15.10: Primary Sources
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Bill Clinton on Free Trade and Financial Deregulation (1993-2000)
During his time in office, Bill Clinton passed the North American Free Trade Act (NAFTA) in 1993, allowing for the free movement of goods between Mexico, the United States, and Canada, signed legislation repealing the Glass-Steagall Act, a major plank of Franklin Roosevelt’s New Deal banking regulation, and deregulated the trading of derivatives, including credit default swaps, a complicated financial instrument that would play a key role in the 2007-2008 economic crash. In the following signing statements, Clinton offers his support of free trade and deregulation.
9/11 Commission Report, “Reflecting On A Generational Challenge” (2004)
On July 22, 2004, the National Commission on Terrorist Attacks Upon the United States—or, the 9/11 Commission—delivered a 500-plus-page report that investigated the origins of the 9/11 attacks and America’s response and offered policy prescriptions for a post-9/11 world.
George W. Bush on the Post-9/11 World (2002)
In his 2002 State of the Union Address, George W. Bush proclaimed that the attacks of September 11 signaled a new, dangerous world that demanded American interventions. Bush identified an “Axis of Evil” and provided a justification for a broad “war on terror.”
Obergefell v. Hodges (2015)
In 2015, the Supreme Court ruled in Obergefell v. Hodges that prohibitions against same-sex marriage were unconstitutional. Gay marriage had been a divisive issue in American politics for well over a decade. Many states passed referendums and constitutional amendments barring same-sex marriages and, in 1996, Bill Clinton signed the Defense of Marriage Act, defining marriage at the federal level as between a man and a woman. In 2003, the Massachusetts Supreme Court struck down Massachusetts’ state’s prohibition, making it the first state to legally marry same-sex couples. More followed and public opinion began to turn. Although President Obama still refused to support it, by 2011 a majority of Americans believed same-sex marriages should be legally recognized. Four years later, the Supreme Court issued its Obergefell decision. The majority opinion, written by Justice Anthony Kennedy, considered the relationship between history and shifting notions of liberty and injustice.
Barack Obama, Howard University Commencement Address (2016)
In 2016, President Barack Obama delivered the commencement address at Howard University, the nation’s most distinguished historically black university. In it, he urged students to be hardworking yet pragmatic in their quest to achieve “justice and equality and freedom” in American life.
Ground Zero (2001)
A worker stands in front of rubble from the World Trade Center at Ground Zero in Lower Manhattan several weeks after the September 11 attacks.
Barack Obama and a Young Boy (2009)
In 2008, Barack Obama became the first African American elected to the presidency. In this official White House photo from May, 2009, 5-year-old Jacob Philadelphia said, “I want to know if my hair is just like yours.”


1.3: Interaction of Individuals, Firms, and Societies
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Introducing the Firm
Firms allow an economy to operate more efficiently and reduce the transaction costs of coordinating production.
Learning objectives
Explain the importance of private companies and firms in the economy
“Firm” is simply another word for company or business. The basic economic marketplace consists of transactions between households and firms. Firms use factors of production – land, labor, and capital – to produce goods that are consumed by households. They may be organized in many different ways – corporations, partnerships, sole proprietorships, and collectives are all examples of firms. Economists who study the theory of the firm attempt to describe, explain, and predict the nature of a firm, including its existence, behavior, structure, and relationship to the market.
The Evolution of the Firm
Not all markets and societies involve firms. In many medieval cities, most production was done by individual craftsmen who were loosely organized into guilds, or by tenant farmers who rented family-sized plots of land. Transactions took place primarily between individuals.
Firms generally appear and become prevalent as an alternative to individual trade when it is more efficient to produce in a non-market environment. For example, in a labor market, it might be too difficult or costly for firms or organizations to engage in production when they have to hire and fire their workers depending on demand/supply conditions. While the advantages of consolidation for efficiency are potentially many and varied, the underlying concept is that integrating operational paradigms enables potential synergy via the construct of a firm.
Firms also allow economic growth, not only for the firm but for the broader society in which it resides. Through separating the business from the individual(s) who starts it, the funding, insurance and liability of a firm can function independently of a person. The separation of a firm from the individual also allows more specifically applicable regulations and laws, broader accumulation of investment capital and more complex strategic alliances. While the detailed implications of a firm and it’s relationship with individuals and society are complex, the important takeaway is that firms play an integral role in economic structure.
The Firm : Organizing production under firms reduces the transaction costs of coordinating production in the market.
The Transaction Theory of the Firm
According to Ronald Coase, people begin to organize their production in firms when the transaction cost of coordinating production through the market exchange is greater than within the firm. He notes that a firm’s interactions with the market may not be under its control (for instance because of sales taxes), but its internal allocation of resources are: “Within a firm, … market transactions are eliminated and in place of the complicated market structure with exchange transactions is substituted the entrepreneur … who directs production.” He asks why alternative methods of production (such as the price mechanism and economic planning), could not either achieve all production, so that either firms use internal prices for all their production, or one big firm runs the entire economy.
For Coase the main reason to establish a firm is to avoid some of the transaction costs of using the price mechanism. These include discovering relevant prices (which can be reduced but not eliminated by purchasing this information through specialists), as well as the costs of negotiating and writing enforceable contracts for each transaction (which can be large if there is uncertainty). Moreover, contracts in an uncertain world will necessarily be incomplete and have to be frequently re-negotiated. The costs of haggling about division of surplus, particularly if there is asymmetric information and asset specificity, may be considerable. Organization into a firm can considerably reduce these costs.
Trade Leads to Gains
Producers and consumers trade because the exchange makes both parties better off.
Learning objectives
Explain why parties trade.
Producers and consumers trade because the exchange makes both parties better off. The benefit of exchange to producers is measured by the amount of profit – that is, the difference between the average cost of producing an item and the price received for that item. The benefit of exchange to a consumer is measured by net utility gained. This is measured by taking the difference between the maximum price a consumer is willing to pay and the actual price they do pay. To understand this, imagine purchasing a car. You would be willing to pay up to $15,000 for a car in good condition, but you are able to buy one for only $12,000. Since you value the car at $3,000 more than you paid for it, $3,000 is the benefit that you gained from the transaction.
Economists refer to these benefits from exchange as producer and consumer surplus. Consumer surplus is the monetary gain obtained by consumers because they are able to purchase a product for a price that is less than the highest price that they would be willing to pay. Producer surplus is the amount that producers benefit by selling at a market price that is higher than the least that they would be willing to sell for.
