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IB Biology/Option D - Evolution

Option D.1 Origin of Life on Earth
D.1.1 Outline the conditions of the pre-biotic Earth, including high temperature, lightning, UV light penetration and a reducing atmosphere.

The energy for forming the molecules was provided by lightning, volcanic activity, meteorite bombardment, high temperatures due to greenhouse gasses and UV radiation.
 * In the pre-biotic Earth (the solar system originated around 4.57 billion years ago) there was little to no oxygen in the atmosphere (any oxygen was absorbed by rocks), so there was no oxygen to steal the energy.
 * At first, the Earth was cold and later melted from heat produced by compaction, radioactive decay and the impact of meteorites. The molten material sorted into layers of varying density with the least dense material solidified into a thin crust. The present continents are attached to plates of crust that float on the mantle. The first seas formed from rain that began when Earth had cooled enough for water in the atmosphere to condense.
 * The prebiotic era dates from 4.4 billion years ago. The earliest life known is dated to 3.8 billion years ago.

D.1.2 Outline the experiments of Miller and Urey into the origin of organic compounds.

 * They simulated conditions on the early Earth by constructing an apparatus that contained a warmed flask of water simulating the primeval sea and an atmosphere of water, hydrogen gas, CH4 (methane), and NH3 (ammonia).
 * Sparks were discharged in the synthetic atmosphere to mimic lightning. A water was boiled while a condenser cooled the atmosphere, raining water and any dissolved compounds back to the miniature sea. The simulated environment produced many types of amino acids and other organic molecules leading them to conclude the prebiotic synthesis of organic molecules was possible.
 * However, the question of the concentration of methane and other chemicals is in doubt so the applicability of the results is uncertain.

D.1.3 Discuss the hypothesis that the first catalysts responsible for polymerization reactions were clay minerals and RNA.

 * Polymers are chains of similar building blocks or monomers, synthesized by condensation reactions (H and OH are removed from polymers and H20 is produced). Early polymerization reactions must have occurred without the help of enzymes.

Clay increases the rate of polymerization in these ways:
 * Monomers bind to charged sites between close clay layers, concentrating amino acids and other monomers
 * Metal ions at binding sites in clay catalyze dehydration reactions
 * The clays provide more stable conditions for the formation of molecules.

Functions of RNA in polymerization:
 * There is a type of RNA called a ribozyme that can catalyze its own replication. This may be very crucial to the origin of RNA and DNA and therefore for the beginning of life.

D.1.4 Discuss the possible role of RNA as the first molecule capable of replicating.

 * Protobionts, or aggregates of abiotically produced molecules, accumulate organic materials like polypeptides from the environment. However, they probably do not possess a mechanism for replicating their particular characteristics so life cannot start. However, an information storage mechanism like RNA would allow them to pass on their characteristics. RNA probably preceded DNA, as DNA is too complex of a molecule to have formed in early protobionts. Short strands of RNA, perhaps most importantly the RNA called ribozymes, could have copied themselves and could have catalysed other reactions. They could then align nucleotides according to a certain pattern when bound to clay. Thus RNA could be replicated and passed on. This is supported by the fact that RNA plays an important role in genetic control in cell life today.

D.1.5 Discuss a possible origin of membranes and prokaryotic cells.

 * Living cells may have been preceded by protobionts, aggregates (groups) of abiotically produced molecules. To form life they need to compartmentalise themselves from the surrounding water.
 * Some protobionts in the presence of lipids form a molecular bilayer (liposome) around the protobiont droplet when shaken. This resembles the lipid bilayer of modern cells. The liposomes can break, reform and merge with others and mix their contents. Within these droplets reactions could be catalysed and RNA could replicate itself.
 * The most successful liposomes at surviving would pass on their characteristics and develop into early prokaryotes.

D.1.6 Discuss the endosymbiotic theory for the origin of eukaryotes.

 * Eukaryotic cells were probably symbiotic groupings of prokaryotic cells with smaller species living within larger prokaryotes. The endosymbiotic theory focuses on the origins of chloroplasts and mitochondria.
 * The ancestors of these organisms (photosynthetic prokaryotes like algae or respiring bacteria) originally entered the host cell as undigested prey or internal parasites. Normally the invaders would be digested but in this case they were not.
 * The host cell would allow them to live and exploit them. These organisms provide food and energy. Those which absorbed prokaryotes which became chloroplast and mitochdria formed plants, those without the chloroplasts became animal cells.
 * The evidence that supports this theory is:
 * Mitochondria and chloroplasts have bacteria-like RNA and ribosomes (70S as opposed to 80S in eukaryote cytoplasms) that enable them to make their own proteins and divide independently of the host cell.
 * Mitochondria and chloroplasts both have double membranes, the inner one probably that of the original cell, and the outer that which was created when the cell was absorbed.
 * Thylakoids resemble structures found in blue-green bacteria.
 * Chloroplasts and mitochondria have naked DNA like prokaryotes.
 * Chlorophyll a is found in both prokaryotes and eukaryotes.
 * Christae in mitochondria resemble mesosomes in bacteria.

Option D.2 Origin of Species
D.2.1 Outline Lamarck's theory of evolution by the inheritance of acquired characteristics.

 * Lamarcks theory of inheritance of acquired characteristics states that the modifications an organism acquired during its lifetime could be passed along to its offspring. The common idea is that if a giraffe stretches up to reach high leaves, it can pass a long neck onto its offspring.

D.2.2 Discuss the mechanism of, and lack of evidence for, the inheritance of acquired characteristics.

 * There is no evidence that acquired characteristics can be inherited. Characteristics that can be passed on are in your genes. If you are not born with genes programming for certain characteristics, you cannot pass down characteristics acquired during your lifetime as these characteristics do not reach sex cells. Example: if you cut off the tails of mice, their offspring will still have tails.

D.2.3 Explain the Darwin-Wallace theory of evolution by natural selection.

 * Populations tend to produce more offspring than an environment can support but populations tend to remain constant
 * There is a struggle for survival among organisms in a species with varied characteristics
 * Those individuals which are best adapted to their environments are most likely to survive
 * Only these pass on their positive characteristics in the form of genes to their offspring who also survive

 * While on his Beagle voyages, Darwin became intrigued with the different types of finches found in the Galapagos. All the species of birds were very similar, just like a species on the mainland of South America but between the islands they differed in size and beak shape. Darwin found that the birds fed on different types of food. Their beaks were adapted to eat different types of leaves, worms and seeds and other types of diets. Darwin explained all his observations and thoughts about the origin of species by the concept of "natural selection". This theory states that great diversity in a species ensures that some members of a population will be more suited to their environment than others. These individuals will be more likely to live long enough to reproduce and pass on their well-suited genes. Therefore, because those that are best suited (the food on an island) are the ones who have the most children, a population will, over time, adapt itself to its environment. The change in the frequency of characteristics (genes) in a population with a common gene pool is evolution.

D.2.4 Discuss other theories for the origin of species including special creation and panspermia.

 * Panspermia is the theory which suggests that life arose elsewhere in the universe and travelled to Earth through space in comets or meteors. Special creation states that a creator(s) formed life directly.

D.2.5 Discuss the evidence for all these theories and the applicability of the scientific method for further investigation.

 * Panspermia - Organic compounds and amino acids have been recovered from modern meteorites. This theory is not incompatible with evolution, which would have occurred once life reached Earth. The theory could be tested in the future by looking for life similar to our own beyond Earth but space travel is not yet sufficiently developed.

 * Special Creation - most of the evidence for this theory is found in religious faith. There is no real scientific evidence for this theory other than a lack of evidence for alternate theories. The theory cannot be falsified (contradicted) by observations, unlike evolution and panspermia. However, some believe that more and more evidence is being found that points towards creation by a divine being.

 * The Lamarckian theory also lacks any evidence in its favour.

 * Evolution fits with commonly held facts and is supported with sufficient amounts of evidence discussed in D.3. It operates based on well understood processes and laws of nature that have not been contradicted by any current observations, and thus is the preferred theory.

Option D.3 Evidence for Evolution
D.3.1 Describe the evidence for evolution as shown by the geographical distribution of living organisms, including the distribution of placental, marsupial and monotreme animals.

 * The two northern continents are only separated by a small sea, the Bering Straight which has been crossable at times in the past. They have very similar mammal life. The three southern continents have been far more isolated from one another and show far greater variety of mammal life.
 * Looking at the way in which continents have drifted over time and examining the fossils in them we can map out the development of different sorts of mammals. Monotremes and marsupials developed in Gondwanaland 165mya before the three southern continents were separated. Placental mammals (developed life young) developed later 135mya and replaced almost all of the monotremes (egg laying mammals) and marsupials (undeveloped young born into pouches) in the other continents.
 * Australia was separated from the other continents so animals there could no longer mate with animals from other continents and Australian animals could not travel elsewhere. This is why marsupials and monotremes are now found only on Australia. Monotremes and marsupials have evolved to fit the niches within the Australian environment and there are no placental mammals. This is simply because of geography not because Australia is unfit for placental mammals. These observations and other similar examples are expected under the theory of evolution.

D.3.2 Outline how remains of past living organisms have been preserved.

 * Study of past evolution - phylogeny
 * Sand eroded from the land is first carried by rivers, seas, swamps where the particles settle to the bottom. These deposits pile up and compress the older sediments into rock. Dead organisms are swept into seas and swamps then settle to the bottom with the sediments. When the sediments turn to rock under pressure the hard parts of the animal may remain as fossils. Minerals also leach into the soft parts of the organism, leaving a petrified cast of its shape within the rock.
 * Fossils can also be stored in:
 * Resins, which become amber
 * Frozen in ice or snow
 * In acid peat in which they cannot decay

D.3.3 Outline the method for dating rocks and fossils using radioisotopes, with reference to 14C and 40K.

 * Fossils contain isotopes of elements that accumulated in the living organisms. If the isotopes are unstable, they will lose protons and break down over time. Since each radioactive isotope has a fixed half-life it can be used to date fossils based on the relative concentrations of the reactant and product of the decay. Half life is the amount of time it takes for half of a sample of a certain substance to break down. Carbon-14 has a half life of 5000 years so useful or dating fossils less than 100,000 years old. Potassium-40 has a half-life of 1.3 billions years so useful for long-term dating. Error of less than 10%.

