list of operators grouped into precedence levels is as follows:

Primary expressions
Identifiers are names of things in C, and consist of either a letter or an underscore ( _ ) optionally followed by letters, digits, or underscores. An identifier (or variable name) is a primary expression, provided that it has been declared as designating an object (in which case it is an lvalue [a value that can be used as the left side of an assignment expression]) or a function (in which case it is a function designator).

A constant is a primary expression. Its type depends on its form and value. The types of constants are character constants (e.g.  is a space), integer constants (e.g.  ), floating-point constants (e.g.  ), and enumerated constants that have been previously defined via.

A string literal is a primary expression. It consists of a string of characters within double quotes ( " ).

A parenthesized expression is a primary expression. It consists of an expression within parentheses ( ( ) ). Its type and value are those of the non-parenthesized expression within the parentheses.

In C11, an expression that starts with _Generic followed by an initial expression, a list of values of the form type: expression where type is either a named type or the keyword default, and constitutes a primary expression. The value is the expression that follows the type of the initial expression or the default if not found.

Postfix operators
First, a primary expression is also a postfix expression. The following expressions are also postfix expressions:

A postfix expression followed by a left square bracket, an expression, and a right square bracket in sequence constitutes an invocation of the array subscript operator. One of the expressions shall have type "pointer to object type" and the other shall have an integer type; the result type is type. Successive array subscript operators designate an element of a multidimensional array.

A postfix expression followed by parentheses or an optional parenthesized argument list indicates an invocation of the function call operator. The value of the function call operator is the return value of the function called with the provided arguments. The parameters to the function are copied on the stack by value (or at least the compiler acts as if that is what happens; if the programmer wanted the parameter to be copied by reference, then it is easier to pass the address of the area to be modified by value, then the called function can access the area through the respective pointer). The trend for compilers is to pass the parameters from right to left onto the stack, but this is not universal.

A postfix expression followed by a dot followed by an identifier selects a member from a structure or union; a postfix expression followed by an arrow  followed by an identifier selects a member from a structure or union who is pointed to by the pointer on the left-hand side of the expression.

A postfix expression followed by the increment or decrement operators ( or   respectively) indicates that the variable is to be incremented or decremented as a side effect. The value of the expression is the value of the postfix expression before the increment or decrement. These operators only work on integers and pointers.

Unary expressions
First, a postfix expression is a unary expression. The following expressions are all unary expressions:

The increment or decrement operators followed by a unary expression is a unary expression. The value of the expression is the value of the unary expression after the increment or decrement. These operators only work on integers and pointers.

The following operators followed by a cast expression are unary expressions:

Operator    Meaning ========    =======    &         Address-of; value is the location of the operand *        Contents-of; value is what is stored at the location -        Negation +        Value-of operator !        Logical negation ( (!E) is equivalent to (0==E) ) ~        Bit-wise complement

The keyword  followed by a unary expression is a unary expression. The value is the size of the type of the expression in bytes. The expression is not evaluated.

The keyword  followed by a parenthesized type name is a unary expression. The value is the size of the type in bytes.

Cast operators
A unary expression is also a cast expression.

A parenthesized type name followed by any expression, including literals, is a cast expression. The parenthesized type name has the effect of forcing the cast expression into the type specified by the type name in parentheses. For arithmetic types, this either does not change the value of the expression, or truncates the value of the expression if the expression is an integer and the new type is smaller than the previous type.

An example of casting an int as a float:

Multiplicative and additive operators
First, a multiplicative expression is also a cast expression, and an additive expression is also a multiplicative expression. This follows the precedence that multiplication happens before addition.

In C, simple math is very easy to handle. The following operators exist: + (addition), - (subtraction), * (multiplication), / (division), and % (modulus); You likely know all of them from your math classes - except, perhaps, modulus. It returns the remainder of a division (e.g. 5 % 2 = 1). (Modulus is not defined for floating-point numbers, but the math.h library has an fmod function.)

Care must be taken with the modulus, because it's not the equivalent of the mathematical modulus: (-5) % 2 is not 1, but -1. Division of integers will return an integer, and the division of a negative integer by a positive integer will round towards zero instead of rounding down (e.g. (-5) / 3 = -1 instead of -2). However, it is always true that for all integer a and nonzero integer b, ((a / b) * b) + (a % b) == a.

There is no inline operator to do exponentiation (e.g. 5 ^ 2 is not 25 [it is 7; ^ is the exclusive-or operator], and 5 ** 2 is an error), but there is a power function.

The mathematical order of operations does apply. For example (2 + 3) * 2 = 10 while 2 + 3 * 2 = 8. Multiplicative operators have precedence over additive operators.

will display the following:

i: 0   j: 1 i: 1   j: 2 i: 2   j: 3 i: 3   j: 4 i: 4   j: 5 At the end they have both equal values: i: 5   j: 5

The shift operators (which may be used to rotate bits)
A shift expression is also an additive expression (meaning that the shift operators have a precedence just below addition and subtraction).

Shift functions are often used in low-level I/O hardware interfacing. Shift and rotate functions are heavily used in cryptography and software floating point emulation. Other than that, shifts can be used in place of division or multiplication by a power of two. Many processors have dedicated function blocks to make these operations fast -- see Microprocessor Design/Shift and Rotate Blocks. On processors which have such blocks, most C compilers compile shift and rotate operators to a single assembly-language instruction -- see X86 Assembly/Shift and Rotate.

shift left
The  operator shifts the binary representation to the left, dropping the most significant bits and appending it with zero bits. The result is equivalent to multiplying the integer by a power of two.

unsigned shift right
The unsigned shift right operator, also sometimes called the logical right shift operator. It shifts the binary representation to the right, dropping the least significant bits and prepending it with zeros. The  operator is equivalent to division by a power of two for unsigned integers.

signed shift right
The signed shift right operator, also sometimes called the arithmetic right shift operator. It shifts the binary representation to the right, dropping the least significant bit, but prepending it with copies of the original sign bit. The  operator is not equivalent to division for signed integers.

In C, the behavior of the  operator depends on the data type it acts on. Therefore, a signed and an unsigned right shift looks exactly the same, but produces a different result in some cases.

rotate right
Contrary to popular belief, it is possible to write C code that compiles down to the "rotate" assembly language instruction (on CPUs that have such an instruction).

Most compilers recognize this idiom:

and compile it to a single 32 bit rotate instruction.

On some systems, this may be "#define"ed as a macro or defined as an inline function called something like "rightrotate32" or "rotr32" or "ror32" in a standard header file like "bitops.h".

rotate left
Most compilers recognize this idiom:

and compile it to a single 32 bit rotate instruction.

On some systems, this may be "#define"ed as a macro or defined as an inline function called something like "leftrotate32" or "rotl32" in a header file like "bitops.h".

Relational and equality operators
A relational expression is also a shift expression; an equality expression is also a relational expression.

The relational binary operators  (less than),   (greater than),   (less than or equal), and   (greater than or equal) operators return a value of 1 if the result of the operation is true, 0 if false. The result of these operators is type.

The equality binary operators  (equals) and   (not equals) operators are similar to the relational operators except that their precedence is lower. They also return a value of 1 if the result of the operation is true and 0 if it is false.

One thing with floating-point numbers and equality operators: Because floating-point operations can produce approximations (e.g. 0.1 is a repeating decimal in binary, so 0.1 * 10.0 is hardly ever 1.0), it is unwise to use the  operator with floating-point numbers. Instead, if a and b are the numbers to compare, compare  to a fudge factor.