The amount of consumer and producer surplus that is gained from a transaction can be seen on a standard supply and demand graph. Consumer surplus is the area (triangular if the supply and demand curves are linear) above the equilibrium price of the good and below the demand curve. This reflects the fact that consumers would have been willing to buy a single unit of the good at a price higher than the equilibrium price, a second unit at a price below that but still above the equilibrium price, etc., yet they in fact pay just the equilibrium price for each unit they buy.
Likewise, in the supply-demand diagram, producer surplus is the area below the equilibrium price but above the supply curve. This reflects the fact that producers would have been willing to supply the first unit at a price lower than the equilibrium price, the second unit at a price above that but still below the equilibrium price, etc., yet they in fact receive the equilibrium price for all the units they sell. The sum of consumer and producer surplus is called economic, or social, surplus, and reflects the total amount of benefit received by society when consumers and producers trade.
Consumer and Producer Surplus : Consumer surplus is the area between the demand line and the equilibrium price, and producer surplus is the area between the supply line and the equilibrium price.
Exchange and Pareto Optimality
An allocation of resources is Pareto efficient when it is impossible to make any one individual better off without making at least one individual worse off. For example, imagine that two individuals prefer peanut butter and jelly sandwiches to a sandwich with only peanut butter or only jelly. A distribution in which Individual A has all of the peanut butter and individual B has all of the jelly is not Pareto efficient, because both parties would be better off if they shared their resources.
Similarly, an action that makes at least one party better off without making any individual worse off is called a Pareto improvement. Any transaction in a free market always produces a Pareto improvement because it makes consumers and/or producers better off without making either party worse off (if this were not the case, the consumer and/or the producer would refuse to participate in the transaction in the first place). It is commonly assumed that outcomes that are not Pareto efficient are to be avoided, and if a Pareto improvement is possible it should always be implemented.
One way to look at whether a transaction is a Pareto improvement is to ask whether it increases consumer or producer surplus without decreasing either party’s surplus. Lowering an item’s price without changing the quantity sold, for example, may increase consumer surplus, but is not a Pareto improvement because producers suffer negative consequences.
Thinking about Efficiency
An efficient market maximizes total consumer and producer surplus.
Learning objectives
Define economic efficiency.
Every economic transaction has a buyer and a seller who will only participate is she is receiving at least a minimum benefit. These benefits are represented as consumer surplus and producer surplus, respectively. In the illustration below, both types of surpluses are displayed graphically. An efficient market maximizes total consumer and producer surplus.
Consumer and Producer Surplus : Consumer and producer surplus are maximized at the market equilibrium – that is, where supply and demand intersect.
The market shown in is one without any distortions such as regulations, taxes, or an inability for buyers to meet sellers. It is subject to what Adam Smith described as the i nvisible hand : if the price is anything except the equilibrium price, market forces will eventually return the market price to equilibrium.
Not all markets are efficient. There are a number of reasons why a market may be inefficient. Perhaps most well known is inefficiency caused by government intervention. Governments can institute any number of policies that prevent markets from achieving the free market equilibrium price and quantity: taxes raise prices, quotas limit the quantity sold, and regulations affect the supply and demand curves. Market inefficiency can also be caused by things such as irrational market actors and barriers to transactions, such as an inability for buyers and sellers to find one another.
Economists often seek to maximize efficiency, but it is important to contextualize such aims. Efficiency is but one of many vying goals in an economic system, and different notions of efficiency may be complementary or may be at odds. Most commonly, efficiency is contrasted or paired with morality, particularly liberty, and justice. Some economic policies may be seen as increasing efficiency at a cost to other goals or values, though this is certainly not a universal tradeoff. For example, taxation will always cause some inefficiency in markets, but many individuals believe that the benefits of programs such as Social Security and public schooling are worth the loss in efficiency.
The Function and Nature of Markets
In a free market, the price and quantity of an item are determined by the supply and demand for that item.
Learning objectives
Summarize the defining characteristics of a free market economy
In economics, a market is defined as a system or institution whereby parties engage in exchange. A market economy is an economy in which decisions regarding investment, production, and distribution are based on supply and demand, and prices of goods and services are determined in a free price system. The major defining characteristic of a market economy is that decisions on investment and the allocation of producer goods are mainly made through markets. This is the opposite of a planned economy, where investment and production decisions are embodied in a plan of production.
A free market is a market structure that is not controlled by a designated authority. Free markets may have different structures: perfect competition, oligopolies, monopolistic competition, and monopolies are all types of markets that may exist in a capitalist economy. The most basic models in economics assume that markets are free and experience perfect competition – there are many buyers and sellers so no individual actor may affect a good’s price; there are no barriers to exit or entry; products are homogeneous; and all actors in the economy have perfect information.
Market Equilibrium
In a free market, the price and quantity of an item is determined by the supply and demand for that item. The market demand function describes the amount of a good that all consumers will purchase at a given price, while the market supply function expresses the amount that producers will supply at a given price. Consider the market for computers. At a price of $1,200, the market may demand 8,000 computers, while producers are willing to supply 15,000 computers. This is not the equilibrium price because at $1,200, supply exceeds demand. In order to reach equilibrium, the price must drop, causing demand to rise and supply to fall until the two are equal. This can be expressed graphically by drawing the market supply function and the market demand function and finding the point where the two curves intersect.
Market Supply and Demand : The market equilibrium exists where the market demand curve and the market supply curve intersect.
Changes to the market supply and market demand will cause changes in the equilibrium price and quantity of the good produced. For example, if a new technology is invented that allows producers to manufacture cars more efficiently, supply will rise and the market supply curve will shift to the right. The new market equilibrium will have a higher number of cars sold at a lower price. When markets are perfectly competitive, the equilibrium outcome of trade in the market is economically efficient. This means that the market is producing the largest net gain possible for society, given consumers’ utility functions and producers’ production functions.
Markets are Typically Efficient
A perfectly competitive market with full property rights is typically efficient.
Learning objectives
Define efficient markets.
An efficient market maximizes total consumer and producer surplus; there is no deadweight loss. An economic system is said to be more efficient than another (in relative terms) if it can provide more goods and services for society without using more resources. In absolute terms, a situation can be called economically efficient if:
Economic Inefficiency : A sign of economic inefficiency in a market is the presence of deadweight loss.
- No one can be made better off without making someone else worse off (commonly referred to as Pareto efficiency),
- No additional output can be obtained without increasing the amount of inputs, and
- Production proceeds at the lowest possible per-unit cost.