D.3.4 Define half-life.

 * Half-life - The number of years it takes for 50% of a sample to decay.

D.3.5 Deduce the approximate age of materials based on a simple decay curve for a radioisotope.

 * Look on the graph at how many times the concentration of the original isotope has halved then multiply that by the half-life.

D.3.6 Outline the palaeontogical evidence for evolution using one example.

 * The existence of the many fossils similar but different to current species is expected under evolution.
 * The Acanthostega fossil shows an amphibian from 365mya which has eight fingers and seven toes so is different from current organisms. It has four legs and a backbone like mammals, reptiles and amphibians but also a fish like tail and gills. It lived in water. It is a ‘missing link’ fossil demonstrating an intermediate point in evolution between current amphibians and fish.
 * In the horse, the number of toes they have has been reduced from four to one. A succession of fossils from the horse ancestor with four toes to a modern horse with one toe shows a trend towards reduced number of toes.
 * A clear progression of species is visible in the fossil record over time, from bacteria, to simple water based organisms, to amphibians, to insects, to reptiles and finally to mammals.

D.3.7 Explain the biochemical evidence provided by the universality of DNA and protein structures for the common ancestry of living organisms.
 * All living organisms have DNA, use the same 20 amino acids and use left, not right handed amino acids. This suggests that all life forms had a common ancestor which did the same.
 * To determine further relationships between organisms, comparing DNA and protein structure can be helpful. DNA Match two single stranded DNA from different species and see how tightly the DNA from one species can bind to DNA from another species. The tighter the bond the greater the similarity and the more closely related the species.
 * Most fundamental enzymes present in most living creatures have a similar amino acid structure.

D.3.8 Explain how variations in specific molecules can indicate phylogeny.
 * Proteins are genetically determined. Thus a close match in the amino acid sequence of two proteins from different species indicates that the genes in those proteins evolved from a common gene present in a shared ancestor.

D.3.9 Discuss how biochemical variations can be used as an evolutionary clock.
 * Mutations are random changes in gene structure but they occur at a roughly predictable rate. In general the more differences between the amino acid sequence of a common protein, the further in the past two species had a common ancestor. For example, the haemoglobin of gorillas only differs by one amino acid from human haemoglobin whereas elephant haemoglobin differs from human haemoglobin by 26 amino acids. Therefore elephants separated as a species from a common ancestor with humans longer ago then did gorillas. Information like this can help to group organisms in trees of descent and suggest how long ago they had a common gene pool.
 * First, check a specific protein and know that mutations occur at a certain rate. Count how many mutations there are in that specific protein and then calculate how many years the organism has evolved. For example, if there are 10 mutations and mutations occur every 5,000 years in this protein - 10 times 5,000 = 50,000 years.

D.3.10 Explain the evidence for evolution provided by homologous anatomical structures, including vertebrate embryos and the pentadactyl limb.

 * Evolutionary transitions leave signs in fossil records in terms of descent with modifications.
 * Homologous anatomical structure is a test for common ancestry. Descent with modification is evident in anatomical similarities between species grouped in the same taxonomical category. For example, the forelimbs of mammals have been modified to fit their function. However similarities in these structures demonstrate that they all originate from a common ancestor.
 * A pentadactyl limb describes the same skeletal elements that make up the forelimbs of humans, cats, whales, bats and all other mammals. They have evolved for different functions but the relationships between the bones in the limb are all remarkably similar.
 * Even distantly related organisms go through similar stages in their embryonic development. Many of them cannot be told apart in the embryonic stage despite looking entirely different as adults. All vertebrate embryos go through a stage in which they have gill pouches on the sides of their throats. This is all easily explained by evolution.

'''D.3.11 Outline two modern examples of observed evolution. One example must be the changes to the size and shape of the beaks of Galapagos finches.'''

 * In 1981 the finch Geospiza fortis had a short, wide beak and had a diet of mostly large, hard seeds. In 1982-3 there was an el Niño event which brought much rain and meant that the vegetation changed and over the next 5 years there were far more soft small fruits. The population boomed because of the rains and increase in food then dropped back when the rains stopped. By 1987 they had longer, narrower beaks than they had before because those beaks helped the birds to survive the population cutback after the rains.
 * In industrial areas of Europe in 1960 more ladybugs showed the black melanic colour allele. The black absorbs more light, and so the black ones have an advantage staying warm when sunlight levels are low. After 1960 as smog levels decreased the proportion of black ladybugs decreased proportionally and both smog and black butterflies reached constant low levels from 1980. Instead they were replaced with a red colour which warned off predators and was more effective at improving survival.

Option D.4 Human Evolution
D.4.1 State the full classification of human beings from kingdom to sub-species.

 * Kingdom - Animalia

 * Phylum - Chordata

 * Class - Mammalia

 * Order - Primata

 * Family - Hominidae

 * Genus - Homo

 * Species - Homo sapiens

 * subspecies - H. s. sapiens

D.4.2 Describe the major physical features, such as the adaptations for tree life, that define humans as primates.

 * Shoulder joints that allow movement in 3 dimension
 * Dexterous hands with opposable thumb and long fingers
 * Sensitive fingers with nails
 * Eyes closer together in front of the face for enhanced depth perception, excellent eye-hand coordination
 * Shoulder joints and skull modified for upright posture
 * Parental care with usually single births and long nurturing of offspring.

D.4.3 Discuss the anatomical and biochemical evidence which suggests that humans are a bipedal and neotenous species of African ape that spread to colonize new areas.

 * Humans, like apes, care for their young for a long time. Offspring also have delayed puberty.
 * Humans show more physical similarities with young apes than mature ones, so we may be neotenous – have evolved to retain juvenile ape characteristics.
 * Humans and apes have dexterous hands and similar hips and muscles.
 * 98% of human and chimpanzee DNA is exactly the same.
 * Fossils have been found of intermediate species in the evolution of humans from apes.

D.4.4 Outline the trends illustrated by the fossils of Australopithecus including A. afarensis, A. africanus and A. robustis and Homo including H. habilis, H. erectus, H. neanderthalensis and H. sapiens.

A. afarensis: 3 - 3.9 million years – ape-like face

A. africanus: 2.3 - 3 million years – flatter face, larger molars for plant based diet

A. robustus: 1.4 - 2.2 million years – very large molars, bones and skull

H. habilis: 1.6 - 2.4 million years – smaller teeth and jaw for meatier diet, first with tools, size like humans

H. erectus: .4 - 1.8 million years – more complex tools so meat significant part of diet and changed teeth.

H. neanderthensis .5 million years – larger brains and bones, larger teeth and jaw, shorter limbs for the cold

H. sapiens: .1 million years – large brain, flat face, reduced teeth, reduced robustness, chin

 * Enlargement of brain and taller and more erect structure.
 * The spike connection to the skull becomes more central to balance centre of gravity.
 * Pelvis changes to support organs in walking.
 * Pelvis shorter and broader to attach walking muscles.
 * Legs become stronger and longer while arms become shorter and weaker.
 * Knee can now be fully straightened.
 * Foot forms more of a platform and rigid shape without opposable toe.

D.4.5 Discuss the possible ecology of these species and the ecological changes that may have prompted their origin.

 * The climate of Africa became drier with thin woodland instead of forests. The depletion of forests encouraged a ground-adapted species that could move on land for long distances and who could carry scattered food. Thus, the adaptation of a bipedal human.
 * Later Africa became much cooler and there was mostly savanna. This may have prompted the evolution of the Homo genus which used tools and group work to hunt large animals for food.

D.4.6 Discuss the incompleteness of the fossil record and the resulting uncertainties with respect to human evolution.

 * The fossil record of human ancestry is incomplete and thus we cannot certainly determine when certain species originated and became extinct.
 * early fossil record is fragmented and scarce because: they were not buried, many killed by predators so bones were spread and few died in location where they would be preserved
 * as there are ‘missing links’ it can not be ensured that the hypothesised evolution of hominids is accurate

D.4.7 Discuss the origin and consequences of bipedalism and increase in brain size.

 * Bipedalism - had to adapt to living on the ground and be able to look for food over longer distances.
 * Allowed carrying of food and water and looking over brush.
 * The increase in brain size allowed the production of tools. This allowed us to hunt larger animals for meat, changing the diet and the teeth structure towards smaller molars.
 * Language becomes possible.
 * We learned to limit the environment's influence (e.g. clothing, fire and housing).
 * This comes at the cost of a longer development period and greater energy use by the brain.

D.4.8 Outline the difference between genetic and cultural evolution.

 * Cultural evolution are changes in the behaviour of a species, for example, what tools they use, as where they live, the things they eat etc.
 * Genetic evolution is changes in the genetic makeup of a species.
 * Cultural evolution has spanned millions of years in three major stages: the nomadic (hunting), agricultural (settled), and industrial ages. However, we have not changed biologically in any significant way. We are probably not any more intelligent than the cave men. Our increased ability is due to the past experience we draw on.

D.4.9 Discuss the relative importance of genetic and cultural evolution in the evolution of humans.

 * Evolution of the brain allowed cultural evolution to take off. Since then cultural evolution has been able to change humans far more quickly than genetic evolution has. Human behaviour and life has changed greatly without much genetic evolution at all. We are now able to change our environment instead of changing to suit our environment.  However, recent research (Voight and Kudaravalli, et al., 2006) indicates humans are in fact evolving under selective environmental pressures based on genetic analysis.

Option D.5 Neo-Darwinism
D.5.1 State that mutations are changes to genes or chromosomes due to chance, but with predictable frequencies.

 * Mutations are changes to genes or chromosomes due to chance, but with predictable frequencies.

D.5.2 Outline phenylketonuria (PKU) and cystic fibrosis as examples of gene mutation, and Klinefelter's syndrome as an example of chromosome mutation.