Bitwise operators
The bitwise operators are  (and),   (exclusive or) and   (inclusive or). The  operator has higher precedence than , which has higher precedence than.

The values being operated upon must be integral; the result is integral.

One use for the bitwise operators is to emulate bit flags. These flags can be set with OR, tested with AND, flipped with XOR, and cleared with AND NOT. For example:

Logical operators
The logical operators are  (and), and   (or). Both of these operators produce 1 if the relationship is true and 0 for false. Both of these operators short-circuit; if the result of the expression can be determined from the first operand, the second is ignored. The  operator has higher precedence than the   operator.

is used to evaluate expressions left to right, and returns a 1 if both statements are true, 0 if either of them are false. If the first expression is false, the second is not evaluated. Here, the  operator checks the left-most expression, then the expression to its right. If there were more than two expressions chained (e.g. ), the operator would check x first, then y (if x is nonzero), then continue rightwards to z if neither x or y is zero. Since both statements return true, the  operator returns true, and the code block is executed.

The && operator checks in the same way as before, and finds that the first expression is false. The && operator stops evaluating as soon as it finds a statement to be false, and returns a false.

is used to evaluate expressions left to right, and returns a 1 if either of the expressions are true, 0 if both are false. If the first expression is true, the second expression is not evaluated. The  operator here checks the left-most expression, finds it false, but continues to evaluate the next expression. It finds that the next expression returns true, stops, and returns a 1. Much how the  operator ceases when it finds an expression that returns false, the   operator ceases when it finds an expression that returns true.

It is worth noting that C does not have Boolean values (true and false) commonly found in other languages. It instead interprets a 0 as false, and any nonzero value as true.

Conditional operators
The ternary  operator is the conditional operator. The expression  has the value of   if   is nonzero,   otherwise.

Example: The expression  evaluates to 0. The ternary operator then looks for the "if-false" value, which in this case, is 6. It returns that, so  is equal to six. Had  been a non-zero, then the expression would have returned a 10.

Assignment operators
The assignment operators are,  ,  ,  ,  ,  ,  ,  ,  ,  , and. The  operator stores the value of the right operand into the location determined by the left operand, which must be an lvalue (a value that has an address, and therefore can be assigned to).

For the others,  is shorthand for. Hence, the following expressions are the same: 1. x += y    -     x = x+y 2. x -= y    -     x = x-y 3. x *= y    -     x = x*y 4. x /= y    -     x = x/y 5. x %= y    -     x = x%y

The value of the assignment expression is the value of the left operand after the assignment. Thus, assignments can be chained; e.g. the expression  would assign the value zero to all three variables.

Comma operator
The operator with the least precedence is the comma operator. The value of the expression  will evaluate both   and , but provides the value of.

This operator is useful for including multiple actions in one statement (e.g. within a for loop conditional).

Here is a small example of the comma operator:


text:
User:Alan Nicoll~enwikibooks

Hi, thanks for your interest.

My interest in Wikibooks is generally twofold: I'd like to contribute by editing a Shakespeare play by adding glosses; and I'd like to publish my thinking about education. Later I may want to do other things, but these are my interests here at the moment.

See my thoughts about glosses at Talk:Study_Guide:Shakespeare.

I've started editing Shakespeare's Sonnets.

If you'd like to know more about me, visit my web page: Al's Web Page

Thanks for stopping by.

Alan

title: User:Alan Nicoll~enwikibooks


text:
User:Seth Ilys~enwikibooks

See my Wikipedia user page.

title: User:Seth Ilys~enwikibooks


text:
User talk:Reddi~enwikibooks

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 Reddi. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Reddi~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 23:40, 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) 05:25, 19 April 2015 (UTC)

title: User talk:Reddi~enwikibooks


text:
Authoring Webpages/Introduction

= Introduction =

The Internet
The World Wide Web is only one of many Internet services. The Internet is a network of computers. ("Net" is short for "network".)

The Internet provides the means to pass packets of data around the globe. In order to make sense of these packets, you need services: protocols (sets of rules) that tell you what a certain piece of data means, and computer programs (software) that allow access to data based on such protocols.

The web is one such service, as are e-mail, FTP, torrents, Instant Messaging, etc. All these services require particular programs that understand their protocols, although some programs can work with multiple services. For instance, a web browser may also be capable of downloading files from an FTP site or managing e-mail.

The World Wide Web
The web was created to make finding and exchanging documents across the Internet easier. Before the web, if you had a document to offer through the Internet (for instance, a scientific paper, or a list of jokes, or a recipe, or your curriculum vitae), you had to put it up on an FTP site and then pass the address of that FTP site to someone else.

With the web, you embedded the address into the page, instead of passing it around. More than that, you could hide the real address from view by wrapping it in descriptive text. Instead of typing in a bunch of numbers in your FTP client, and then moving from folder to folder until you stumbled on a file called ppr_john.ps, you could create a 'link' called "John's paper on astroturf" that would take you directly to the desired file.

Hypertext
HTML stands for "HyperText Markup Language". But what is "hypertext"?

Webpages are documents. They are files on computers. When displayed in a browser, they are displayed as "hypertext". The definition of files and computers is outside the scope of this textbook, but hypertext needs to be explained before we can continue.

On a computer, a text file behaves like print: like a book or a magazine. You can scroll through the file instead of having to turn pages, but the text behaves the same way it would if it were printed. It simply sits on the page and you read it. Hypertext, on the other hand, has additional functionality; mainly, the use of "hyperlinks". ("Hyperlinks" are commonly referred to simply as "links".) A link is a part of the page that, when clicked on by a mouse (or tapped with a finger on a touchscreen), takes the reader to a different part of the document, or to an entirely different document. Typically, links are clickable parts of text, but they can also be images. For instance, clicking a chapter heading in the table of contents could take you to that chapter; a linked phrase could take you to a footnote or a reference document for further information.

Probably the first description of "hypertext" appeared in 1945, when Vannevar Bush wrote an article in The Atlantic Monthly called "As We May Think," about a futuristic device he called a "Memex". He described the device as electronically linked to a library and able to display books and films from the library. The Memex also gave the reader the ability to automatically follow references to the work referenced.

The Memex did more than offer linked information to a user, though. It was a tool for establishing links as well as following them.

HTML was designed by Tim Berners-Lee with similar goals in mind: to provide a way for scientists to create a huge library of interlinked works and to provide a way for the users of this library to alter certain documents: for instance, to add annotations or links.

The latter part of Berners-Lee's dream never really materialized until the invention of the first wiki, Wiki-Wiki Web. Wiki pages are special types of web pages that allow the reader to edit them. For example, this textbook is part of a wiki. Anybody can change the contents of this textbook on its website.

Since hypertext is so different from normal text, there are certain things that need to be considered when writing it.

Where hypertext links to another document, the author needs to make clear what it links to. (The dreaded "click here" is, without a doubt, the worst way to create a hyperlink, as it tells the user absolutely nothing about the linked document.)

Where a hypertext document is part of a larger whole (say, a website), it is often helpful to the users if they can find out which part of a larger whole it is. The document should link to a home page. The home page is a web page that forms the "front", or table of contents, of a website. The home page usually contains information about the website and has menus of links that allow the user to navigate to various parts of the website.

For instance, a home page might say: "This is the personal home page of Clarence Wiley". This suggests to the visitor that the web pages found there are probably of a personal nature, most likely created by Clarence, and of some value to Clarence.