Economists refer to two types of market efficiency. A market has productive efficiency when units of goods are being supplied at the lowest possible average cost. This condition is satisfied if the equilibrium quantity is at the minimum point of the average total cost curve. For example, if a farm can produce 10,000 bushels of corn with 20 employees, but is currently producing 10,000 bushels with 25 employees, it is not achieving productive efficiency.
A market has allocative efficiency if the price of a product that the market is supplying is equal to the value consumers place on it. This is equivalent to saying that the marginal cost of an item is equal to its price. If a market is not allocatively efficient, then it is creating too much of something that consumers value less than other goods, or not enough of something that consumers value more. A market that produces 500 loaves of bread but only one gallon of milk is probably not allocatively efficient.
As you study economics further, it is usually safe to assume that markets are efficient unless you’re dealing with a distortion (e.g. regulations, imperfect information sharing).
It is important to note that achieving economic efficiency is not always the most important goal for a society. A market can be perfectly efficient but highly unequal, for example. If 1% of the population controls virtually all the income, then the market will efficiently allocate virtually all of its production to those same people. While this is economically efficient, many would argue that it is not desirable. Efficient markets may have negative effects on those that exist outside of the market; for example, the energy market may cause environmental harm that is not captured in the economic notion of efficiency.
Government Intervention May Fix Inefficient Markets
Governments can intervene to make a market more efficient when a market failure, such as externalities or asymmetric information, exists.
Learning objectives
Discuss the role of government in addressing common market failures
In an efficient market, firms can produce goods at the lowest possible cost while individuals can access the goods and services they desire, all while utilizing the least resources possible. A market can be said to be economically efficient if it has certain qualities:
- perfectly competitive
- mobile resources
- accurate and freely available information
- individuals directly receive the costs and benefits of their transactions
Market failure is the name for when a market is not efficient; that is, when it deviates from one or more of the above conditions. However, in reality no market is perfectly efficient. In general, minor inefficiencies do not dramatically affect society. But when society is adversely affected by economic inefficiency, such as when a monopoly firm raises prices to a point where people cannot afford a basic good, the government will sometimes intervene.
Consider the problem of externalities, the phenomenon of when a transaction occurs that affects people who were not directly involved. For example, when a coal plant producing electricity causes pollution, there is a transaction between the company and the resident who purchases the product. But if you live near the coal plant and suffer from asthma due to the smog it produces, you are encountering a negative externality. You had no choice in the transaction, but are experiencing its effects.
Externalities are an example of economic inefficiency, since those involved in the economic transaction do not bear the full costs of the transaction. In this case, governments can intervene by taxing the transaction and using the money to negate the harmful effects or to compensate those affected by the negative externality. Similarly, when a transaction produces positive externalities, efficiency is achieved when the government subsidizes the transaction. Education is an example of a transaction that has a positive effect on society.
Another case in which markets do not operate efficiently on their own is the market for public goods. Public goods are nonrival, which means that more than one (and sometimes many!) individual can use the good at one time. They are also nonexcludable, which means that their use cannot be prevented. For example, consider a beautiful fountain in a public park. The company that built the fountain cannot force people to pay money in order to enjoy it, since its in a public area; and since one person looking at the fountain doesn’t prevent others from looking at it, it is a nonrival good.
Free markets will generally produce less than the optimal amount when a good is nonexcludable and nonrivalrous, which means that a government can make the market more efficient by producing the public good itself. By using tax revenue, governments can avoid the problem of free riders and produce an efficient quantity of public goods even when the free market cannot.
National Defense as a Public Good : National defense is a classic example of a good that is nonexcludable and nonrivalrous. It will be under-produced unless the government provides it.
Full Economy Interactions
Variables that describe the full economy, such as GDP and unemployment, are determined by the decisions of individual economic actors.
Learning objectives
Explain how the macroeconomy is the sum of many individual economic actors’ decisions.
In the most basic economic model, the economy consists of interactions between households, which provide labor and purchase goods, and firms, which employ labor and produce goods. Macroeconomics studies the aggregate effects of the actions of many individual households and firms. While microeconomists might study how a market with one producer and one consumer reaches equilibrium, macroeconomists combine the demand of all consumers in a market (aggregate demand) and the supply from all producers in a market (aggregate supply) to look at the way these groups interact on a large scale.
Circular Flow : The economy consists of interactions between firms and households.
Consider the market for CDs. Each individual consumer has a demand function for CDs that determines how many he will buy at a particular price – for example, one consumer may only buy a single album if they cost $15 each, but would buy two if the price dropped to $10 each. Likewise, each producer has a production function that determines how many CDs it will produce at a given price; it may produce 10,000 CDs if they can be sold for $10, but will increase production to 12,000 if the price rises to $15. In order to understand the entire market for CDs, economists add the demand of all consumers at each possible price, creating an aggregate demand curve, and the total quantity supplied by producers at each possible price, creating an aggregate supply curve. The point at which these two curves intersect shows the market equilibrium for CDs.
The Macroeconomy
Just as the choices made by individual consumers and producers can be aggregated to describe an entire industry, their combined effects can also influence a nation’s overall economic activity. Macroeconomists study a variety of these effects, but three are central to macroeconomic research:
- Gross domestic product (GDP) – the size of an entire economy’s output – is measured by adding together all the production undertaken by a nation’s firms. Individual firms affect GDP every time they choose to produce more or less. Consumers affect GDP whenever they increase or decrease demand for goods.
- Inflation occurs when many individual consumers increase demand for a good, raising the equilibrium price for the economy as a whole.
- Unemployment rises when firms choose to produce less or when consumers decrease their demand at a given price.
Key Points
- Firms generally appear and become prevalent as an alternative to individual trade when it is more efficient to produce in a non-market environment.
- Limited liability separates the management of a firm from its ownership, allowing companies to raise money easily because owners do not need to risk everything in the case of bankruptcy.
- Most industries experience increasing returns to scale up to a point, which means that more goods can be produced using fewer resources.
- According to Ronald Coase, the main reason to establish a firm is to avoid some of the transaction costs of using the price mechanism.
- The benefit of exchange to producers is measured by the profit the producer makes. The benefit of exchange to a consumer is measured by net utility gained.
- Consumer surplus is the monetary gain obtained by consumers because they are able to purchase a product for a price that is less than the highest price that they would be willing to pay.
- Producer surplus is the amount that producers benefit by selling at a market price that is higher than the least that they would be willing to sell for.