 * PKU is caused by a gene mutation that ruins the protein that breaks down the amino acid phenylalanine. It is caused by many possible recessive alleles all at very low frequencies. This causes toxic levels of this amino acid in the blood, which causes mental retardation and death.
 * Most instances of cystic fibrosis are caused by a specific mutation on a recessive allele. The mutation changes and amino acid and ruins the body’s protein that moves chloride ions in epithelial cells. A lot of thick mucus builds up inside airways, which leads to infection. Untreated, most children will die by age 5.
 * Klinefelter's syndrome results from an extra X chromosome due to trisomy (non-disjunction). Thus a person has an XXY pattern of sex chromosomes. Such a person has male sex organs but low testosterone, reduced male characteristic and abnormally small testes. Not significant in evolution as all are sterile.

D.5.3 Explain that variation in a population results from the recombination of alleles during meiosis and fertilisation.

 * Meiosis includes independent assortment of different alleles and also crossing over. These two processes make the total possible gene combinations from an individual very large and produce considerable variation in sexual reproduction by mixing up genes in new combinations.
 * In fertilisation the genes of two members of a species are mixed, increasing variation further.

D.5.4 State that adaptations (or micro-evolutionary steps) may occur as the result of an allele frequency increasing in a population over a period of time.

D.5.5 Describe how the evolution of one species into another species involves the accumulation of many advantageous alleles in the gene pool of a population over a period of time.
 * Evolution involves the change in heritable traits of a population over successive generations, as determined by shifts in allele frequencies of genes
 * Over time this results in speciation

D.5.6 State that a species is a potentially interbreeding population having a common gene pool.

 * A species is a potentially interbreeding population having a common gene pool.
 * A gene pool is all the genes in an interbreeding population.

D.5.7 Discuss the definition of the term species.

 * Most species are identified by their physical characteristics alone, but sometimes these are not sufficient.
 * Accurately, a species is a groups of organism that are potentially interbreeding and produce fertile offspring. They have a common gene pool and are reproductively isolated from other such groups.
 * This definition is somewhat ambiguous, but the main defining factor is the ability to produce fertile offspring. This is why donkeys and horses are different species. They produce mules when they breed, but because their chromosomes don't match completely, mules are infertile.
 * However, some species that are clearly different can interbreed and produce hybrids, like some ducks and plants.
 * Species which reproduce asexually cannot be classified in this way.
 * Fossils cannot be classified this way as we do not know how they bred.
 * Some species are almost identical physically but cannot interbreed like the Pipistrelle bat in Britain.
 * Ring species like the herring gulls and lesser black backed gulls around the Arctic circle are able to interbreed with the population adjacent to them, but not with those further around the circle.

D.5.8 Discuss the process of speciation in terms of migration, geographical or ecological isolation and adaptation, leading to reproductive or genetic isolation of gene pools.

 * As organisms become separated geographically and cannot breed (their gene pools do not mix) these organisms become different enough to be unable to breed. Such organisms have become different species.
 * Differences between two populations that have been geographically or ecologically isolated are caused by natural selection. Such speciation usually occurs because of genetic variation. They have different habitats to evolve to, so they evolve differently, and eventually become so different that they cannot breed. Then, even if they are not geographically isolated, they are reproductively or genetically isolated.

D.5.9 Discuss ideas on the pace of evolution including gradualism and punctuated equilibrium.

 * Gradualism suggests that evolutionary change occurs at a constant slow rate.
 * Punctuated equilibrium implies long periods with no change and short periods with rapid evolution.
 * The latter is gaining popularity as being more important than once thought. Volcanic eruptions and meteor impacts and climate changes speed up evolution at certain times as they create rapid changes in the environment which require rapid adaptation to survive. Only certain species will survive and evolve. For example, when the Dinosaurs died, there was a sudden punctuation in the evolutionary path of all other organisms because they were now able to adapt to fill newly available niches.

Option D.6 The Hardy-Weinberg Principle
Only Higher Level

D.6.1 Describe an adaptation in terms of the change in frequency of a gene's alleles.

 * If an allele increases the chances of survival and reproduction of individuals that possess it, the frequency of the allele in the gene pool will tend to increase.
 * If a moth inherits an allele of a gene that would make it bright pink in a dark brown environment - it would stand out in its environment and be eaten. This would cause the bright pink allele to decline in frequency.

D.6.2 Explain how the Hardy-Weinberg equation (p squared + 2pq +q squared = 1) is derived.

p + q = 1
 * In a population where two alleles occur, p represents the frequency of the dominant allele and q the frequency of the recessive allele. Thus the combined frequency of the alleles must account for 100% of the genes for that particular locus in the population.

p² + 2pq + q² = 1 AA + Aa + aA + aa = 1
 * When gametes combine their alleles to form zygotes, the probability of AA (A being the dominant allele) is p squared. The probability of aa is q squared. There are two ways in which an Aa genotype can arise, depending on which parent contributes the dominant allele. Thus, the frequency of heterozygous individuals is 2pq. The frequency of genotypes must add to 1. Thus,

D.6.3 Calculate allele, genotype and phenotype frequencies for two alleles of a gene, using the Hardy-Weinberg equation.

 * If a population is known to be following the Hardy-Weinberg Principle, the Hardy-Weinberg can be used to calculate unknown frequencies. An example of this is a gene in which two alleles controls the ability to taste phenylthiocarbamide (PTC). The ability to taste PTC is due to the dominant allele (T) and non-tasting is due to the recessive allele (t).

1600 people were tested in a survey. 461 were non-tasters - a frequency of 0.288. Their genotype was homozygous recessive (tt). If q = frequency of t allele, q squared = .288 so q = .537 If p = frequency of T allele, p = (1 - q) = .463 The frequency of homozygous dominants (TT) and heterozygotes (Tt) can be calculated. p squared = frequency of homozygous dominants p squared = (.463)(.463) = .214 2pq = frequency of heterozygotes 2pq = 2 (.463)(.537) = .497

D.6.4 State that the Hardy-Weinberg principle can also be used to calculate allele, genotype and phenotype frequencies for genes with two alleles.

 * The Hardy-Weinberg principle can also be used to calculate allele, genotype and phenotype frequencies for genes with two alleles.

D.6.5 State the Hardy-Weinberg principle and the conditions under which it applies.

p + q = 1 If there is random mating in a population, the chance of inheriting two copies of the first of the two alleles is (p)(p). The chance of inheriting two copies of the second of the two alleles is (q)(q). The expected frequency of the heterozygous genotype is 2pq. The sum of     p² + 2pq + q² = 1
 * If there are two alleles of a gene in a population, there are three possible genotypes - homozygous for each of the two alleles and heterozygous. The frequency of the two alleles in the population is usually represented by the letters p and q. The total frequency of the alleles in the population is 1.0, so

 * If gene frequencies change, then evolution occurs

D.6.6 Describe one example of transient polymorphism and sickle cell anemia as an example of balanced polymorphism.

 * Polymorphism is when two or more forms of a phenotype are represented in high enough frequencies to be readily noticeable.
 * A balanced polymorphism is a polymorphism in which the frequencies of the characteristics remain fairly constant over time. For example, the recessive allele for sickle cell anemia is very damaging when it is present in the homozygous recessive form because the person only has sickle shaped red blood cells. However, in the heterozygous form, it protects the person from malaria. That is why it continues to exist instead of being bred out. In populations exposed to malaria it will increase in the population until the homozygous recessive form becomes too common. Then it will remain at a balanced frequency.
 * A transient polymorphism is one that is changing in frequency over time. For example, if a population with a high incidence of the sickle cell anemia gene moves to a location where malaria is no longer present the gene will begin to disappear because the selective pressure in its favour has disappeared but the pressure against it (the homozygous form) continues. Over time its frequency will fall.

title: IB Biology/Option D - Evolution


text: User talk:219.140.228.13

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

title: User talk:219.140.228.13


text:
User talk:Jleonar

Copyright problem with Image:1Dwave graph2.png
--Derbeth 11:09, 5 October 2005 (UTC)

 * How did you get Image:Cherrybomb muffler2.jpg? It's a very nice picture, but if you add license to an image, you should either write that you're the author or provide link to source. BTW, you have some older images (like Image:Resistor11.jpg - look at your "my contributions" page) that are still not tagged. Can you fix it if you have some time? Thank you. --Derbeth talk 12:28, 27 November 2005 (UTC)

 * Please stop uploading images without copyright information. You must have seen warning on the upload page: we delete images with unknown status and may block notorious uploaders of such images. Please complete information about all of your images. --Derbeth talk 06:58, 8 December 2005 (UTC)

title: User talk:Jleonar


text:
User talk:Darkonc

Copyright problem with Image:Ginger-smoothbody.png

 * No problem, it's not an urgent task. --Derbeth 11:49, 5 October 2005 (UTC)

I don't think you have to insert such note in your images. Look f.e. at Image:Ginger-smoothbody.png. There is "File history" section and everyone can see, who's the author, so you don't need to write it. If you want to add some information about yourself, write it on your user page.

The second thing: CC-BY-SA is a good licence, but unfortunately, it's incompatibile with GFDL. The best soution is either dual-licensing (GFDL + cc-by-sa-2.5) or using cc-by-2.5. You can also use PD-self, but I understand that not everyone is keen to release his work to public domain ;-)

If you want to be able to change description of all of your images at once, you can make a template like Darkonc/ImageCopyright and insert license and you notes there. But, as I said before, I don't think you need it --Derbeth 20:35, 5 October 2005 (UTC)

title: User talk:Darkonc


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User talk:ArtFusion

Copyright problem with Image:Les Demoiselles d'Avignon.jpg
--Derbeth 11:13, 5 October 2005 (UTC)

Pictures
As you can see at the top of this page, "All unlicensed images uploaded to wikibooks before November 1, 2006 will soon be deleted." You have uploaded images and some of them are untagged. You can see them here. Please review them and add appropriate tags. If you have any questions, feel free to ask me. --Think Fast 04:34, 13 November 2006 (UTC)

title: User talk:ArtFusion


text:
User talk:Mkpapp

Copyright problem with Image:Eq2.gif
--Derbeth 11:17, 5 October 2005 (UTC)

title: User talk:Mkpapp


text:
User talk:Henrikmk

Copyright problem with Image:Rebolbutton1.png
--Derbeth 11:33, 5 October 2005 (UTC)

 * Please stop uploading images without license information. If you don't correct descriptions of the images, they will be deleted. Reapeating cases of uploading images without proper license information may result in blocking you from editing. --Derbeth 15:32, 5 November 2005 (UTC)

title: User talk:Henrikmk


text:
User talk:Barsathi

Copyright problem with Image:Pt14.jpg
--Derbeth 11:34, 5 October 2005 (UTC)

Images
According to my records, you have uploaded at least 294 images to wikibooks, all of which are not properly licensed. All these images are going to be deleted unless you have them properly licensed. --Whiteknight (talk) (projects) 19:23, 3 November 2006 (UTC)

Image copyright problem with Image:Lat4.jpg
Thanks for uploading Image:Lat4.jpg. However, the image may soon be deleted unless we can determine the copyright holder and copyright status. The Wikimedia Foundation is very careful about the images included in Wikibooks because of copyright law (see Wikibooks' Copyright policy).