Similarly, web pages that are part of a website often use a uniform style. Since arriving at a uniform writing style is not always easy or convenient (think of a web page with dozens of authors), other hints may be employed that tell visitors where they are. One of these hints is explained in the following chapter.

Dangerous metaphors
Calling web documents "pages" is a metaphor. They aren't really "pages", as in a book. However, it is useful to explain characteristics of the new and unknown by comparing them with similar characteristics of the old and known. As such, metaphors can be a useful and powerful device. However, the new and unknown has traditionally suffered harshly from a heavy stamping of the metaphor on the subject.

It is said by some that when television was invented, it took fifty years for the new medium to escape from being a stage in a box. Far into the twentieth century, television was made by aiming a camera at a stage (a 'set' in television and cinema terms) and just passing on to the viewer what went on.

Oddly enough, the things that are possible on television, but not on stage, were already possible when the first consumer grade television sets were being produced: broadcasting with a time-delay (record now, broadcast later), editing a program, using unusual viewpoints, animation, overlaying images et cetera. Of course there were a few 'revolutionaries' who used these techniques, but in general the metaphor (television is like a stage) held the new medium captive.

Even today, news anchors can sit behind their desk clad in nothing but underwear below the waist, safe in the knowledge that the camera will never do anything the audience of a stage play wouldn't do.

Today, the web has been struck similarly hard by failing metaphors. Since the web is clearly tied to computers, a lot of people confuse authoring web pages with programming. Since a lot of web content was written from the start by people using graphical web browsers, designing a web page is often primarily considered a graphical design task. (Contrary to popular belief, the first web browser displayed images and used an early form of stylesheets. )

False metaphor #1: Programming
Programming is the art of creating a computer program. A computer program is something that tells the computer what to do. Usually, a computer program is a list of instructions. For instance, a computer programmer can write a program that tells the computer to open a window on a screen, and display a large, bold text in the top-left corner.

A hypertext document can be implemented as a computer program. A modern day example of programming hypertext would be PDF, the Adobe originated format for distributing print documents across computer networks and to printers. However, HTML, the hypertext language for the web, is not a programming language. Instead, it is a Markup language: it allows one to "mark up" the structure of a document. (You may want to revisit this section later, once you have "marked up" a few web pages yourself.) HTML is a way to tell a web browser what the different parts of a document are. For example, one part of a web page could be a paragraph and another part could be a list.

Viewing HTML as a programming language means that you view its constructs, its labels, its mark-up, as instructions to the browser. For instance, you may want to indicate that a particular piece of text should be printed in large and bold letters. You could use the HTML code for headings for this, because most graphical browsers will display a heading as large, bold text. However, you may get visitors with browsers who don't display a heading as large bold text. That's the moment when the trap of the false metaphor closes around you.

The important thing to remember is that HTML tells a web browser what the different parts of a page are, not how they should look.

False metaphor #2: web authoring as graphical design
To view the web as a graphical medium is a much more insidious problem, because it is largely a correct view. Most web pages are browsed through a graphical browser. On such occasions, the graphical design of a web page can have a decisive influence on how well the content of that page is perceived and received by the visitor.

The problem lies not so much in seeing the web as a graphical medium, but in our assumptions on what a "graphical medium" is. Since web pages are often displayed on a computer screen, web page authors often design a layout grid with certain dimensions. Not everybody may be using the same dimensions, though, and visitors are hardly ever willing to change the dimensions of their windowing system to accommodate the wishes of a website's author.

The web can be displayed on a vast array of devices, some of which are not graphical at all: think of speaking browsers or touch browsers (Braille rules for the blind). You need to adjust your assumptions on what a graphical medium means to write good web pages.

The practice of creating web pages that can be accessed by a wide variety of types of browsers is called "accessibility". As the web grows, accessibility becomes more and more important.

Other, less damaging metaphors
It is clear that when we wish to see the web in the light of another, better-understood invention, we need to do this with care, and clearly delineating the limits of our comparisons.

I would like to propose a few metaphors that are just as useful as the ones before, but that have less potential for damage.

The media that are perhaps the most natural candidates for comparing the web to are unsurprisingly other Internet services. They all share with each other that they provide a way for people to talk directly to other people without intervention of middlemen such as editors and publishers. This possibility for direct contact stems from the underlying low-level protocols of the Internet. On the Internet, every computer can talk to any other computer.

Other systems for sharing information in the free-form manner the web allows, are abundant in real life. Just try to imagine all the possibilities you have when you want to announce a neighborhood party to strangers, or when you want to share your daily troubles with relative strangers: letter pages, bulletin boards, pen pal magazines, etc. spring to mind.

Conclusion
The important idea to take away from this chapter is that the web is a way of sharing information. There is nothing wrong with running programs on the web, or with presenting graphical design on it. These are well-understood and accepted uses of the web. What you should make a distinction between, though, is the web as a way to present information and the form and shape that information takes.

Your need to share --> Your way of sharing it --> An audience.

Questions and Exercises
1. Collect examples of typical web pages. How do they fit into the web? Which role do they play?

2. Find a web page. Make a list of all of the page's possible users, and write down how they would experience the page. Would they find the information they were looking for? Which information would they not be able to find? Would they have an easy way of finding more information through hyperlinks?

If you follow this course as part of a class, let the teacher pick a web page and see how many different audiences the students can come up with.

If you follow this course by yourself, you could go to http://www.google.com and enter a random word in its search box, then activate the "I'm Feeling Lucky" link. I find that the names of kitchen things often make great 'random' words ('ladle', 'cinnamon', 'fridge', 'stove', etc).

3. Try to view a web page using a non-graphical browser, or a browser, such as Opera, with all graphics capabilities switched off. If this exercise is done as part of a class, form duos: let one student face away, let the other read out what's going on in the web page. Let the student who cannot read the web page give instructions and see how hard or easy it is to use the website.

4. Try to think of a subject you would like to create a web page or a website about. Go to a search engine and try to find web pages about this or similar subjects. For instance, if you would like to create a website for your football club, try to find the websites of other local and internationally famous clubs. According to your personal taste and opinions, what have the authors of these sites done right? What have they done wrong?

Answers
For answers, see Answers to Questions and Exercises.

Previous: Requirements - Up: Table of Contents - Next: Creating a simple page

title: Authoring Webpages/Introduction


text:
Wikibooks:WikiProject Mathematics

Scope
This WikiProject aims to organize books in the area of mathematics.

 * Provide community support for writing individual mathematics texts.
 * For example links to tools and resources, links to templates, information on producing mathematical images using gnuplot and other software and mathematical formulas using wikified TeX

 * Discuss our approach for mathematics books in general on wikibooks.
 * For example coming to decisions on merging and splitting books, consistent approach to appendices, examples, exercises. Monographs versus textbooks.

To join this project, add your name to the list of participants and the line onto your user page.

Soon to be Active in 2017-2018
User:Merbst MediaWiki extension developer & Mathematics hobbyist with a reader-oriented Glossarizing extension under development. User:Tsirel Retired professor of mathematics.

Active in 2011

 * 1) Adrignola (discuss • contribs) - High school: Trigonometry, Geometry, Calculus, Probability and Statistics - (transcriptions)
 * 2) Wisden (discuss • contribs) - Trigonometry, A-level Mathematics/Edexcel/Core 1
 * 3) JamesCrook (discuss • contribs) - Trigonometry, Modular Arithmetic
 * 4) Nfgdayton (discuss • contribs) - Algebra
 * 5) Whoop whoop pull up (discuss • contribs) -

No activity seen in 2011

 * 1) Thenub314 (discuss • contribs) - Trigonometry, Ordinary Differential Equations
 * 2) Adam Majewski
 * 3) WeijiBaikeBianji
 * 4) Freiberg
 * 5) CyborgTosser
 * 6) Dan Gardner
 * 7) Perl

Mathematical Picture of the Month
We use the traditional for this page definition of a month as one twelfth of the orbital period, where the planet in question is unspecified.