- An allocation of resources is Pareto efficient when it is impossible to make any one individual better off without making at least one individual worse off.
- Economists assume that firms seek to maximize their profits – defined as the difference between total revenue and total cost – while consumers seek to maximize their utility – which is roughly defined as the total satisfaction gained from goods, services, or actions.
- An efficient allocation of resources maximizes total consumer and producer surplus.
- Because they produce efficient outcomes, the seemingly haphazard workings of the marketplace can promote the common good.
- Efficiency is but one of many vying goals in an economic system, and different notions of efficiency may be complementary or may be at odds.
- A market is defined as a system or institution whereby parties engage in exchange. A market economy is an economy in which decisions regarding investment, production, and distribution are based on supply and demand, and prices of goods and services are determined in a free price system.
- In a perfectly competitive market there are many buyers and sellers so no individual actor may affect a good’s price; there are no barriers to exit or entry; products are homogeneous; and all actors in the economy have perfect information.
- Changes to the market supply and market demand will cause changes in the equilibrium price and quantity of the good produced.
- When markets are perfectly competitive, the equilibrium outcome of trade in the market is economically efficient. This means that the market is producing the largest net gain possible for society, given consumers’ utility functions and producers’ production functions.
- A market has productive efficiency when units of goods are being supplied at the lowest possible average cost.
- A market has allocative efficiency if the price of a product that the market is supplying is equal to the value consumers place on it.
- It is important to note that achieving economic efficiency is not always the most important goal for a society. A market can be perfectly efficient but highly unequal.
- A smoothly functioning market requires that producers possess property rights to the goods and services they produce and that consumers possess property rights to the goods and services they buy.
- Economic efficiency occurs under the following conditions: competitive markets with accurate exchange of information and mobile resources, in which individuals bear the full costs and benefits of their transactions.
- The criteria for economic efficiency are rarely fully met.
- If a transaction affects individuals not involved in the transaction (either positively or negatively), that transaction is said to have an externality.
- Governments can intervene by taxing negative externalities or subsidizing positive externalities.
- Free markets will generally produce less than the optimal amount when a good is nonexcludable and nonrivalrous, which means that a government can make the market more efficient by producing the public good itself.
- Macroeconomists combine the demand of all consumers in a market ( aggregate demand ) and the supply from all producers in a market ( aggregate supply ) to look at the way these groups interact on a large scale.
- Just as the choices made by individual consumers and producers can be aggregated to describe an entire industry, their combined effects can also influence a nation’s overall economic activity.
- GDP is measured by adding together all the production undertaken by a nation’s firms. Individual firms affect GDP every time they choose to produce more or less. Consumers affect GDP whenever they increase or decrease demand for goods.
Key Terms
- increasing returns to scale : The characteristic of production in which output increases by more than the proportional increase in inputs.
- firm : A business enterprise, however organized.
- utility : The ability of a commodity to satisfy needs or wants; the satisfaction experienced by the consumer of that commodity.
- consumer surplus : The difference between the maximum price a consumer is willing to pay and the actual price they do pay.
- producer surplus : The amount that producers benefit by selling at a market price that is higher than the lowest price at which they would be willing to sell.
- producer surplus : The amount that producers benefit by selling at a market price that is higher than the lowest price at which they would be willing to sell.
- consumer surplus : The difference between the maximum price a consumer is willing to pay and the actual price they do pay.
- market economy : An economy in which goods and services are exchanged in a free market, as opposed to a state-controlled or socialist economy; a capitalistic economy.
- equilibrium : The condition of a system in which competing influences are balanced, resulting in no net change.
- Pareto efficiency : The state in which no one can be made better off by making another worse off.
- public good : A good that is both non-excludable and non-rivalrous in that individuals cannot be effectively excluded from use and where use by one individual does not reduce availability to others.
- externality : An impact, positive or negative, on any party not involved in a given economic transaction or act.
- free rider : One who obtains benefit from a public good without paying for it directly.
- inflation : An increase in the general level of prices or in the cost of living.
- aggregate : A mass, assemblage, or sum of particulars; something consisting of elements but considered as a whole.
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- Deadweight-loss-price-ceiling. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...ce-ceiling.svg . License : CC BY: Attribution
- free rider. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/free_rider . License : CC BY-SA: Attribution-ShareAlike
- externality. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/externality . License : CC BY-SA: Attribution-ShareAlike
- public good. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/public%20good . License : CC BY-SA: Attribution-ShareAlike
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- F-15C. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:F-15C.jpg . License : Public Domain: No Known Copyright
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4.R: Further Applications of Trigonometry (Review)
4.1: Non-right Triangles: Law of Sines
For the exercises 1-5 assume \(\alpha \) is opposite side \(a\), \(\beta \) is opposite side \(b\), and \(\gamma \) is opposite side \(c\). Solve each triangle, if possible. Round each answer to the nearest tenth.
1) \(\beta =50^{\circ}, a=105, b=45\)
- Answer
-
Not possible
2) \(\alpha =43.1^{\circ}, a=184.2, b=242.8\)
3) Solve the triangle.
- Answer
-
\(C=120^{\circ}, a=23.1, c=34.1\)
4) Find the area of the triangle.
5) A pilot is flying over a straight Highway. He determines the angles of depression to two mile posts \(2.1\) km apart to be \(25^{\circ}\) and \(49^{\circ}\), as shown in the figure below. Find the distance of the plane from point \(A\) and the elevation of the plane.
- Answer
-
distance of the plane from point \(A:2.2\) km, elevation of the plane: \(1.6\) km
4.2: Non-right Triangles - Law of Cosines
1) Solve the triangle, rounding to the nearest tenth, assuming \(\alpha \) is opposite side \(a\), \(\beta \) is opposite side \(b\), and \(\gamma \) s opposite side \(c: a=4, b=6,c=8\)
2) Solve the triangle in the Figure below, rounding to the nearest tenth.
- Answer
-
\(B=71.0^{\circ},C=55.0^{\circ},a=12.8\)
3) Find the area of a triangle with sides of length \(8.3\), \(6.6\), and \(9.1\).
4) To find the distance between two cities, a satellite calculates the distances and angle shown in the Figure below (not to scale). Find the distance between the cities. Round answers to the nearest tenth.
- Answer
-
\(40.6\) km
4.3: Polar Coordinates
1) Plot the point with polar coordinates \(\left ( 3,\dfrac{\pi }{6} \right )\).
2) Plot the point with polar coordinates \(\left ( 5,\dfrac{-2\pi }{3} \right )\).