The copyright holder is usually the creator, the creator's employer, or the last person who was transferred ownership rights. Copyright information on images is signified using copyright templates. The three basic license types on Wikibooks are open content, public domain, and fair use. Find the appropriate template in Image copyright tags and place it on the image page like this:.

Please signify the copyright information on any other images you have uploaded or will upload. Remember that images without this important information or giving false information can be deleted by an administrator. If you have any questions, feel free to contact me, or ask them at the Study help desk. Thank you. --Derbeth 12:52, 9 November 2005 (UTC)

Image copyright
Thanks for uploading images on Wikibooks. However, the images may soon be deleted unless we can determine the copyright holder and copyright status. The Wikimedia Foundation is very careful about the images included in Wikibooks because of copyright law (see Wikibooks' Copyright policy).

The copyright holder is usually the creator, the creator's employer, or the last person who was transferred ownership rights. Copyright information on images is signified using copyright templates. The three basic license types on Wikibooks are open content, public domain, and fair use. Find the appropriate template in Image copyright tags and place it on the image page like this:.

Please signify the copyright information on all images you have uploaded or will upload. Remember that images without this important information or giving false information can be deleted by an administrator. If you have any questions, ask them at the Staff lounge. Thank you.--Az1568(Talk) 07:02, 20 January 2007 (UTC)

All your uploaded images were deleted
All the images you uploaded without license information in 2005 were deleted. You and the other authors of Diagnostic Radiology were given amply opportunity to either provide license information or find alternative images. The redlinks in Diagnostic Radiology were left behind as placeholders so you know what images are missing and where they go. Hopefully you'll be able to use that information to find alternatives. If you have concerns about these deletions, please contact me.  – Mike.lifeguard  | talk 15:11, 17 July 2007 (UTC)

title: User talk:Barsathi


text:
User talk:Apina

Copyright problem with Image:Electrolysis anode.png
--Derbeth 11:35, 5 October 2005 (UTC)

Pictures
As you can see at the top of this page, "All unlicensed images uploaded to wikibooks before November 1, 2006 will soon be deleted." You have uploaded images and some of them are untagged. You can see them here. Please review them and add appropriate tags. If you have any questions, feel free to ask me. --Think Fast 04:25, 13 November 2006 (UTC)

title: User talk:Apina


text: User talk:Shin31

title: User talk:Shin31


text:
User talk:Chrismo~enwikibooks

Copyright problem with Image:128px-Headlock leg.png
--Derbeth 11:36, 5 October 2005 (UTC)

Wikiversity
I saw your name on the Wikiversity participant's list, you may be very experienced in mathematics already, but I'm trying out instructing a Calculus course if you're interested, go to Wikiveristy, check out the mathematics department, pure mathematics, and in the course list is included Calculus. Fephisto 22:54, 9 July 2006 (UTC)

Your account will be renamed
Hello,

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

Unfortunately, your account clashes with another account also called Chrismo. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Chrismo~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 23:06, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 04:48, 19 April 2015 (UTC)

title: User talk:Chrismo~enwikibooks


text:
Template:Circuit Theory/Page

 Circuit Theory

This template serves as a page-header template for the  book. This template should appear at the top of every page in that book. It automatically includes the host page in :. This template changes shape for the Print Version of the circuit theory book. In the print version, the template appears only as two horizontal lines. Also, through some CSS, this template forces a page break before itself, so that each page starts on a fresh printed page.

title: Template:Circuit Theory/Page


text:
Metroid

Metroid is the first game in the of, and was released for the  in. The game was produced by, one of Nintendo's most famous game and hardware designers, and directed by. The game is also very notable for its music, which was composed by Hirokazu "Hip" Tanaka.

Table of Contents

 * /Gameplay/
 * /Controls/
 * /Walkthrough/

title: Metroid


text:
Topology/Product Spaces

Before we begin
We quickly review the set-theoretic concept of Cartesian product here. This definition might be slightly more generalized than what you're used to.

Definition
Let $$\Lambda$$ be an indexed set, and let $$X_\lambda$$ be a set for each $$\lambda \in \Lambda$$. The Cartesian product of each $$X_\lambda$$ is

$$\prod_{\lambda \in \Lambda}X_\lambda = \{x:\Lambda\rightarrow\bigcup_{\lambda \in \Lambda} X_{\lambda} | x(\lambda) \in X_\lambda\}$$.

Example
Let $$\Lambda = \mathbb{N}$$ and $$X_\lambda = \mathbb{R}$$ for each $$n \in \mathbb{N}$$. Then

$$\prod_{\lambda \in \Lambda} X_\lambda = \mathbb{R}^\mathbb{N} = \{x: \mathbb{N} \rightarrow \mathbb{R} \mid x(n) \in \mathbb{R}\, \forall\, n \in \mathbb{N}\} = \{(x_1, x_2, \ldots) \mid x_n \in \mathbb{R}\, \forall\, n \in \mathbb{N}\}$$.

Product Topology
Using the Cartesian product, we can now define products of topological spaces.

Definition
Let $$X_\lambda$$ be a topological space. The product topology of $$\prod_{\lambda \in \Lambda} X_\lambda$$ is the topology with base elements of the form $$\prod_{\lambda \in \Lambda} U_\lambda$$, where $$U_\lambda = X_\lambda$$ for all but a finite number of $$\lambda$$ and each $$U_\lambda$$ is open.

Examples

 * Let $$\Lambda = \{1,2\}$$ and $$X_\lambda = \mathbb{R}$$ with the usual topology. Then the basic open sets of $$\mathbb{R}^2$$ have the form $$(a,b) \times (c,d)$$:

 * Let $$\Lambda = \{1,2\}$$ and $$X_\lambda = R_l$$ (The Sorgenfrey topology). Then the basic open sets of $$\mathbb{R}^2$$ are of the form $$[a,b)\times [a,b)$$:

title: Topology/Product Spaces


text:
Cookbook talk:Chicken and Andouille Sausage Gumbo

Search
Why doesn't this show up if you do a search for Gumbo in the Cookbook??
 * Probably because the search indexes for wikibooks aren't updated that regularly. This is a relatively new page. Kellen T 17:22, 5 October 2005 (UTC)

 * Someone will have to choose if that's a soup or a stew, and why it's claiming to be a gumbo sans orka. Nmontague 18:49, 5 October 2005 (UT C)

Okra & Mysterious Ingredient
Not sure what you mean 'claiming to be a gumbo sans okra'? Most gumbos don't have okra but some do. My vote, and most from Lousiana would say gumbo is more soup than stew. For what its worth. How do you get that entry attribution after your comment? Is it automatic? (user Tchoup, unsigned)

I'll have to ask a co-worker from there. Currently, the recipe suggests a mysterious ingredient that I believe has been banned by the FDA as a poison. (cancer or kidney problems if I remember right) AlbertCahalan 00:48, 7 October 2005 (UTC)

You sign comments with 4 tilde ('~') characters in a row. On a qwerty keyboard, look to the left of the exclamation point. AlbertCahalan 00:48, 7 October 2005 (UTC)

I am from Lousiana. Filé, made from Sassafras root is your mysterious ingredient. The research is debatable, cancer inlarge doses in rats. In any case, the base of gumbo is the trinity and a roux (dark or light/oil or butter). Okra, or any other ingredient is just a style of gumbo. Seafood, okra, etc. File, as well, and it is only added to one's bowl. Never to the gumbo. I made the original entry for this gumbo. Pretty exciting. I no longer feel I can change it though. Who am I to make changes to something like a recipe.Tchoup 11:06, 7 October 2005 (UTC)

 * As far as I can remember, I've never heard of a gumbo without okra, because okra is gumbo. That's where the name comes from, that's how you know it's a gumbo.  Course, being Canadian, my views of food are bias towards how I've been taught foods are supposed to be and I've never been served a gumbo by an actual cajun.  Gumbos I've seen boarder between the definitions of soup and stew that I know, stews being the thicker, simpler, slow cooked, less spiced and more ingredient-flavour based meals and soups being lighter, spiced and a side dish.  The gumbo I've had was not so think but not really spiced and didn't really take any definate, "oh, that's X," kind of feel to it.  Nmontague 04:40, 17 October 2005 (UTC)

 * Originally, the name did come from it being a soup of gumbo (okra), but it obviously has taken on another meaning. Agree about file only being added to the bowl -- it's just wrong to add it to the entire soup! File is also only used in gumbo without okra (i.e. chicken and andouille); okra itself is the thickener in gumbo. Alainna 05:21, 26 November 2006 (UTC)

Should I create a different page for seafood gumbo and gumbo z'herbes? Alainna 05:21, 26 November 2006 (UTC)

Filé powder is made from sassafrass leaves, not the roots. The leaves do not contain significant amounts of the carcinogenic safrole. And as a long-time neighbor of Louisiana who has eaten many gallons of gumbo, I can say that okra is not a necessary or even preferred ingredient. -- 66.25.244.7 07:36, 15 July 2007 (UTC)

List of Ingredients
Am amused to see that a recipe for gumbo that duly mentions okra in the intro and has plenty discussions re here on the talk page does not have okra listed in the ingredients for the recipe! OoEtMePasty (talk) 15:16, 21 March 2008 (UTC)
 * Same for chicken stock, which the recipe first stipulates, never to mention it again. I'm sorry, this recipe sucks, please fix ASAP! Maikel (talk) 00:37, 29 April 2008 (UTC)