Picture is from Pictures of Julia and Mandelbrot Sets. The mathematically interesting point about the picture is the smooth colouring, based on distance to the set, rather than the more usual banding of colour based on number of iterations before leaving the unit circle. The distance is converted to a height, which then makes a 3D surface that can be lit.

Selected as Mathematical picture of the month from Jan 16th 2011.

Things to do
Want to help with an existing mathematics text?

There are mathematics books at every level from preschool to Graduate Level and there is room for anyone to help, no matter your mathematics background. Here are some ideas how you can get involved:
 * You can expand on stubs. Mathematics pages will appear in Mathematics stubs, or in Stubs
 * Look at Recent Changes to see which mathematics books are being actively worked on and who by, and get involved.
 * WikiProject Languages templated the Language pages with, perhaps we should consider making a similar template and implementing it.

Book Merging and Splitting
Current thinking on how content should be split between books:
 * Work has been done to combine two Algebra books that were at the same level and covered similar material.
 * High School Trigonometry is a wikibooks compatible licensed version of a Californian textbook. It is not being merged with Trigonometry whose origins pre-date it.
 * One of the reasons is that the pedagogical approach is different. Trigonometry explains how to remember formulas, check them, relate them and think about them - making memorisation easier.  Similarly through puzzle-like examples it shows how proofs can break down, explains how someone could arrive at a proof and gives alternative proofs, rather than present proofs as things to be rote learned.  It also has pedagogical innovations such as links to video.

 * Trigonometry is a three volume book, currently all in one name space. It is being split for different levels.  In particular algebra is done step by step in book one.
 * Modular Arithmetic is a subset of number theory. The book is intended as a monograph, and is unlikely to be merged into a larger number theory book any time soon.

Templates
Some templates which authors might like to re use:

Examples and Exercises
An example that is worked through step by step.

An exercise where the student is asked to do something like check a result or answer a question.

Info Boxes
For use of the pigeonhole principle:

Table of Contents Markers

 * Some maths-for-computing topic with a special icon beside it

Summary Facts
In any triangle the angles always sum to $$180^{\circ}\,$$

Typesetting
There are several guides to help provide information of how to typeset mathematics, they are the following: Keep in mind that there is no correct choice of how to typeset mathematics on a Wikibook. The goal should be consistency both within an individual module, and within the modules of a given book. It is a good rule of thumb to take a look at how typesetting is done in a particular book when you start to edit it and to try to emulate that style. This should hopefully give the readers a consistent presentation.
 * Help:Formulas

Consider using \displaystyle on all formulas, even when in line in text. That way the shape and appearance of symbols will be consistent across the html and the set off text. Without this there can be confusions when the same symbol looks different in the two forms.

Some LaTeX to Plunder

 * $$\displaystyle f:\mathbb{R}^2\to\mathbb{C} \text{ defined as  } f:(x,y)\mapsto x+iy$$

 * $$(x+y)^n = \sum_{k=0}^n {n \choose k}x^{n-k}y^k.$$

Images
The best way to upload files is to upload them to Commons, which transparently makes the files available in all of the Wikimedia projects.

You can also browse that repository by category and by search term, and there are many useful images already there. It is also possible to go to a related wikipedia page for a topic, follow the images to wikicommons, then look for other images in the same category or by the same uploader.

Vector images
There is a strong preference on Wikimedia for SVG graphics files. Programs that can output graphics in svg format :
 * Gnuplot (A general mathematical plotting program)
 * Thenub314 can give information on producing animated graphs.

 * Inkscape (An svg image editor)
 * The current stable version of inkscape has problems reading svg graphs produced by Gnuplot. A good width for lines in geometric figures is 1.7px.  A good font to use is Lucida Bright.  Greek letters like 'alpha' can be inserted by cutting and pasting from Word.  Fonts should be converted to paths in case not present on user's machine, even though it makes i8ln harder.  Check svg images in firefox before uploading.  If you see unwanted black regions it is because of 'rotated text' tags that are Inkscape specific.  If you've converted the text to paths the rotated text tags can be removed (use the XML editor feature) to fix this problem. Many geometric images, especially of tiling and subdivision, look a lot better with solid fill of pastel colours.

 * Octave (Which actually calls gnuplot.)
 * Maxima CAS (Which can call gnuplot, but also other programs like VTK )
 * C.a.R (Compass and Ruler) (produces fixed-scale graphs)

Note that one can also directly create SVG file.

Image Requests
If you need a diagram to illustrate a mathematical text, describe what you need here. The mathematical project has recently been producing diagrams for geometry (for the trigonometry book) and even fake road signs to help teach mathematical concepts.

Transwiki
It can be useful to start new mathematics topic pages by importing the corresponding page from wikipedia as a starter. Discuss possible imports here.

title: Wikibooks:WikiProject Mathematics


text:
User:KirbyMeister:FlashDev

Possible outline for creating a WikiBook on Flash Development, feel free to modify at will

Chapter 1: Flash Background Small history of Flash Explain what Flash is Explain what it isnt Pros and Cons Uses of Flash Examples (uses for flash, ads, webcartoons (H*R), webgames) Chapter 2: Your First Flash Animation Section 1: Animation Techniques Explain various animation techniques Frame by Frame vs. Tweening Section 2: First Frame Explain how to draw in Flash Warn about stickman movies Explain symbols Section 3: Animating Creating keyframes Using Layers Tweening Shape vs. Motion tweens Section 4: Music Importing music Stream vs. Event sound Compression upon export A note about music sources Section 5: Preloaders What is a Preloader Why use a Preloader When NOT to use a Preloader How to make a Preloader Preloader Alternatives Section 6: Distribution Exporting from flash (.swf, .avi, among others) When to export to an SWF file When to export to an AVI file AVI specifics (compression, lack of actionscript) SWF vs. AVI comparison Chapter 3: Your First Interactive Flash Animation Section 1: Actionscript basics Explain movieclips, buttons Explain various features of actionscript Section 2: Making webgames Why make a webgame Explain specifically about games Explain different types of games Section 3: Making entire websites in Flash Why make a site in Flash Backend technologies How to update the site Chapter 4: Where to go from here to be discussed...

title: User:KirbyMeister:FlashDev


text:
Electronics/Resonance

Resonance is where the reactance of capacitors and inductors cancel at a specific frequency.
 * $$Z_C = Z_L$$
 * $$\frac {1}{j\omega C} = j\omega L$$
 * $$\omega = \frac{1}{\sqrt{LC}}$$

This frequency is called Resonance Frequency usually, denoted as $$\omega_o = \frac{1}{\sqrt{LC}}$$

title: Electronics/Resonance


text:
Geometry/Differential Geometry

 * Introduction
 * Basic Curves

title: Geometry/Differential Geometry


text:
Geometry/Differential Geometry/Introduction

Differential geometry studies geometry by considering differentiable paramaterizations of curves, surfaces, and higher dimensional objects. Prerequisites include vector calculus, linear algebra, analysis, and topology.