- Answer
-
3) Convert \(\left ( 6,\dfrac{-3\pi }{4} \right )\) to rectangular coordinates.
4) Convert \(\left ( -2,\dfrac{3\pi }{2} \right )\) to rectangular coordinates.
- Answer
-
\((0,2)\)
5) Convert \((7,-2)\) to polar coordinates.
6) Convert \((-9,-4)\) to polar coordinates.
- Answer
-
\((9.8489,203.96^{\circ})\)
For the exercises 7-9, convert the given Cartesian equation to a polar equation.
7) \(x=-2\)
8) \(x^2+y^2=64\)
- Answer
-
\(r=8\)
9) \(x^2+y^2=-2y\)
For the exercises 10-11, convert the given polar equation to a Cartesian equation.
10) \(r=7\cos \theta\)
- Answer
-
\(x^2+y^2=7x\)
11) \(r=\dfrac{-2}{4\cos \theta +\sin \theta }\)
For the exercises 12-13, convert to rectangular form and graph.
12) \(\theta =\dfrac{3\pi }{4}\)
- Answer
-
\(y=-x\)
13) \(r=5\sec \theta\)
4.4: Polar Coordinates - Graphs
For the exercises 1-5, test each equation for symmetry.
1) \(r=4+4\sin \theta\)
- Answer
-
symmetric with respect to the line \(\theta =\dfrac{\pi }{2}\)
2) \(r=7\)
3) Sketch a graph of the polar equation \(r=1-5\sin \theta\). Label the axis intercepts.
- Answer
-
4) Sketch a graph of the polar equation \(r=5\sin (7\theta )\).
5) Sketch a graph of the polar equation \(r=3-3\cos \theta\)
- Answer
-
4.5: Polar Form of Complex Numbers
For the exercises 1-2, find the absolute value of each complex number.
1) \(-2+6i\)
2) \(4-3i\)
- Answer
-
\(5\)
Write the complex number in polar form.
3) \(5+9i\)
4) \(\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i\)
- Answer
-
\(\mathrm{cis}\left (-\dfrac{\pi }{3} \right )\)
For the exercises 5-6, convert the complex number from polar to rectangular form.
5) \(z=5\mathrm{cis}\left (\dfrac{5\pi }{6} \right )\)
6) \(z=3\mathrm{cis}(40^{\circ})\)
- Answer
-
\(2.3+1.9i\)
For the exercises 7-8, find the product \(z_1 z_2\) in polar form.
7) \(\begin{align*} z_1 &= 2\mathrm{cis}(89^{\circ})\\ z_2 &= 5\mathrm{cis}(23^{\circ}) \end{align*}\)
8) \(\begin{align*} z_1 &= 10\mathrm{cis}\left ( \dfrac{\pi }{6} \right )\\ z_2 &= 6\mathrm{cis}\left ( \dfrac{\pi }{3} \right ) \end{align*}\)
- Answer
-
\(60\mathrm{cis}\left ( \dfrac{\pi }{2} \right )\)
For the exercises 9-10, find the quotient \(\dfrac{z_1}{z_2}\) in polar form.
9) \(\begin{align*} z_1 &= 12\mathrm{cis}(55^{\circ})\\ z_2 &= 3\mathrm{cis}(18^{\circ}) \end{align*}\)
10) \(\begin{align*} z_1 &= 27\mathrm{cis}\left ( \dfrac{5\pi }{3} \right )\\ z_2 &= 9\mathrm{cis}\left ( \dfrac{\pi }{3} \right ) \end{align*}\)
- Answer
-
\(3\mathrm{cis}\left ( \dfrac{4\pi }{3} \right )\)
For the exercises 11-12, find the powers of each complex number in polar form.
11) Find \(z^4\) when \(z=2\mathrm{cis}(70^{\circ})\)
12) Find \(z^2\) when \(z=5\mathrm{cis}\left ( \dfrac{3\pi }{4} \right )\)
- Answer
-
\(25\mathrm{cis}\left ( \dfrac{3\pi }{2} \right )\)
For the exercises 13-14, evaluate each root.
13) Evaluate the cube root of \(z\) when


14.7: Nervous System Worksheet Answers
1. The diagram below is of a nerve cell or neurone.
- i. Add the following labels to the diagram.
- Axon; Myelin sheath; Cell body; Dendrites; Muscle fibres ;
- ii. If you like, colour in the diagram as suggested below.
- Axon - purple;
- Myelin sheath - yellow;
- Cell body - blue;
- Dendrites - green;
- Muscle fibres – red;
- iii. Now indicate the direction that the nerve impulse travels.
2. There are three different kinds of neurone or nerve cell. Match each kind with its function.
- A. Motor neuron; B. Sensory neuron; C. Relay neuron;
| Kind of neurone | Function |
|---|---|
| B. Sensory neuron |
The nerve cell that carries impulses from a sense receptor to the brain or spinal cord. |
| C. Relay neuron | The nerve cell that connects sensory and motor neurons |
| A. Motor neuron |
The nerve cell that transmits impulses from the brain or spinal cord to a muscle or gland |
3. Match the descriptions in the table below with the terms in the list.
| Term | Function |
|---|---|
| B. Axon | 1. The long fibre that carries the nerve impulses. |
| J. Nerve | 2. A bundle of axons. |
| A. Synapse | 3. The connection between adjacent neurons. |
| K. Neurotransmitter | 4. The chemical secreted into the gap between neurons at a synapse. |
| G.Reflex | 5. A rapid automatic response to a stimulus. |
| C. Myelin sheath |
6. The covering of fatty material that speeds up the passage of nerve impulses. |
| L. Axon terminal |
7. The structure at the end of an axon that produces neurotransmitters to transmit the nerve impulse across the synapse. |
| D. Nerve impulse | 8. The high speed signals that pass along the axons of nerve cells. |
| I. Dendrites | 9. The branching filaments that conduct nerve impulses towards the cell. |
| E. Sense receptor |
10. The sense organ or cells that receive stimuli from within and outside the body. |
| F. Response | 11. The reaction to a stimulus by a muscle or gland. |
| H. Cell body | 12.The part of the nerve cell containing the nucleus. |
4. The diagram below shows a cross-section of the spinal cord. Add the following labels to the diagram.
- Central canal; White matter; Dorsal root; Grey matter; Ventral root; Skin;
- Muscle; Sensory neuron; Relay neuron; Motor neuron; Pain receptors in skin
5.