Rant
I've moved this unwikified edit here from the "Notes" segment of the article: ''Experiment with onion powder, garlic powder, seasoned salt, black pepper, white pepper, bay leaves, even cumin. Gumbo is NEVER about recipes. Look in your fridge, there is something thats will go bad in the next couple of days. Throw it in the gumbo, trust me. You will never learn to make "for real" gumbo from a recipe. Experiment!. Its like our music, improvisation is the ideal, not some perfect recipe. Sometimes, when you are young, your gumbo won't always be awesome, but when you get to be 30 or 40, you will be able to create a bitchin' gumbo anywhere in the world from available ingredients. Please, experiment. Down here, we argue all the time about what a proper gumbo is. Put in some potatoes or tomatoes or carrots (To represent the Irish and Spanish!). Potatoes in particular will soak up the flavor of the gumbo. By the way, the cooking time of the gumbo listed above as one hour is ridiculous. Gumbo should cook for at least 3 to 4 hours. TRUST ME!'' Maikel (talk) 00:45, 29 April 2008 (UTC)


text:
User talk:194.80.20.10

Welcome to Wikibooks. We invite everyone to contribute constructively to our collection. Take a look at the welcome page if you would like to learn more about contributing. However, unconstructive edits are considered vandalism, and if you continue in this manner you may be blocked from editing without further warning. Please stop, and consider improving rather than damaging the hard work of others. Thanks. -- Herby talk thyme 08:46, 9 May 2007 (UTC)

title: User talk:194.80.20.10


text:
Circuit Theory/Resistors

Resistors
Mechanical engineers seem to model everything with a spring. Electrical engineers compare everything to a Resistor. Resistors are circuit elements that resist the flow of current. When this is done a voltage appears across the resistor's two wires.

A pure resistor turns electrical energy into heat. Devices similar to resistors turn this energy into light, motion, heat, and other forms of energy.

Current in the drawing above is shown entering the + side of the resistor. Resistors don't care which leg is connected to positive or negative. The + means where the positive or red probe of the volt meter is to be placed in order to get a positive reading. This is called the "positive charge" flow sign convention. Some circuit theory classes (often within a physics oriented curriculum) are taught with an "electron flow" sign convention.

In this case, current entering the + side of the resistor means that the resistor is removing energy from the circuit. This is good. The goal of most circuits is to send energy out into the world in the form of motion, light, sound, etc.

Resistance
Resistance is measured in terms of units called "Ohms" (volts per ampere), which is commonly abbreviated with the Greek letter Ω ("Omega"). Ohms are also used to measure the quantities of impedance and reactance, as described in a later chapter. The variable most commonly used to represent resistance is "r" or "R".

Resistance is defined as:

 * $$r = {\rho L \over A}$$

where ρ is the resistivity of the material, L is the length of the resistor, and A is the cross-sectional area of the resistor.

Conductance
Conductance is the inverse of resistance. Conductance has units of "Siemens" (S), sometimes referred to as mhos (ohms backwards, abbreviated as an upside-down Ω). The associated variable is "G":

 * $$G = \frac{1}{r}$$

Before calculators and computers, conductance helped reduce the number of hand calculations that had to be done. Now conductance and it's related concepts of admittance and susceptance can be skipped with matlab, octave, wolfram alpha and other computing tools. Learning one or more these computing tools is now absolutely necessary in order to get through this text.

Resistor terminal relation

The drawing on the right is of a battery and a resistor. Current is leaving the + terminal of the battery. This means this battery is turning chemical potential energy into electromagnetic potential energy and dumping this energy into the circuit. The flow of this energy or power is negative.

Current is entering the positive side of the resistor even though a + has not been put on the resistor. This means electromagnetic potential energy is being converted into heat, motion, light, or sound depending upon the nature of the resistor. Power flowing out of the circuit is given a positive sign.

The relationship of the voltage across the resistor V, the current through the resistor I and the value of the resistor R is related by ohm's law:

 * $$V=R*I$$

A resistor, capacitor and inductor all have only two wires attached to them. Sometimes it is hard to tell them apart. In the real world, all three have a bit of resistance, capacitance and inductance in them. In this unknown context, they are called two terminal devices. In more complicated devices, the wires are grouped into ports. A two terminal device that expresses Ohm's law when current and voltage are applied to it, is called a resistor.

Resistor Safety
Resistors come in all forms. Most have a maximum power rating in watts. If you put too much through them, they can melt, catch fire, etc. Resistance is an electrical passive element which oppose the flow of electricity. Air can be considered as one of the resistances. Air is used as a resistance in Lightning Arrestors. Also vacuum also is used as a resistor.

Example
Suppose the voltage across a resistor's two terminals is 10 volts and the measured current through it is 2 amps. What is the resistance?

If $$ v=iR $$ then $$ R = v/i = 10V/2A = 5 ohms$$

title: Circuit Theory/Resistors


text:
Japanese/Lessons/Simultaneous action

Simultaneous action is designated by replacing the "〜ます" ending of the polite present positive tense of the first verb with "〜ながら". The tense is then determined by the latter verb. Both actions should be performed by the same person.

Those more comfortable thinking in terms of the dictionary form can see this as adding "〜ながら" to the stem of vowel-stem verbs, or the い-row of consonant-stem verbs (e.g. 飲む -> 飲みながら example below). The irregular verbs are also put in the い-row making them identical to the consonant-stem verbs.

The tense of the sentence is given with the second verb.

title: Japanese/Lessons/Simultaneous action


text:
Wikibooks talk:Book of the month/November 2005 voting

It is not cleart how to vote. Where is ' the appropriate section? Do you actually put the characters #````. Where do they go? I'm confused.

 * I'll try to clear that up on the instruction page. You can use the # character to make an ordered list, and to "sign" you name, use the tilda characters (~).  You can type in  to sign your name from what you are logged in as, and use four tildas ~ to put a date and time stamp on the signature automatically.  Yes, this can be confusing as the MediaWiki software changes these symbols after you do the edit, and a new user can't find out how to do this.  Specific details are found on the editing help window that is linked at the bottom of the edit window.  --Rob Horning 04:21, 9 October 2005 (UTC)


text:
Talk:Trigonometry/Radians

Suitability for those new-to-Radians
This page in its current form is unsuitable for anyone new to radians. It does not explain why we would want to use radians (admittedly a difficult sell at this stage, with no calculus). Also I have never seen the crazy $$\displaystyle\theta^c$$ notation. JamesCrook (talk) 21:04, 27 October 2010 (UTC)

Picture is wrong
It is obvious that the picture of an angle of 1 radian doesn't really show a 1 radian angle.130.89.219.179 15:46, 5 October 2005 (UTC)

Keeping the Units Right
I changed the formulas for converting, because I didn't understand them. Let's look at an example:

 * $$\theta=60^\circ$$

Now converting, using the original formula:

 * $$\theta^\circ \frac {\pi}{180}=(60^\circ)^\circ \frac {\pi}{180} = (\frac {\pi}{30})^{\circ\circ}$$

Let's try it another way:

 * $$\alpha =\theta^\circ = 60^\circ$$

Converting:

 * $$\theta^\circ \frac {\pi}{180}=60^\circ \frac {\pi}{180} = (\frac {\pi}{30})^\circ$$

The answer is in degrees!?

Now suppose the answer should be the number of radians. What is the meaning of the degree symbol in the formula?

130.89.219.179 16:25, 5 October 2005 (UTC)

When you changed 60 degrees to radians, your formula was wrong. The part:

Theta*pi/180

is wrong. It is supposed to be

Theta*pi/180°, so that the degree symbol is eliminated.

Correcting your equation,

Theta*pi/180°=60° *pi/180°=pi/30

Centesimal system
The principal unit in system is grade and is denoted by (g).One right angle is divided in 100 equal parts,called grades,and each grade is subdivided into 100 minutes,and each minutes,and each minutes into 100 seconds.

IN MATHEMATICAL FORM : ONE RIGHT ANGLE = 100grade 1 grade = 100' 1'     = 100".

title: Talk:Trigonometry/Radians


text:
Talk:How to Pass a Course/Studying

Just added some comments on taking notes while reading. This is a subject matter that I think isn't talked about enough, and perhaps should have an entire section to it's self because of it's importance.

title: Talk:How to Pass a Course/Studying


text: Template:NPOV note

title: Template:NPOV note


text:
User talk:Benjamin Gatti~enwikibooks

Please capitalize VB code as in the VB IDE. Also, remember that the book is about programming in VB Classic not VB.Net. Nonetheless, thanks for contributing. --kwhitefoot 16:51, 5 October 2005 (UTC)

Your account will be renamed
Hello,

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

Unfortunately, your account clashes with another account also called Benjamin Gatti. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Benjamin Gatti~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 23:02, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 04:38, 19 April 2015 (UTC)

title: User talk:Benjamin Gatti~enwikibooks


text: User talk:216.120.190.137

title: User talk:216.120.190.137


text: User talk:212.45.45.13

title: User talk:212.45.45.13


text:
How to Pass a Course/Taking notes

Taking notes is an extremely important aspect of studying that is often neglected, despite the fact that it can make the difference between a moderate grade and an excellent one. Taking notes can involve more cognitive or "thought work" than the common alternative of merely highlighting important terms or phrases in a text, but it also produces better learning. Its benefits happen by focusing attention on the material in two ways: initially at the time of reading it, and later at the time of reviewing the notes.

Why Take Notes?
Takings notes provides a person with a way to review material learned. Reviewing and reading notes allows a person a chance to know what is on an exam, test, or pop quiz. One of the more important reasons to take notes, is to create a placemark. This placemark allows the note taker to know when the lecturer stopped giving a lecture for the day.

For example: ''Did the teacher stop midway between Chapter 10.. or are we on Chapter 11 now?''

In this make-believe situation: You review the notes and notice how the ending material of the notes are similar to the material seen before the ending part of Chapter 10. Therefore, you figure the teacher did not lecture on Chapter 11 yet... or else you dazed out really bad the last half of class.

Knowing this gives you a chance to read ahead or study certain materials, and place emphasis on studying a particular section of notes and book reading.