One goal of differential geometry is to classify and represent differentiable curves in ways which are independent of their paramaterization. For example consider the curve represented by $$y=3x$$. Although $$(x,y)=(t,3t)$$ and $$(x,y)=(3t,9t)$$ are different paramterizations, they both represent the same curve. More generally, we consider the slope of the curve $$3=\frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{1}{\frac{dx}{dt}}=\frac{3}{1}$$.

We call this type of curve a line. We can even rotate, and move it around, but it is still a line. The goal of Differential Geometry will be to similarly classify, and understand classes of differentiable curves, which may have different paramaterizations, but are still the same curve.

By adding sufficient dimensions, any equation can become a curve in geometry. Therefore, the ability to discern when two curves are unique also has the potential for applications in distinguishing information from noise. There may be multiple ways of receiving the same information--in different paramterizations, but we want to distinguish if the information is actually unique.


text:
Template:High School Mathematics Extensions/Matrices

Introduction
"Matrix" may be more popularly known as a giant computer simulation, but in mathematics it is a totally different thing. To be more precise, a matrix (plural matrices) is a rectangular array of numbers. For example, below is a typical way to write a matrix, with numbers arranged in rows and columns and with round brackets around the numbers:

\begin{pmatrix} 1 & 5&10&20 \\ 1&-3&-5&9\\ 3&-1&-1&-1\\ 3&2&4&-5 \end{pmatrix} $$ The above matrix has 4 rows and 4 columns, so we call it a 4 × 4 (4 by 4) matrix. Also, we can have matrices of many different shapes. The shape of a matrix is the name for the dimensions of the matrix (m by n, where m is the number of rows and n the number of columns). Here are some more examples of matrices:

This is an example of a 3 × 3 matrix:

\begin{pmatrix} 1&2&3\\ 4&5&6\\ 7&8&9\\ \end{pmatrix} $$

This is an example of a 5 × 4 matrix:

\begin{pmatrix} a&b&c&d\\ h&g&f&e\\ i&j&k&l\\ p&o&n&m\\ q&r&s&t\\ \end{pmatrix} $$

This is an example of a 1 × 6 matrix:

\begin{pmatrix} 1&2&3&4&5&6\\ \end{pmatrix} $$

The theory of matrices is intimately connected with that of (linear) simultaneous equations. The ancient Chinese had established a systematic way to solve simultaneous equations. The theory of simultaneous equations was furthered in the east by the Japanese mathematician, Seki and a little later by Leibniz, Newton's greatest rival. Later, Gauss (1777 - 1855), one of the three giants of modern mathematics, popularised the use of Gaussian elimination, which is a simple step by step algorithm for solving any number of linear simultaneous equations. By then the use of matrices to represent simultaneous equation neatly on paper (as discussed above) had become quite common1.

Consider the simultaneous equations:
 * $$x + y = 10$$
 * $$x - y = 4$$

it has the solution x = 7 and y = 3, and the usual way to solve it is to add the two equations together to eliminate the y. Matrix theory offers us another way to solve the above simultaneous equations via matrix multiplication (covered below). We will study the widely accepted way to multiply two matrices together. In theory with matrix multiplication we can solve any number of simultaneous equations, but we shall mainly restrict our attention to 2 × 2 matrices. But even with that restriction, we have opened up doors to topics simultaneous equations could never offer us. Two such examples are
 * 1) using matrices to solve linear recurrence relations which can be used to model population growth, and
 * 2) encrypting messages with matrices.

We shall commence our study by learning some of the more fundamental concepts of matrices. Once we have a firm grasp of the basics, we shall move on to study the real meat of this chapter, matrix multiplication.

Elements
An element of a matrix is a particular number inside the matrix, and it is uniquely located with a pair of numbers. E.g. let the following matrix be denoted by A, or symbolically:
 * $$A =

\begin{pmatrix} 1&2&3\\ 4&5&6\\ 7&8&9\\ \end{pmatrix} $$ the (2,2)th entry of A is 5; the (1,1)th entry of A is 1, the (3,3) entry of A is 9 and the (2,3)th entry of A is 8. The (i, j)th entry of A is usually denoted ai,j and the (i , j)th entry of a matrix B is usually denoted by bi,j and so on.

Summary

 * A matrix is an array of numbers
 * A m×n matrix has m rows and n columns
 * The shape of a matrix is determined by its number of rows and columns
 * The (i,j)th element of a matrix is located in ith row and jth column

Matrix addition & Multiplication by a scalar
Matrices can be added together. But only the matrices of the same shape can be added. This is very natural. E.g.

A = \begin{pmatrix} 1&x&2x\\ 1&3x&5x\\ 1&3&4\\ \end{pmatrix} $$

B = \begin{pmatrix} 2&9&8\\ 0&-1&8\\ 4&6&7\\ \end{pmatrix} $$ then

A + B = \begin{pmatrix} 1&2&3\\ 4&5&6\\ 7&8&9\\ \end{pmatrix} + \begin{pmatrix} 2&9&8\\ 0&-1&8\\ 4&6&7\\ \end{pmatrix} = \begin{pmatrix} 1+2&2+9&3+8\\ 4+0&5+(-1)&6+8\\ 7+4&8+6&9+7\\ \end{pmatrix} = \begin{pmatrix} 3&11&11\\ 4&4&14\\ 11&14&16\\ \end{pmatrix} $$

Similarly matrices can be multiplied by a number. We call the number a scalar to distinguish it from a matrix. The reader need not worry about the definition here, just remember that a scalar is simply a number.

5A = A + A + A + A + A = 5\begin{pmatrix} 1&2&3\\ 4&5&6\\ 7&8&9\\ \end{pmatrix} = \begin{pmatrix} 5&10&15\\ 20&25&30\\ 35&40&45\\ \end{pmatrix} $$ in this case the scalar value is 5. In general, when we do s × A, where s is a scalar and A a matrix, we multiply each entry of A by s.

Matrix Multiplication
The widely accepted way to multiply two matrices together is definitely non-intuitive. As mentioned above, multiplication can help with solving simultaneous equations. We will now give a brief outline of how this can be done. Firstly, any system of linear simultaneous equations can be written as a matrix of coefficients multiplied by a matrix of unknowns equaling a matrix of results. This description may sound a little complicated, but in symbolic form it is quite clear. The previous statement simply says that if A, x and b are matrices, then Ax = b, can be used to represent some system of simultaneous equations. The beautiful thing about matrix multiplications is that some matrices can have multiplicative inverses, that is we can multiply both sides of the equation by A-1 to get x = A-1b, which effectively solves the simultaneous equations.

The reader will surely come to understand matrix multiplication better as this chapter progresses. For now we should consider the simplest case of matrix multiplication, multiplying vectors. We will see a few examples and then we will explain process of multiplication

\begin{matrix}

A_{2\times 1} = \begin{pmatrix} 2\\ 9\\ \end{pmatrix} &, & B_{1\times 2} = \begin{pmatrix} 3 & 5 \end{pmatrix}

\end{matrix} $$

then

B_{1\times 2} \times A_{2\times 1} = \begin{pmatrix} 3 & 5 \end{pmatrix} \times \begin{pmatrix} 2\\ 9\\ \end{pmatrix} = \begin{pmatrix} (3 \times 2) + (5 \times 9) \end{pmatrix} = \begin{pmatrix} 51 \end{pmatrix}

$$

Similarly if:
 * $$ \begin{matrix}

A_{3\times 1} = \begin{pmatrix} 1\\ 2\\ 3\end{pmatrix} &, & B_{1\times 3} = \begin{pmatrix} 4 & 5 & 6\end{pmatrix}

\end{matrix} $$

then

B_{1\times 3} \times A_{3\times 1}= \begin{pmatrix} 4 & 5 &6 \end{pmatrix} \times \begin{pmatrix}1\\2\\3\\\end{pmatrix} = \begin{pmatrix}(4 \times 1) + (5 \times 2) + (6 \times 3)\end{pmatrix} = \begin{pmatrix} 32 \end{pmatrix} $$

A matrix with just one row is called a row vector, similarly a matrix with just one column is called a column vector. When we multiply a row vector A, with a column vector B, we multiply the element in the first column of A by the element in the first row of B and add to that the product of the second column of A and second row of B and so on. More generally we multiply a1,i by bi,1 (where i ranges from 1 to n, the number of rows/columns) and sum up all of the products. Symbolically:

 * $$ A_{1\times n} \times B_{n\times 1} = (\sum_{i=1}^na_{1,i}\times b_{i,1} )$$ (for information on the $$\sum$$ sign, see Summation_Sign)
 * where n is the number of rows/columns.