- a) List in order the 3 different neurons involved in a reflex arc from the stimulus to the response.
| Stimulus | sensory neuron | relay neuron | motor neuron | Response |
|---|
- b) Name 3 different reflexes found in animals.
Reflex 1. Blink reflex .
Reflex 2. Paw pinch refle x.
Reflex 3. Swallowing reflex, plus many others .
6. The diagram below shows the nervous system of a horse. Add the following labels.
- Brain; Spinal cord; Cranial nerves; Spinal nerves; Sciatic nerve; Nerves of the autonomic nervous system; Vagus nerve; Network of nerves to forelimb.
7. Indicate whether the following parts of the nervous system are part
of the Central Nervous System CNS) or the Peripheral Nervous System (PNS).
| Part of nervous system | CNS or PNS? |
|---|---|
| Brain | CNS |
| Autonomic nervous system | PNS |
| Spinal nerves | PNS |
| Spinal cord | CNS |
| Cranial nerves | PNS |
8. The diagram below shows a section of a dog’s brain. Add the labels
in the list below and, if you like, colour in the diagram as suggested.
- Cerebellum - blue;
- Spinal cord - green;
- Medulla oblongata - orange;
- Hypothalamus - purple;
- Pituitary gland - red;
- Cerebral hemispheres – yellow.
9. Match the descriptions below with the terms in the list. You may need to use some terms more than once.
| Term | Description |
|---|---|
| E. Hypothalamus | 1. Controls water balance and body temperature. |
| D.Medulla oblongata | 2. Where the respiratory rate is controlled. |
| C. Cerebellum | 3. Where posture, balance and voluntary muscle movements are controlled. |
| A. Cerebral hemispheres |
4. Contains centres governing mental activity, including intelligence, memory, and learning. |
| H. Meninges | 5. The tough fibrous envelope enclosing the brain and spinal cord. |
| F. Pituitary | 6. The “master” gland of the endocrine system. |
| Cerebral hemispheres/cortex | 7. Responsible for instigating voluntary movements. |
| J. Cerebrospinal fluid | 8. The fluid that surrounds the brain and spinal cord. |
| G. Grey matter | 9. Composed of cell bodies and nuclei. |
| B. White matter | 10. Composed of axons. |
| Cerebral hemispheres/cortex | 11. Where the sensations of sight, sound, taste etc. are interpreted. |
| I. Ventricles | 12. Spaces in the brain filled with cerebral spinal fluid. |
| K. Sulcus | 13. A fold in the cerebral cortex. |
| L. Carotid artery | 14. The artery that supplies the brain with oxygenated blood. |
10. Match the descriptions below with the parts of the nervous
system in the list. You may need to use some terms more than once.
| Description | Part of the nervous system |
|---|---|
|
1. Part of the nervous system that is composed of the brain and the spinal cord. |
B. Central nervous system |
|
2. Part of the nervous system that is composed of the cranial and spinal nerves. |
C. Peripheral nervous system |
|
3. The part of the peripheral nervous system that regulates the activity of the heart and smooth muscle. |
A. Autonomic nervous system |
|
4. The part of the autonomic nervous system that increases heart and respiratory rates, increases blood flow to the skeletal muscles and dilates the pupils of the eye. |
E. Sympathetic nervous system |
|
5. The part of the autonomic nervous system that increases gut activity and decreases heart and respiratory rates. |
D. Parasympathetic nervous system |
11. Name the nerves described below using the choices in the list.
| Nerve | Description |
|---|---|
| Vestibular nerve |
1. The 8th cranial nerve that carries impulses from the organs of balance and hearing to the brain. |
| Optic nerve |
2. The 2nd cranial nerve that carries nervous impulses from the retina of the eye to the brain. |
| Sciatic nerve | 3. The largest nerve in the body serving the muscles of the leg. |
| Olfactory nerve |
4. The 1st cranial nerve that carries impulses from the organ of smell in the nose to the brain. |
| Vagus nerve |
5. The 10th cranial nerve that supplies the pharynx, lungs, heart, stomach and most of the abdominal organs. |


7.8: Introduction to Linguistics
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Language is such a special topic that there is an entire field, linguistics, devoted to its study. Linguistics views language in an objective way, using the scientific method and rigorous research to form theories about how humans acquire, use, and sometimes abuse language. There are a few major branches of linguistics, which it is useful to understand in order to learn about language from a psychological perspective.
Major Branches of Linguistics
This diagram outlines the various subfields of linguistics, the study of language. These include phonetics, phonology, morphology, syntax, semantics, and pragmatics.
Phonetics and Phonology
A phoneme is the smallest unit of sound that makes a meaningful difference in a language. The word “bit” has three phonemes, /b/, /i/, and /t/ (in transcription, phonemes are placed between slashes), and the word “pit” also has three: /p/, /i/, and /t/. In spoken languages, phonemes are produced by the positions and movements of the vocal tract, including our lips, teeth, tongue, vocal cords, and throat, whereas in sign languages phonemes are defined by the shapes and movement of the hands. English contains about 45 phonemes.
Whereas phonemes are the smallest units of sound in language, phonetics is the study of individual speech sounds; phonology is the study of phonemes, which are the speech sounds of an individual language. These two heavily overlapping subfields cover all the sounds that humans can make, as well as which sounds make up different languages.
Morpheme and Morphology
A morpheme is a string of one or more phonemes that makes up the smallest units of meaning in a language. Some morphemes, such as one-letter words like “I” and “a,” are also phonemes, but most morphemes are made up of combinations of phonemes. Some morphemes are prefixes and suffixes used to modify other words. For example, the syllable “re-” as in “rewrite” or “repay” means “to do again,” and the suffix “-est” as in “happiest” or “coolest” means “to the maximum.”
Morphology is the study of words and other meaningful units of language like suffixes and prefixes. A morphologist would be interested in the relationship between words like “dog” and “dogs” or “walk” and “walking,” and how people figure out the differences between those words.
Syntax
Syntax is the set of rules of a language by which we construct sentences. Each language has a different syntax. The syntax of the English language requires that each sentence have a noun and a verb, each of which may be modified by adjectives and adverbs. Some syntaxes make use of the order in which words appear, while others do not.
Syntax is the study of sentences and phrases, or how people put words into the right order so that they can communicate meaningfully. All languages have underlying rules of syntax, which, along with morphological rules, make up every language’s grammar. An example of syntax coming into play in language is “Eugene walked the dog” versus “The dog walked Eugene.” The order of words is not arbitrary—in order for the sentence to convey the intended meaning, the words must be in a certain order.51
Semantics and Pragmatics
Semantics , generally, is about the meaning of sentences. Someone who studies semantics is interested in words and what real-world object or concept those words denote, or point to.