Notes are also a way of reviewing what you have learned in the lecture. During the lecture, a teacher may put emphasis on a particular term or word. These particular terms or words may be found on an oncoming test or quiz. Studying materials on notes gives a person a chance to know what the teacher expects you to know, now that the teacher has presented you with that material.

If the material is on the notes, the teacher expects you to know the material is on the notes.
 * Therefore, there is a good chance the material will be on a test.

The Voice Recorder
Besides the obvious paper and pencil, many technological additions of the past decades allow for someone to take notes with deadly accuracy.

 * Digital Voice Recorder
 * OR Analog Voice Recorder (recorders that use casette tapes)

These additions save effort (and reduce writers' cramp) at the time of listening to a lecture, but they also have two kinds of limitations. One kind is practical--getting good quality sound (see below for more discussion of this challenge).

Another kind of limitation is the time needed, sooner or later, to listen to recorded notes. Since speech proceeds at a relatively fixed rate, it is hard to do "speed listening" equivalent to "speed reading" of notes that are written. This fact of life puts a premium on abbreviating voice recordings of notes--though it should also be noted that the physical work of writing puts a premium on abbreviating written as well. Since taking notes is work either by writing or by recording, the best advice is to use whichever medium feels most comfortable. Learning recorder

Buying a Voice Recorder
The voice recorder is essential to any student who wants to take accurate notes. More importantly is a voice recorder with the ability to upload voice files to a computer console. Often when looking into buying a voice recorder, the choice can be complicated and indefinite. Fore, one must buy and try before deciding on one to buy. However, due to the technical limitations provided by manufacturers and stores, reviews and other word of mouth advice seems to be the supplies provided in obtaining a decent voice recorder.

Many things should be considered when buying a voice recorder.

 * How long does the voice recorder record?
 * * Can it record for two hours or more?
 * Some classes last two hours and thirty minutes (2:30 hours)
 * A recorder that can record for this long is essential.

 * Can the batteries be replaced or do the batteries need to be recharged?
 * * What type of batteries does the recorder take?
 * Many devices "eat up" battery electrolytes (power) quickly.
 * Getting a recorder with easy to replace batteries found in common stores is suitable for replenishing the supply of power.

 * Can the recordings be saved to a computer?
 * Being able to put files on a computer allows for a person to sort lectures according to dates.

Using the Voice Recorder
Getting a voice recorder saves a person a lot of time taking notes. If a teacher is mostly lecture, as in, the teacher rarely writes on the chalkboard/whiteboard, then it would be best to get a voice recorder to save your hand from developing arthritis.

Positioning inside a Class Room
The typical area of a placement is one of the greatest factors to consider when using a voice recorder. Observation is the key to finding a good place to set the recorder.

If you notice that a teacher does not past a certain line of boundary, say past his or her desk, then you could possibly put the recorder on his or her desk. (ask if you may first)

However, if the teacher walks around the room this creates problems. If a teacher walks past the desk, and his or her back is facing towards the desk, then the teacher's voice will not reach the voice recorder as effective--and the voice may not be recorded at all.

Often placing the recorder one's own desk is suitable if better positioning cannot be found elsewhere.

However, to have a recorder sitting on a desk while jotting down notes can cause interference when voice and other sounds are being recorded. Often the writings and scribbles cause a sound of scratching on the paper which will be picked up on the recorder. At other times, the sound of flipping a page on a binder--to go to the next clean sheet for writing notes--will be picked up on the recorder. Another thing to consider is the sound of the arm moving against a desk, while the hand goes downwards on a sheet of paper, as notes are being written. Often the rubbing sound of an arm against a desk and the hand against the paper will be recorded.

The remedies recommended for these problems are as follows:

 * Use an inkpen.
 * Inkpens glide on paper. Also, pencil writings tend to fade with time.
 * Pencil is often a bad writing utensil for notes. This is why I suggest an erasable pen.

 * Wear long sleeves
 * Human skin is like leather at times. Clothing will create less friction on the desk.

 * Use an ID card under your hand.
 * This is an art technique I learned long ago. It allows the hand to be placed on the ID, as the ID easily slides on paper. Sometimes it works; sometimes it does not.

 * Listen carefully to the person giving a lecture.
 * I've found that professors at times go off at a tangent. The best thing to do is notice when there is a break in his or her logic being conversated, and quickly flip the page. It is at times best to bring the notebook off the desk, away from the recorder, and then flip the page to reduce recorded noise.

Uploading the Files
I personally like to upload all files to a computer. With one or more file belong to a particular date, say Feb-26-07, I put those files in a computer folder labeled: Feb-26-07 From there, I make a backup copy of the files inside the folders and put them on a CD.

 * CD-Drives are more common to find around schools than DVD-Drives. Therefore, a CD is more suitable than a DVD storage medium.

Also, I create a backup of that CD.
 * Take the files inside the CD and put those onto a storage medium: considerably another CD.
 * After burning a CD, make a backup on a DVD (if possible -- if not, then a CD) and then store that disc somewhere safe. I like to put things in my top dresser drawer. Always make backups of a lecture, this is very important. Backups are gravely important!
 * Another thing to consider is if you use a program to listen to the audio files. If you use a program to listen to the audio files, it would be best to find the setup program of that recorder. Take the setup program and put the program on the CD that you will carry around with you. Also, put a copy of the audio listening setup program on the backup to always have the setup program on hand.
 * Sometimes the setup programs can be found on the installation CD that comes with a voice recorder.
 * Other times, a person could call Technical Help to help them find the program from the Internet and download it.

Never take the original with you! Always take the Backup.

Take the backup with you around the campus and listen to the audio files at a campus computer when you want to study/review.

Uploading with an Analog Recorder
This is a guide if someone wishes to create a backup. Otherwise, a person could simply find a safe place to store his or her cassette tape and access when he or she wishes to listen to it.

Make a backup of the cassette tape recording.

Typically one uses the Line Out feature of a recorder and plugs a wire into the Line Out hole.

The wire then extends to the Line In hole of another device (computers, boombox, etc.) and then inserts that part of the wire into the Line In.

(Recorder) Line Out <> Line In (Boombox)

Typically one pushes play on the analog voice recorder and pushes record on the boombox. Label the cassette tape with the date of the recording and set that somewhere safe.

You don't technically need to make a backup, however, I always like the idea of making backups.

I could be wrong, if I am, please someone correct this entry.

Getting the teacher to repeat something
Often times a person in the classroom will cough harshly. Coughs are typically very loud and will block out any voice coming into the recorder. Many times, other students could not hear what the teacher said during the time someone in the class coughed. In consideration, what a person should do is ask the teacher to repeat what he or she said. Do not fear asking the teacher the question, "Could you please repeat that?" because many other students may not have heard the teacher over the other student's sneeze, cough, or disruptive movements.

With the Chalkboard
A method that is very well used when taking notes and using a voice recorder can help a person backtrack and review notes. Often times a teacher will give lectures and write on the board rarely. When a teacher writes on the board, it is best to look at the voice recorder and take note of how much time has expired and write the current recording time on the note page.

For example:

Let's say the teacher has been talking about studies in psychology without writing anything on the board for a while.

Finally, the teacher writes something on the board.

 * Phallic Stage and male idea of swords

The recorder has been on for about one hour and thirty minutes. The recorder displays (1:30:15) with :15 being 15 seconds.

 * Write the time displayed on the notepage. (1:30:15)
 * Then, copy the material on the chalkboard onto the notepage.

Without the Chalkboard
Sometimes the teacher doesn't even use the chalkboard. However insane that may be, some may not. In that situation, a voice recorder would be highly regarded as necessary.

As the teacher talks, you need to find placemarks that help you backtrack to certain parts of the lecture.

For instance, the teacher breaks the class lecture into three parts:
 * Intro to baskets
 * how to choose weaving material
 * weaving the basket

Subparts are included, however, he or she does not describe them to you. Thus, as the lecture goes on, you'll need to find a way to backtrack to certain material that he or she has spoken about.

If you don't want to listen to the whole recording again, it would be wise write that time down. That way, you can backtrack and review that material.

Let's say the teacher talks about Native American weaving techniques. You know this part is going to be on the test.

You look at the voice recorder displaying (1:45:19)

You jot down on your notebook

 * (1:45:19) Native American Techniques.

Problems with recording Lectures
There are many disadvantages to using a recorder. When recording a lecture, one must re-listen to the entire lecture. This means using another 2 and 1/2 hours. However, you could use a tape recorder if you intend to re-use a small clip in you're private study.

Taking notes while reading the text book
Taking notes while reading your text can be the most critical part of passing a class. In most classes the majority of information learned is taken from the text book, thus it is critical you retain as much of the information as possible. The mechanics of taking notes while reading a text book can be broken down further into various techniques.

Type or write?
Typing your notes or writing your notes is a matter of personal preference. If you can type faster than you write, you might prefer typing over writing. Also typing your notes makes sure that they are legible. However if you do plan to type your notes make sure that you have your book, monitor, and keyboard in a comfortable position to avoid neck or back cramps while note taking. Writing out your notes has the advantage of allowing free-form drawing. Something that is possible when typing, but can be cumbersome.

The Bookmark Technique
Often words/terms in bold--or words/terms that are important--are read. The best thing a person can do is take a sheet of writing paper and fold it up until it forms a makeshift bookmark. Then, when coming across terms, write the word found in bold on the bookmark writing line with the page number. Sometimes putting down names read and the page number the information about the person is on--does well.

This way, a person can track what page he or she is on. Also, it provides a way for someone to glance quickly at the bookmark and try to remember what the term means. If he or she does not remember what the word/term means, he or she can simply turn back to the page and regrasp the material.

The best part of this technique is that you can fold the bookmark up and put it in your pocket while going from place to place. All someone has to do is take the bookmark out of his or her pocket, look at it, and think hard about the terms and words on the bookmark. This is a great way to quickly review.

Copying the text verbatim
Copying text from the text book verbatim is the least perferred method of taking notes, but it can be somewhat helpful if it is the only option available.

Copying definitions verbatim
When reading a Math or Science text, copying definitions verbatim is appropriate, and required. Science and Math are build upon precise definitions, so modifying in any way will distort key concepts. The definitions are so important that copying them to flash cards for memorization is encouraged if time permits.