 * In words: the product of a column vector and a row vector is the sum of the product of item 1,i from the row vector and i,1 from the column vector where i is from 1 to the width/height of these vectors.

Note: The product of matrices is also a matrix. The product of a row vector and column vector is a 1 by 1 matrix, not a scalar.

Exercises
Multiply:
 * $$\begin{pmatrix}1&2\end{pmatrix}\begin{pmatrix}1\\2\end{pmatrix}$$

 * $$\begin{pmatrix}1\\2\end{pmatrix}\begin{pmatrix}1&2\end{pmatrix}$$

 * $$\begin{pmatrix}\frac{1}{8}&9\end{pmatrix}\begin{pmatrix}16\\2\end{pmatrix}$$

 * $$\begin{pmatrix}a&b\end{pmatrix}\begin{pmatrix}d\\e\end{pmatrix}$$

 * $$\begin{pmatrix}6 + 6b&3 - b\end{pmatrix}\begin{pmatrix}0\\0\end{pmatrix}$$

 * $$\begin{pmatrix}0&abc\end{pmatrix}

\begin{pmatrix}a\\0\end{pmatrix}$$

Multiplication of non-vector matrices
Suppose $$A_{m \times n}B_{n \times p} = C_{m \times p}$$ where A, B and C are matrices. We multiply the ith row of A with the jth column of B as if they are vector-matrices. The resulting number is the (i,j)th element of C. Symbolically:
 * $$c_{i,j} = \sum_{k=1}^{n}a_{i,k}\times b_{k,j}$$

 Example 1

Evaluate AB = C and BA= D, where
 * $$ A =

\begin{pmatrix} 3&2\\ 5&6\\ \end{pmatrix} $$ and
 * $$ B =

\begin{pmatrix} 2&6\\ 8&7\\ \end{pmatrix} $$

 Solution
 * $$c_{1,1} = \begin{pmatrix} 3&2 \end{pmatrix} \begin{pmatrix}2\\ 8\end{pmatrix} = (3\times 2 + 2\times 8) = 22$$
 * $$c_{1,2} = \begin{pmatrix} 3&2 \end{pmatrix} \begin{pmatrix}6\\ 7\end{pmatrix} = (3\times 6 + 2\times 7) = 32$$
 * $$c_{2,1} = \begin{pmatrix} 5&6 \end{pmatrix} \begin{pmatrix}2\\ 8\end{pmatrix} = (5\times 2 + 6\times 8) = 58$$
 * $$c_{2,2} = \begin{pmatrix} 5&6 \end{pmatrix} \begin{pmatrix}6\\ 7\end{pmatrix} = (5\times 6 + 6\times 7) = 72$$

i.e.
 * $$C =

\begin{pmatrix} 22&32\\58&72 \end{pmatrix} $$

 * $$d_{1,1} = \begin{pmatrix} 2&6 \end{pmatrix} \begin{pmatrix}3\\ 5\end{pmatrix} = (2\times 3 + 6\times 5) = 36$$
 * $$d_{1,2} = \begin{pmatrix} 2&6 \end{pmatrix} \begin{pmatrix}2\\ 6\end{pmatrix} = (2\times 2 + 6\times 6) = 40$$
 * $$d_{2,1} = \begin{pmatrix} 8&7 \end{pmatrix} \begin{pmatrix}3\\ 5\end{pmatrix} = (8\times 3 + 7\times 5) = 59$$
 * $$d_{2,2} = \begin{pmatrix} 8&7 \end{pmatrix} \begin{pmatrix}2\\ 6\end{pmatrix} = (8\times 2 + 7\times 6) = 58$$

i.e.
 * $$D =

\begin{pmatrix} 36&40\\59&58 \end{pmatrix} $$

Example 2  Evaluate AB and BA where
 * $$A =

\begin{pmatrix} 5&17\\ 2&7 \end{pmatrix} $$
 * $$B =

\begin{pmatrix} 7&-17\\ -2&5 \end{pmatrix} $$

 Solution

\begin{pmatrix} 5&17\\ 2&7 \end{pmatrix} \begin{pmatrix} 7&-17\\ -2&5 \end{pmatrix} = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} $$

\begin{pmatrix} 7&-17\\ -2&5 \end{pmatrix} \begin{pmatrix} 5&17\\ 2&7 \end{pmatrix} = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} $$

Example 3  Evaluate AB and BA where
 * $$A =

\begin{pmatrix} 2&6\\ 0&5 \end{pmatrix} $$
 * $$B =

\begin{pmatrix} 5&-6\\ 0&2 \end{pmatrix} $$

 Solution

\begin{pmatrix} 2&6\\ 0&5 \end{pmatrix} \begin{pmatrix} 5&-6\\ 0&2 \end{pmatrix} = \begin{pmatrix} 10&0\\ 0&10 \end{pmatrix} $$

\begin{pmatrix} 5&-6\\ 0&2 \end{pmatrix} \begin{pmatrix} 2&6\\ 0&5 \end{pmatrix} = \begin{pmatrix} 10&0\\ 0&10 \end{pmatrix} $$

Example 4  Evaluate the following multiplication:

\begin{pmatrix} a\\ b \end{pmatrix} \begin{pmatrix} c&d\\ \end{pmatrix} $$

 Solution

Note that:

\begin{pmatrix} a\\ b \end{pmatrix} $$ is a 2 by 1 matrix and

\begin{pmatrix} c&d\\ \end{pmatrix} $$ is a 1 by 2 matrix. So the multiplication makes sense and the product should be a 2 by 2 matrix.

\begin{pmatrix} a\\ b \end{pmatrix} \begin{pmatrix} c&d\\ \end{pmatrix} = \begin{pmatrix} ac&ad\\ bc&bd\\ \end{pmatrix} $$

Example 5  Evaluate the following multiplication:

\begin{pmatrix} 1\\ 2 \end{pmatrix} \begin{pmatrix} 3&4\\ \end{pmatrix} $$

 Solution

\begin{pmatrix} 1\\ 2 \end{pmatrix} \begin{pmatrix} 3&4\\ \end{pmatrix} = \begin{pmatrix} 1 \times 3& 1 \times 4\\ 2 \times 3& 2 \times 4 \\ \end{pmatrix} = \begin{pmatrix} 3& 4\\ 6& 8 \\ \end{pmatrix} $$

Example 6  Evaluate the following multiplication:

\begin{pmatrix} a&0\\ 0&b \end{pmatrix} \begin{pmatrix} c&0\\ 0&d \end{pmatrix} $$

 Solution  $$ \begin{pmatrix} a&0\\ 0&b \end{pmatrix} \begin{pmatrix} c&0\\ 0&d \end{pmatrix} = \begin{pmatrix} ac&0\\ 0&bd \end{pmatrix} $$

Example 7  Evaluate the following multiplication:

\begin{pmatrix} a&b\\ c&d \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix} $$

 Solution  $$ \begin{pmatrix} a&b\\ c&d \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} ax+by\\ cx+dy \end{pmatrix} $$

Note Multiplication of matrices is generally not commutative, i.e. generally AB ≠ BA.