Pragmatics is an even broader field that studies how the context of a sentence contributes to meaning. For example, someone shouting “Fire!” has a very different meaning if they are in charge of a seven-gun salute than it does if they are sitting in a crowded movie theater. Every language is different. In English, an adjective comes before a noun (“red house”), whereas in Spanish, the adjective comes after (“casa [house] roja [red].”) In German, you can put noun after noun together to form giant compound words; in Chinese, the pitch of your voice determines the meaning of your words. in American Sign Language, you can convey full, grammatical sentences with tense and aspect by moving your hands and face. But all languages have structural underpinnings that make them logical for the people who speak and understand them.52
Cognitive Language and Communication
When learning one or more languages in middle childhood, children are able to understand that there are many complex parts including comprehension, fluency, and meaning when communicating. The following are areas of cognitive language and communication.
Lexicon
Every language has its rules, which act as a framework for meaningful communication. But what do people fill that framework up with? The answer is, of course, words. Every human language has a lexicon —the sum total of all of the words in that language. By using grammatical rules to combine words into logical sentences, humans can convey an infinite number of concepts.
Grammar
Because all language obeys a set of combinatory rules, we can communicate an infinite number of concepts. While every language has a different set of rules, all languages do obey rules. These rules are known as grammar. Speakers of a language have internalized the rules and exceptions for that language’s grammar. There are rules for every level of language—word formation (for example, native speakers of English have internalized the general rule that -ed is the ending for past-tense verbs, so even when they encounter a brand-new verb, they automatically know how to put it into past tense); phrase formation (for example, knowing that when you use the verb “buy,” it needs a subject and an object; “She buys” is wrong, but “She buys a gift” is okay); and sentence formation.
Older children are also able to learn new rules of grammar with more flexibility. While younger children are likely to be reluctant to give up saying "I goed there", older children will learn this rather quickly along with other rules of grammar.
Vocabulary
One of the reasons that children can classify objects in so many ways is that they have acquired a vocabulary to do so. By fifth grade, a child's vocabulary has grown to 40,000 words. It grows at a rate that exceeds that of those in early childhood. This language explosion, however, differs from that of younger children because it is facilitated by being able to associate new words with those already known, and because it is accompanied by a more sophisticated understanding of the meanings of a word.
Context
Words do not possess fixed meanings but change their interpretation as a function of the context in which they are spoken. We use contextual information —the information surrounding language—to help us interpret it. Context is how everything within language works together to convey a particular meaning. Context includes tone of voice, body language, and the words being used. Depending on how a person says something, holds his or her body, or emphasizes certain points of a sentence, a variety of different messages can be conveyed. For example, the word “awesome,” when said with a big smile, means the person is excited about a situation. “Awesome,” said with crossed arms, rolled eyes, and a sarcastic tone, means the person is not thrilled with the situation.54
New Understanding
Those in middle and late childhood are also able to think of objects in less literal ways. For example, if asked for the first word that comes to mind when one hears the word "pizza", the younger child is likely to say "eat" or some word that describes what is done with a pizza. However, the older child is more likely to place pizza in the appropriate category and say "food". This sophistication of vocabulary is also evidenced by the fact that older children tell jokes and delight in doing so. They may use jokes that involve plays on words such as "knock-knock" jokes or jokes with punch lines. Young children do not understand play on words and tell "jokes" that are literal or slapstick, such as "A man fell down in the mud! Isn't that funny?"56
Contributors and Attributions
51. Beginning Psychology – Intelligence and Language by Charles Stangor is licensed under CC BY-NC-SA 3.0
52. Child Development – Unit 6: Language Development references Psychology .by Boundless, licensed under CC BY-SA 4.0
54. Beginning Psychology – Intelligence and Language by Charles Stangor is licensed under CC BY-NC-SA 3.0
56. Lifespan Development - Module 6: Middle Childhood by Lumen Learning references Psyc 200 Lifespan Psychology by Laura Overstreet, licensed under CC BY 4.0
Beginning Psychology – Intelligence and Language by Charles Stangor is licensed under CC BY-NC-SA 3.0


8.1: Bifurcation of Equilibria I
We will now study the topic of bifurcation of equilibria of autonomous vector fields, or ‘’what happens as an equilibrium point loses hyperbolicity as a parameter is varied?” We will study this question through a series of examples, and then consider what the examples teach us about the ‘’general situation” (and what this might be).
ExAMPLE \(\PageIndex{22}\): The Saddle-Node Bifurcation
Consider the following nonlinear, autonomous vector field on \(\mathbb{R}^2\):
\(\dot{x} = \mu x^2\),
\[\dot{y} = y, (x, y) \in \mathbb{R}^2, \label{8.1}\]
where \(\mu\) is a (real) parameter. The equilibrium points of (8.1) are given by:
\[(x, y) = (\sqrt{μ}, 0), (-\sqrt{μ}, 0). \label{8.2}\]
It is easy to see that there are no equilibrium points for \(\mu < 0\), one equilibrium point for \(\mu = 0\), and two equilibrium points for \(\mu > 0\).
The Jacobian of the vector field evaluated at each equilibrium point is given by:
\[(\sqrt{μ}, 0): \begin{pmatrix} {-2\sqrt{μ}}&{0}\\ {0}&{-1} \end{pmatrix}, \label{8.3}\]
from which it follows that the equilibria are hyperbolic and asymptotically stable for \(\mu > 0\), and nonhyperbolic for \(\mu = 0\).
\[(-\sqrt{μ}, 0): \begin{pmatrix} {2\sqrt{μ}}&{0}\\ {0}&{1} \end{pmatrix} \label{8.4}\]
from which it follows that the equilibria are hyperbolic saddle points for \(\mu > 0\), and nonhyperbolic for \(\mu = 0\). We emphasize again that there are no equilibrium points for \(\mu < 0\).
As a result of the "structure" of (8.1) we can easily represent the behavior of the equilibria as a function of \(\mu\) in a bifurcation diagram. That is, since the x and y components of (8.1) are "decoupled", and the change in the number and stability of equilibria us completely captured by the x coordinates, we can plot the x component of the vector field as a function of \(\mu\), as we show in Fig. 8.1.
In Fig. 8.2 we illustrate the bifurcation of equilibria for (8.1) in the \(x - y\) plane.