Paraphrasing concepts and paragraphs
Paraphrasing concepts is the ideal way to take notes. It will force you to actually think about what you read. For Math and Science texts, extract ALL definitions and relevant theorems from paragraphs, and organize them into lists and sublists. This will make sure you know they exist. When reading all other texts, try to summarize ALL concepts. Summarizing paragraphs will help too, but isn't as effective because the author of the text has already done some of the organization for you. You may think that summarizing all concepts is a bit extreme, but often in a class you don't know what will be on the test and what won't, so if you want a high grade, this is the only way to be sure.

What if I don't understand something in the text?
The following is just a partial list of what you can do in this situation.

 * 1) Ask your instructor for clarification.  You can call or email your instructor before the next time class meets, or just wait until class.  This is usually a very good option if you only have a small number of questions.  Unfortunately instructors tend to be very busy and are difficult to get in touch with.
 * 2) Ask a friend or class mate.  This is a good option only if your friend knows the material better than you do.  If not then you will have a situation of the blind leading the blind.
 * 3) Get another text book from the library that covers the same subject.  This is an excellent option because text book explanations are usually well thought out,  and there are often many text books in the library on the same subject, which all describe the same concepts, but from different points of view, and in different wording.

Asking yourself, and answering questions
This is an EXTREMELY good technique. Once you read a section of text, ask your self as many questions as time allows about what you have read. Try to answer them based on the content of the book. It is good to do this as you go along, because it will make sure that your truly understood read before you go on to the next section. This technique is essential if you are reading from texts that don't have a lot of questions in them, or have questions that don't have answers in the back.

Taking notes during lecture
Shorthand can be helpful. Following along in the derivation of a solution to a math problem is a very good way to stay focused.

Helpful Links

 * Notemaking Tools
 * Study Skills Library California Polytechnic State University, San Luis Obispo

title: How to Pass a Course/Taking notes


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Circuit Theory/Laplace Transform

Laplace Transform
The Laplace Transform is a powerful tool that is very useful in Electrical Engineering. The transform allows equations in the "time domain" to be transformed into an equivalent equation in the Complex S Domain. The laplace transform is an integral transform, although the reader does not need to have a knowledge of integral calculus because all results will be provided. This page will discuss the Laplace transform as being simply a tool for solving and manipulating ordinary differential equations.

Laplace transformations of circuit elements are similar to phasor representations, but they are not the same. Laplace transformations are more general than phasors, and can be easier to use in some instances. Also, do not confuse the term "Complex S Domain" with the complex power ideas that we have been talking about earlier. Complex power uses the variable $$\mathbb{S}$$, while the Laplace transform uses the variable s. The Laplace variable s has nothing to do with power.

The transform is named after the mathematician Pierre Simon Laplace (1749-1827). The transform itself did not become popular until Oliver Heaviside, a famous electrical engineer, began using a variation of it to solve electrical circuits.

Laplace Domain
The Laplace domain, or the "Complex s Domain" is the domain into which the Laplace transform transforms a time-domain equation. s is a complex variable, composed of real and imaginary parts:

 * $$s = \sigma + j\omega$$

The Laplace domain graphs the real part (σ) as the horizontal axis, and the imaginary part (ω) as the vertical axis. The real and imaginary parts of s can be considered as independent quantities.

The similarity of this notation with the notation used in Fourier transform theory is no coincidence; for $$\sigma=0$$, the Laplace transform is the same as the Fourier transform if the signal is causal.

The Transform
The mathematical definition of the Laplace transform is as follows:

 * $$F(s)

= \mathcal{L} \left\{f(t)\right\} = \int_{0^-}^\infty e^{-st} f(t)\,dt$$

The transform, by virtue of the definite integral, removes all t from the resulting equation, leaving instead the new variable s, a complex number that is normally written as $$s=\sigma+j\omega$$. In essence, this transform takes the function f(t), and "transforms it" into a function in terms of s, F(s). As a general rule the transform of a function f(t) is written as F(s). Time-domain functions are written in lower-case, and the resultant s-domain functions are written in upper-case.

we will use the following notation to show the transform of a function:

 * $$f(t) \Leftrightarrow F(s)$$

We use this notation, because we can convert F(s) back into f(t) using the inverse Laplace transform.

The Inverse Transform
The inverse laplace transform converts a function in the complex S-domain to its counterpart in the time-domain. Its mathematical definition is as follows:

\mathcal{L}^{-1} \left\{F(s)\right\} = {1 \over {2\pi}}\int_{c-i\infty}^{c+i\infty} e^{ft} F(s)\,ds = f(t)$$

where $$c$$ is a real constant such that all of the poles $$s_1,s_2,...,s_n$$ of $$F(s)$$ fall in the region $$\mathfrak{R}\{s_i\} < c$$. In other words, $$c$$ is chosen so that all of the poles of $$F(s)$$ are to the left of the vertical line intersecting the real axis at $$s=c$$.

The inverse transform is more difficult mathematically than the transform itself is. However, luckily for us, extensive tables of laplace transforms and their inverses have been computed, and are available for easy browsing.

Transform Properties
The most important property of the Laplace Transform (for now) is as follows:

 * $$\mathcal{L} \left\{ f'(t) \right\} = sF(s) - f(0)$$

Likewise, we can express higher-order derivatives in a similar manner:

 * $$\mathcal{L} \left\{f''(t)\right\} = s^2F(s) - s f(0) - f'(0) $$

Or for an arbitrary derivative:

 * $$\mathcal{L} \left\{f^{(n)}(t)\right\} = s^nF(s) - \sum_{i=0}^{n-1} s^{(n-1-i)} f^{(i)}(0) $$

where the notation $$ f^{(n)}(t) $$ means the nth derivative of the function $$ f $$ at the point $$ t $$, and $$ f^{(0)}(t) $$ means $$ f(t) $$.

In plain English, the laplace transform converts differentiation into polynomials. The only important thing to remember is that we must add in the initial conditions of the time domain function, but for most circuits, the initial condition is 0, leaving us with nothing to add.

For integrals, we get the following:

 * $$\mathcal{L}\left\{ \int_0^t f(t)\, dt \right\} = {1 \over s}F(s)$$

Initial Value Theorem
The Initial Value Theorem of the laplace transform states as follows:

 * $$f(0) \Leftrightarrow \lim_{s \to \infty} sF(s)$$

This is useful for finding the initial conditions of a function needed when we perform the transform of a differentiation operation (see above).

Final Value Theorem
Similar to the Initial Value Theorem, the Final Value Theorem states that we can find the value of a function f, as t approaches infinity, in the laplace domain, as such:

 * $$\lim_{t \to \infty} f(t) \Leftrightarrow \lim_{s \to 0} sF(s)$$

This is useful for finding the steady state response of a circuit. The final value theorem may only be applied to stable systems.

Transfer Function
If we have a circuit with impulse-response h(t) in the time domain, with input x(t) and output y(t), we can find the Transfer Function of the circuit, in the laplace domain, by transforming all three elements:

In this situation, H(s) is known as the "Transfer Function" of the circuit. It can be defined as both the transform of the impulse response, or the ratio of the circuit output to its input in the Laplace domain:

 * $$H(s) = \mathcal{L} \left\{h(t) \right\} = \frac{Y(s)}{X(s)}$$

Transfer functions are powerful tools for analyzing circuits. If we know the transfer function of a circuit, we have all the information we need to understand the circuit, and we have it in a form that is easy to work with. When we have obtained the transfer function, we can say that the circuit has been "solved" completely.

Convolution Theorem
Earlier it was mentioned that we could compute the output of a system from the input and the impulse response by using the convolution operation. As a reminder, given the following system:

We can calculate the output using the convolution operation, as such:

 * $$y(t) = x(t) * h(t)$$

Where the asterisk denotes convolution, not multiplication. However, in the S domain, this operation becomes much easier, because of a property of the laplace transform:

 * $$\mathcal{L} \left\{ a(t) * b(t) \right\} = A(s)B(s)$$

Where the asterisk operator denotes the convolution operation. This leads us to an English statement of the convolution theorem:

Now, if we have a system in the Laplace S domain:

We can compute the output Y(s) from the input X(s) and the Transfer Function H(s):

 * $$Y(s) = X(s)H(s)$$

Notice that this property is very similar to phasors, where the output can be determined by multiplying the input by the network function. The network function and the transfer function then, are very similar quantities.

Resistors
The laplace transform can be used independently on different circuit elements, and then the circuit can be solved entirely in the S Domain (Which is much easier). Let's take a look at some of the circuit elements:

Resistors are time and frequency invariant. Therefore, the transform of a resistor is the same as the resistance of the resistor:

 * $$R(s) = r$$

Compare this result to the phasor impedance value for a resistance r:

 * $$Z_r = r \angle 0$$

You can see very quickly that resistance values are very similar between phasors and laplace transforms.

Ohm's Law
If we transform Ohm's law, we get the following equation:

 * $$V(s) = I(s)R$$

Now, following ohms law, the resistance of the circuit element is a ratio of the voltage to the current. So, we will solve for the quantity $$\frac{V(s)}{I(s)}$$, and the result will be the resistance of our circuit element:

 * $$R = \frac{V(s)}{I(s)}$$

This ratio, the input/output ratio of our resistor is an important quantity, and we will find this quantity for all of our circuit elements. We can say that the transform of a resistor with resistance r is given by:

 * $$\mathcal{L}\{\text{resistor}\} = R = r$$

Capacitors
Let us look at the relationship between voltage, current, and capacitance, in the time domain:

 * $$i(t) = C\frac{dv(t)}{dt}$$

Solving for voltage, we get the following integral:

 * $$v(t) = \frac{1}{C}\int_{t_0}^{\infty} i(t)dt$$

Then, transforming this equation into the laplace domain, we get the following:

 * $$V(s) = \frac{1}{C} \frac{1}{s} I(s)$$

Again, if we solve for the ratio $$\frac{V(s)}{I(s)}$$, we get the following:

 * $$\frac{V(s)}{I(s)} = \frac{1}{sC}$$

Therefore, the transform for a capacitor with capacitance C is given by:

 * $$\mathcal{L}\{\mbox{capacitor}\} = \frac{1}{sC}$$

Inductors
Let us look at our equation for inductance:

 * $$v(t) = L \frac{di(t)}{dt}$$

putting this into the laplace domain, we get the formula:

 * $$V(s) = sLI(s)$$

And solving for our ratio $$\frac{V(s)}{I(s)}$$, we get the following:

 * $$\frac{V(s)}{I(s)} = sL$$

Therefore, the transform of an inductor with inductance L is given by:

 * $$\mathcal{L}\{\text{inductor}\} = sL$$

Impedance
Since all the load elements can be combined into a single format dependent on s, we call the effect of all load elements impedance, the same as we call it in phasor representation. We denote impedance values with a capital Z (but not a phasor $$\mathbb{Z}$$).