Diagonal matrices
A diagonal matrix is a matrix with zero entries everywhere except possibly down the diagonal. Multiplying diagonal matrices is really convenient, as you need only to multiply the diagonal entries together.

Examples

The following are all diagonal matrices $$ \begin{pmatrix} a&0\\ 0&b \end{pmatrix} \begin{pmatrix} c&0\\ 0&d \end{pmatrix} \begin{pmatrix} 1&0\\ 0&2 \end{pmatrix} \begin{pmatrix} 0&0\\ 0&0 \end{pmatrix} \begin{pmatrix} a&0&0\\ 0&c&0\\ 0&0&0 \end{pmatrix} $$

 Example 1  $$ \begin{pmatrix} a&0\\ 0&b \end{pmatrix} \begin{pmatrix} e&0\\ 0&f \end{pmatrix} \begin{pmatrix} h&0\\ 0&i\\ \end{pmatrix} = \begin{pmatrix} aeh&0\\ 0&bfi\\ \end{pmatrix} $$

 Example 2  $$ \begin{pmatrix} a&0\\ 0&b \end{pmatrix} \begin{pmatrix} a&0\\ 0&b \end{pmatrix} \begin{pmatrix} a&0\\ 0&b \end{pmatrix} = \begin{pmatrix} a^3&0\\ 0&b^3 \end{pmatrix} $$

The above examples show that if D is a diagonal matrix then Dk is very easy to compute, all we need to do is to take the diagonal entries to the kth power. This will be an extremely useful fact later on, when we learn how to compute the nth Fibonacci number using matrices.

Exercises
1. State the dimensions of C
 * a) C = An×pBp×m
 * b) $$C =

\begin{pmatrix} 10^{10}&20\\ 5000&0 \end{pmatrix} \begin{pmatrix} 1&2&3&4\\ 2&5&6&6 \end{pmatrix} $$

2. Evaluate. Please note that in matrix multiplication (AB)C = A(BC) i.e. the order in which you do the multiplications does not matter (proved later).
 * a)

\begin{pmatrix} 1&1\\ 0&1\\ \end{pmatrix} \begin{pmatrix} 1&1\\ 0&1\\ \end{pmatrix} \begin{pmatrix} 1&1\\ 0&1\\ \end{pmatrix} \begin{pmatrix} 1\\ 1\\ \end{pmatrix}$$

 * b)

\begin{pmatrix} 3&1\\ 2&8\\ \end{pmatrix} \begin{pmatrix} 1&1\\ 0&2\\ \end{pmatrix} \begin{pmatrix} 1&1\\ 0&1\\ \end{pmatrix} \begin{pmatrix} 1\\ 1\\ \end{pmatrix}$$

3. Performing the following multiplications:

C = \begin{pmatrix} 1&2\\ 4&5 \end{pmatrix} \begin{pmatrix} 1&0\\ 0&1\\ \end{pmatrix}$$

D = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} \begin{pmatrix} 1&2\\ 4&5\\ \end{pmatrix} $$ What do you notice?

The Identity & multiplication laws
The exercise above showed us that the matrix:

\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} $$ is a very special. It is called the 2 by 2 identity matrix. An identity matrix is a square matrix, whose diagonal entries are 1's and all other entries are zero. The identity matrix, I, has the following very special properties for all matrices A. We don't usually specify the shape of the identity because it's obvious from the context, and in this chapter we will only deal with the 2 by 2 identity matrix. In the real number system, the number 1 satisfies: r × 1 = r = 1 × r, so it's clear that the identity matrix is analogous to "1".
 * 1) $$A \times I = A$$
 * 2) $$I \times A = A$$

Associativity, distributivity and (non)-commutativity

Matrix multiplication is a great deal different to the multiplication we know from multiplying real numbers. So it is comforting to know that many of the laws the real numbers satisfy also carries over to the matrix world. But with one big exception, in general AB ≠ BA.

Let A, B, and C be matrices. Associativity means
 * (AB)C = A(BC)

i.e. the order in which you multiply the matrices is unimportant, because the final result you get is the same regardless of the order which you do the multiplications.

On the other hand, distributivity means
 * A(B + C) = AB + AC

and
 * (A + B)C = AC + BC

Note: The commutative property of the real numbers (i.e. ab = ba), does not carry over to the matrix world.

 Convince yourself

For all 2 by 2 matrices A, B and C. And I the identity matrix.

1. Convince yourself that in the 2 by 2 case:
 * A(B + C) = AB + AC

and
 * (A + B)C = AC + BC

2. Convince yourself that in the 2 by 2 case:
 * A(BC) = (AB)C

3. Convince yourself that:
 * $$AB\ne BA$$

in general. When does AB = BA? Name at least one case.

Note that all of the above are true for all matrices (of any dimension/shape).

Determinant and Inverses
We shall consider the simultaneous equations:
 * ax + by = α (1)
 * cx + dy = β (2)

where a, b, c, d, α and β are constants. We want to determine the necessary conditions for (1) and (2) to have a unique solution for x and y. We proceed:
 * Let (1') = (1) × c
 * Let (2') = (2) × a

i.e.
 * acx + bcy = cα (1')
 * acx + ady = aβ (2')

Now
 * let (3) = (2') - (1')
 * (ad - bc)y = aβ - cα (3)

Now y can be uniquely determined if and only if (ad - bc) ≠ 0. So the necessary condition for (1) and (2) to have a unique solution depends on all four of the coefficients of x and y. We call this number (ad - bc) the determinant, because it tells us whether there is a unique solution to two simultaneous equations of 2 variables. In summary
 * if (ad - bc) = 0 then there is no unique solution
 * if (ad - bc) ≠ 0 then there is a unique solution.

Note: Unique, we can not emphasise this word enough. If the determinant is zero, it doesn't necessarily mean that there is no solution to the simultaneous equations! Consider:
 * x + y = 2
 * 7x + 7y = 14

the above set of equations has determinant zero, but there is obviously a solution, namely x = y = 1. In fact there are infiinitely many solutions! On the other hand consider also:
 * x + y = 1
 * x + y = 2

this set of equations has determinant zero, and there is no solution at all. So if determinant is zero then there is either no solution or infinitely many solutions.

 Determinant of a matrix   We define the determinant of a 2 × 2 matrix
 * $$A = \begin{pmatrix}

a&b\\ c&d \end{pmatrix}$$ to be
 * $$\det (A) = ad - bc \!$$

Inverses
It is perhaps, at this stage, not very clear what the use is of the det(A). But it's intimately connected with the idea of an inverse. Consider in the real number system a number b, it has (multiplicative) inverse 1/b, i.e. b(1/b) = (1/b)b = 1. We know that 1/b does not exist when b = 0.

In the world of matrices, a matrix A may or may not have an inverse depending on the value of the determinant det(A)! How is this so? Let's suppose A (known) does have an inverse B (i.e. AB = I = BA). So we aim to find B. Let's suppose further that
 * $$A

= \begin{pmatrix} a&b\\ c&d \end{pmatrix}$$ and
 * $$B

= \begin{pmatrix} w&x\\ y&z \end{pmatrix}$$ we need to solve four simultaneous equations to get the values of w, x, y and z in terms of a, b, c, d and det(A).
 * aw + by = 1
 * cw + dy = 0
 * ax + bz = 0
 * cx + dz = 1

the reader can try to solve the above by him/herself. The required answer is
 * $$B = \frac{1}{\det(A)}\begin{pmatrix}

d&-b\\ -c&a \end{pmatrix} $$ In here we assumed that A has an inverse, but this doesn't make sense if det(A) = 0, as we can not divide by zero. So A-1 (the inverse of A) exists if and only if det(A) ≠ 0.

Summary

If AB = BA = I, then we say B is the inverse of A. We denote the inverse of A by A-1. The inverse of a 2 × 2 matrix
 * $$A =

\begin{pmatrix} a&b\\ c&d \end{pmatrix} $$ is
 * $$A^{-1} = \frac{1}{\det(A)}\begin{pmatrix}

d&-b\\ -c&a \end{pmatrix} $$ provided the determinant of A is not zero.

Solving simultaneous equations
Suppose we are to solve:
 * ax + by = α
 * cx + dy = β

We let
 * $$ A = \begin{pmatrix}a&b\\c&d\end{pmatrix}$$
 * $$ w = \begin{pmatrix}x\\y\end{pmatrix}$$
 * $$ \gamma = \begin{pmatrix}\alpha\\ \beta\end{pmatrix}$$

we can translate it into matrix form
 * $$ \begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}\alpha\\ \beta\end{pmatrix}$$

i.e
 * $$ Aw = \gamma$$

If A's determinant is not zero, then we can pre-multiply both sides by A-1 (the inverse of A)

\begin{matrix} A^{-1}Aw &=& A^{-1}\gamma\\ Iw &=& A^{-1}\gamma\\ w &=& A^{-1}\gamma\\ \end{matrix} $$ i.e.
 * $$\begin{pmatrix}x\\y\end{pmatrix} = \frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\begin{pmatrix}\alpha\\ \beta\end{pmatrix}$$

which implies that x and y are unique.

 Examples

Find the inverse of A, if it exists
 * a) $$ A = \begin{pmatrix}1&5\\2&3\end{pmatrix}$$
 * b) $$ A = \begin{pmatrix}10&2\\2&7\end{pmatrix}$$
 * c) $$ A = \begin{pmatrix}a&b\\3a&3b\end{pmatrix}$$
 * d) $$ A = \begin{pmatrix}3&5\\5&3\end{pmatrix}$$

 Solutions
 * a) $$A^{-1} = \frac{1}{-7}\begin{pmatrix}3&-5\\-2&1\end{pmatrix}$$
 * b) $$A^{-1} = \frac{1}{66}\begin{pmatrix}7&-2\\-2&10\end{pmatrix}$$
 * c) No solution, as det(A) = 3ab - 3ab = 0
 * d) $$A^{-1} = \frac{1}{-16}\begin{pmatrix}3&-5\\-5&3\end{pmatrix}$$

 Exercises

1. Find the determinant of
 * $$ A = \begin{pmatrix}\frac{2}{5}&\frac{2}{3}\\ \\ \frac{3}{2}& \frac{5}{2}\end{pmatrix}$$. Using the determinant of A, decide whether there's a unique solution to the following simultaneous equations

\begin{matrix} \frac{2}{5}x + \frac{2}{3}y = 0\\ \frac{3}{2}x + \frac{5}{2}y = 0 \end{matrix} $$

2. Suppose
 * C = AB

show that
 * det(C) = det(A)det(B)

for the 2 × 2 case. Note: it's true for all cases.

3. Show that if you swap the rows of A to get A' , then det(A) = -det(A' )

4. Using the result of 2

a) Prove that if:
 * $$A = P^{-1}BP$$

then det(A) = det(B)

b) Prove that if:
 * Ak = 0

for some positive integer k, then det(A) = 0.

5. a) Compute A5, i.e. multiply A by itself 5 times, where
 * $$A =

\begin{pmatrix} -1&6\\ -1&4\\ \end{pmatrix} $$

b) Find the inverse of P where

P = \begin{pmatrix} 1&-2\\ -1&3\\ \end{pmatrix} $$

c) Verify that
 * $$A =

P^{-1} \begin{pmatrix} 1&0\\ 0&2\\ \end{pmatrix} P $$

d) Compute A5 by using part (b) and (c).

e) Compute A100

Other Sections
Next Section > High_School_Mathematics_Extensions/Matrices/Linear_Recurrence_Relations_Revisited

Problem Set > High_School_Mathematics_Extensions/Matrices/Problem Set

Project > High_School_Mathematics_Extensions/Matrices/Project/Elementary_Matrices


text:
GCSE Science/Edexcel/Chemical Patterns

This edexcel module is splt into three sub-models. You should study all three of them to complete all the material in the module.

 * Atomic Structure
 * The Periodic Table
 * Rates of Reaction

title: GCSE Science/Edexcel/Chemical Patterns


text: GCSE Science/The Periodic Table

title: GCSE Science/The Periodic Table


text:
User:Mountain

Hi, everyone! I'm Mountain, a graduate student from Beijing, China.

I wrote the first meaningful page in zh.wikipedia.org, and I'm the first man translate mediawiki's localization file into Chinese. Now I'm an administrator in Chinese Wikipedia.

I'm very interested in programming language and Programming practice.

 * My user page: Chinese Wikipedia-English Wikipedia-Meta-Wikipedia-Wikisource.
 * My blog: http://conch.blogbus.com (in Chinese)
 * My bookmark: del.icio.us-furl.net

title: User:Mountain


text:
Cookbook talk:Paprika Chicken

This recipe was the Featured Recipe on the main Cookbook page from March 30 2004 - April 7 2004. Gentgeen

moved from recipe page

We would like to spread the word though Paprika is great with almost everything!

Paprika Rules!

end moved section

sour cream?
I make something similar, chicken paprikash, which has a sour cream, paprika, and onion sauce. Here is the recipe:

http://www.recipesource.com/ethnic/europe/hungarian/00/rec0019.html
 * "Chicken paprikash" and "paprika chicken" are exactly the same thing, which Hungarians call "paprikás csirke".

problems
If you follow this recipe as given you'll end up with a scorched mess which you'll have to throw away. It should have some chicken stock or water added in which to simmer the meat and vegetables until they're cooked, otherwise the paprika and veggies will burn and the chicken will be dry and undercooked in places. motterj

True...and also, regarding the sour cream question....authentic Chicken Paprikas is indeed finished in the last minute or two of cooking with sour cream and sometimes (optionally) a little bit of white "rantas" (roux) if you like a slightly thiker sauce (just don't overdo the thickening). Some folks use tomato as well and some don't but most Hungarians agree that if you leave out the sour cream, you haven't really made chicken paprikas. Period. It is essential to the dish. AnonymousHunkie

No problem
True. Let's make this recipe work. Warrington (talk) 21:24, 1 September 2008 (UTC)

Done.

Warrington (talk) 10:11, 2 September 2008 (UTC)

title: Cookbook talk:Paprika Chicken


text:
User talk:Mountain

Welcome to the wikibooks project! When you're free from programming, perhaps try and help out with the Chinese wikibook?

title: User talk:Mountain