Determining electric current in circuits

In the network shown, determine the character of the currents $$I_1(t)$$, $$I_2(t)$$, and $$I_3(t)$$ assuming that each current is zero when the switch is closed.

===Solution ===

Current flow at a joint in circuit
Since the algebraic sum of the currents at any junction is zero, then

$$ I_1(t)-I_2(t)-I_3(t) = 0 $$.........(182)

Voltage balance on a circuit
Applying the voltage law to the circuit on the left we get

$$ I_1(t)R_1 + L_2 \frac{dI_2(t)}{dt}\ =E(t) $$......... (182-1)

Applying again the voltage law to the outside circuit, given that E is constant, we get

$$ I_1(t)R_1+I_3(t)R_3 + L_3 \frac{dI_3(t)}{dt}\ =E(t) $$......... (182-2)

Laplace Transforms of current and voltage equations
Transforming (182), (182-1) and (182-2), we get

$$ i_1(s)-i_2(s)-i_3(s)=0$$.........(182-3)

$$ i_1(s)R_1 +sL_2i_2(s)=\frac{E}{s}\ $$......... (182-4)

$$ i_1(s)R_1 +( R_3 + sL_3 ) i_3(s)= \frac{E}{s}\ $$	       ......... (182-5)

Review on implementing Laplace Transformation
The three Laplace transformed equations (182-3), (182-4), and (182-5) show the benefits of integral transformation in converting differential equations into linear algebraic equations that could be solved for the dependent variables (the three currents in this case), then inverse transformed to yield the required solution.

 * In equation (182-3), we utilized the sum property of Laplace transforms.
 * In equation (182-4), we utilized the transform of differential derivative as follows.

$$ si_2(s)-I_2(0)= \mathcal{L}\left\lbrace\frac {dI_2}{dt}\right\rbrace $$.........(182-4.1) Since, substituted by the given initial condition: $$ I_2(0)=0$$

 * In equation (182-5), we also utilized the transform of differential derivative

$$ si_3(s)-I_3(0)= \mathcal{L}\left\lbrace\frac {dI_3}{dt}\right\rbrace $$.........(182-5.2)

Again, we substituted by the given initial condition: $$ I_3(0)=0$$

The fact that the applied voltage was a step function, implied the use of Laplace transform of a step function, as follows:

$$ \frac {E}{s}= \mathcal{L}\left\lbrace E \right\rbrace $$.........(182-5.3)

Solution linear simultaneous equations
The three linear simultaneous equations (182-3), (182-4), and (182-5) have the three unknown $$i_1(s)$$, $$i_2(s)$$, and $$i_3(s)$$ and can be solved by Cramer’s rule of matrices among other simple methods of elimination, as follows.

$$ i_1(s) = \frac{ \begin{vmatrix} 0 & -1 & -1 \\ \frac{E}{s}\ & sL_2 & 0 \\ \frac{E}{s}\ & 0 &R_3 + sL_3 \\ \end{vmatrix}}{ \Delta}= \frac{E}{s}\frac{R_3+s(L_2+L_3)}{\Delta} $$ .........    (182-6)

Where, the determinant ∆ for the matrix is determined as follows $$ \Delta = \begin{vmatrix} 1 & -1 & -1 \\ R_1 & sL_2 & 0 \\ R_1 & 0 &R_3 + sL_3 \\ \end{vmatrix} = \begin{vmatrix} 1 & 0 & 0 \\ R_1 & sL_2+R_1 & R_1 \\ R_1 & R_1 & R_1+R_3 + sL_3 \\ \end{vmatrix} $$

$$ \Delta = s^2L_2L_3+s(R_1L_2+R_3L_2 +R_1R_3)+R_1R_3$$......... (182-6.1)

Since we are interested in the factors of Δ, we consider the equation Δ =0. Since all coefficients of this equation are positive, hence it cannot have any positive roots. Its discriminant is

$$(R_1L_2+R_3L_2 +R_1R_3)^2-4L_2L_3R_1R_3$$ ........ (182-6.1.1)

which can be written

 * $$R^2_1L^2_2+2R_1L_2(R_3L_2+R_1L_3)+(R_3L_2-R_1L_3)^2$$........ (182-6.1.2)

which is positive. Hence the equation Δ = 0 has two negative distinct roots $$-\alpha_1$$ and $$-\alpha_2$$, say.

Therefore, $$\Delta = L_2L_3(s+\alpha_1)(s+\alpha_2)$$ ......... (182-6.2)

Where,$$-\alpha_1$$ and $$-\alpha_2$$ are the roots of the quadratic equation (182-6.1) as follows

$$\alpha_1=\frac{1}{2}\left\lbrace\frac{R_1L_2+R_3L_2+R_1R_3}{L_2L_3}+\sqrt {\left(\frac{R_1L_2+R_3L_2+R_1R_3}{L_2L_3}\right)^2-4\frac{R_1R_3}{L_2L_3}}\right\rbrace $$ ......... (182-6.2.1)

$$\alpha_2=\frac{1}{2}\left\lbrace\frac{R_1L_2+R_3L_2+R_1R_3}{L_2L_3}-\sqrt {\left(\frac{R_1L_2+R_3L_2+R_1R_3}{L_2L_3}\right)^2-4\frac{R_1R_3}{L_2L_3}}\right\rbrace $$ ......... (182-6.2.2)

Therefore, equations (182-6) and (186-6.2) give

$$i_1(s)=\frac{E}{s}.\frac{R_3+s(L_2+L_3)}{L_2L_3(s+\alpha_1)(s+\alpha_2)} $$

$$i_1(s)=\frac{A_0}{s}+\frac{A_1}{s+\alpha_1}+\frac{A_1}{s+\alpha_2} $$ 	 .........(182-7)

The constants $$A_0$$, $$A_1$$, and $$A_2$$ are obtained in terms of $$R_1$$, $$L_2$$, $$L_3$$, and $$R_3$$ and are given as:

$$A_0=-\frac{ER_3}{L_2L_3\alpha_1\alpha_2}$$ .........(182-7.1)

$$A_1=E\frac{R_3\alpha_2-\alpha_1\alpha_2(L_2+L_3)}{L_2L_3\alpha_1\alpha_2(\alpha_1-\alpha_2)}$$.........(182-7.2)

$$A_2=E\frac{\alpha_2(L_2+L_3)-R_3}{L_2L_3\alpha_2(\alpha_1-\alpha_2)}$$.........(182-7.3)

Inverse Laplace Transforms of current equations
The inverse Laplace transform of (182-7) is therefore, $$I_1(t)=\mathcal{L}^{-1}\left\lbrace\frac{A_0}{s}+\frac{A_1}{s+\alpha_1}+\frac{A_1}{s+\alpha_2}\right\rbrace =A_0+A_1e^{-\alpha_1t}+A_2e^{-\alpha_2t} $$.........(182-8)

The remaining variables $$I_2(t)$$ and $$I_3(t)$$ and the corresponding voltages are determined by equations (182), (182-1) and (182-2)

Analysis of circuit dynamics
The electric current $$I_1(t)$$ in equation (182-8) shows a time-independent component $$A_0$$ and two decay terms, which reach asymptotic values as t reaches ∞. In other words, the currents in the three circuits lack sinusoidal osculation, mainly because: (1) the applied voltage is constant and (2) the circuit does not have capacitance components.

Notes: This example could be modified in various ways to involve voltage impulse, sinusoidal voltage source, capacitance,and various boundary and initial conditions of charges and currents.

Generalization of the method
In the above example, the following modifications can be made:

(1) The applied voltage in the Krichhoff's equation can take many forms such as

 * :$$E(t)=E_o\delta(t)$$
 * :$$E(t)=E_o\sin(\omega t)$$
 * :$$E(t)=E_of(t)$$

(2) Capacitance add the integral term of current over the duration as

 * : $$ \frac{1}{C}\int_0^t I(\tau)d\tau$$

title: Circuit Theory/Laplace Transform


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Talk:How to Pass a Course

I think a page dedicated to calculator use in exams or during studying is sorely missing.

Orphaning pages
I dont think you can have a complete book on passing classes without a section on cheating.

Please don't delete links to Wikibooks unless you are doing a full reorganization of the content here. I'm restoring the "passing college" section because it wasn't put into another wikibook. We don't need more junk in the orphan bin please. --Rob Horning 18:23, 30 March 2006 (UTC)

 * I got the whole situation fixed. I found the content where it was supposed to have been moved to, and I notified the user who did the page moves.  It is now a part of the College Survival Guide, a totally seperate Wikibook.  --Rob Horning 03:14, 31 March 2006 (UTC)

title: Talk:How to Pass a Course


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Circuit Theory/Contributors

This page will serve as a common discussion point for contributors to the Circuit Theory Wikibook. Please use this page to keep a list of Templates used on each page as well as conventions and notations used in the book. Contributors are encouraged to sign their name here.

Formatting
When referring to values like RTH and RL (like listed above) should we use $$R_TH, R_L$$ ( $$R_TH, R_L$$ )?

 * That's the preferred way, I think. It's more effort, but it certainly looks better. --Whiteknight (talk) (projects) 02:59, 12 June 2006 (UTC)

Templates
The following templates are in use:

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Contributors

 * Shellreef 04:17, 1 November 2005 (UTC)
 * Whiteknight 19:22, 5 October 2005 (UTC)
 * Jcarroll 04:27, 19 March 2006 (UTC)

title: Circuit Theory/Contributors


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