This type of bifurcation is referred to as a saddle-node bifurcation (occasionally it may also be referred to as a fold bifurcation or tangent bifurcation, but these terms are used less frequently).
The key characteristic of the saddle-node bifurcation is the following. As a parameter (\(\mu\)) is varied, the number of equilibria change from zero to two, and the change occurs at a parameter value corresponding to the two equilibria coalescing into one nonhyperbolic equilibrium.
\(\mu\) is called the bifurcation parameter and \(\mu = 0\) is called the bifurcation point.
Example \(\PageIndex{23}\) (The Transcritical Bifurcation)
Consider the following nonlinear, autonomous vector field on \(\mathbb{R}^2\):
\(\dot{x} = \mu x-x^2\),
\[\dot{y} = y, (x, y) \in \mathbb{R}^2, \label{8.5}\]
where \(\mu\) is a (real) parameter. The equilibrium points of (8.5) are given by:
\[(x, y) = (0, 0), (\mu, 0). \label{8.6}\]
The Jacobian of the vector field evaluated at each equilibrium point is given by:
\[(0,0) \begin{pmatrix} {\mu}&{0}\\ {0}&{-1} \end{pmatrix} \label{8.7}\]
\[(\mu, 0) \begin{pmatrix} {-\mu}&{0}\\ {0}&{1} \end{pmatrix} \label{8.8}\]
from which it follows that (0, 0) is asymptotically stable for \(\mu < 0\), and a hyperbolic saddle for \(\mu > 0\), and \((\mu, 0)\) is a hyperbolic saddle for \(\mu < 0\) and asymptotically stable for \(\mu > 0\). These two lines of fixed points cross at \(\mu = 0\), at which there is only one, nonhyperbolic fixed point.
In Fig. 8.3 we show the bifurcation diagram for (8.5) in the \(\mu - x\) plane.
In Fig. 8.4 we illustrate the bifurcation of equilibria for (8.5) in the \(x - y\) plane for \(\mu < 0\), \(\mu = 0\), and \(\mu > 0\).
This type of bifurcation is referred to as a transcritical bifurcation.
The key characteristic of the transcritical bifurcation is the following. As a parameter (\(\mu\)) is varied, the number of equilibria change from two to one, and back to two, and the change in number of equilibria occurs at a parameter value corresponding to the two equilibria coalescing into one nonhyperbolic equilibrium.
ExAMPLE \(\PageIndex{24}\) (The (Supercritical) Pitchfork Bifurcation) .
Consider the following nonlinear, autonomous vector field on \(\mathbb{R}^2\):
\(\dot{x} = \mu x-x^3\),
\[\dot{y} = -y, (x, y) \in \mathbb{R}^2, \label{8.9}\]
where \(\mu\) is a (real) parameter. The equilibrium points of (8.9) are given by:
\[(x, y) = (0, 0), (\sqrt{\mu}, 0), (-\sqrt{\mu}, 0) \label{8.10}\]
The Jacobian of the vector field evaluated at each equilibrium point is given by:
\[(0, 0) \begin{pmatrix} {\mu}&{0}\\ {0}&{1} \end{pmatrix} \label{8.11}\]
\[(\pm \sqrt{μ}, 0) \begin{pmatrix} {-2\mu}&{0}\\ {0}&{1} \end{pmatrix} \label{8.12}\]
from which it follows that (0, 0) is asymptotically stable for \(\mu < 0\), and a hyperbolic saddle for \(\mu > 0\), and \(( \pm \sqrt{μ}, 0)\) are asymptotically stable for \(\mu > 0\), and do not exist for \(\mu < 0\). These two curves of fixed points pass through zero at \(\mu = 0\), at which there is only one, nonhyperbolic fixed point.
In Fig. 8.5 we show the bifurcation diagram for (8.9) in the \(\mu - x\) plane.
In Fig. 8.6 we illustrate the bifurcation of equilibria for (8.9) in the \(x - y\) plane for \(\mu < 0\), \(\mu = 0\), and \(\mu > 0\).
Example \(\PageIndex{25}\) (The (Subcritical) Pitchfork Bifurcation)
Consider the following nonlinear, autonomous vector field on \(\mathbb{R}^2\):
\(\dot{x} = \mu x+x^3\),
\[\dot{y} = y, (x, y) \in \mathbb{R}^2, \label{8.13}\]
where \(\mu\) is a (real) parameter. The equilibrium points of (8.9) are given by:
\[(x, y) = (0, 0), (\sqrt{μ}, 0), (-\sqrt{μ}, 0). \label{8.14}\]
The Jacobian of the vector field evaluated at each equilibrium point is given by:
\[(0, 0) \begin{pmatrix} {\mu}&{0}\\ {0}&{-1} \end{pmatrix} \label{8.15}\]
\[(\pm \sqrt{\mu}, 0) \begin{pmatrix} {-2\mu}&{0}\\ {0}&{1} \end{pmatrix} \label{8.16}\]
from which it follows that (0, 0) is asymptotically stable for \(\mu < 0\), and a hyperbolic saddle for \(\mu > 0\), and \((\pm \sqrt{μ}, 0)\) are hyperbolic saddles for \(\mu < 0\), and do not exist for \(\mu > 0\). These two curves of fixed points pass through zero at \(\mu = 0\), at which there is only one, nonhyperbolic fixed point. In Fig. 8.7 we show the bifurcation diagram for (8.13) in the \(\mu - x\) plane.
In Fig. 8.8 we illustrate the bifurcation of equilibria for (8.9) in the \(x - y\) plane for \(\mu < 0\), \(\mu = 0\), and \(\mu > 0\).
We note that the phrase supercritical pitchfork bifurcation is also referred to as a soft loss of stability and the phrase subcritical pitchfork bifurcation is referred to as a hard loss of stability. What this means is the following. In the supercritical pitchfork bifurcation as \(\mu\) goes from negative to positive the equilibrium point loses stability, but as \(\mu\) increases past zero the trajectories near the origin are bounded in how far away from the origin they can move. In the subcritical pitchfork bifurcation the origin loses stability as \(\mu\) increases from negative to positive, but trajectories near the unstable equilibrium can become unbounded.
It is natural to ask the question,
"what is common about these three examples of bifurcations of fixed points of one dimensional autonomous vector fields?" We note the following.
- A necessary (but not sufficient) for bifurcation of a fixed point is nonhyperbolicity of the fixed point.
- The "nature" of the bifurcation (e.g. numbers and stability of fixed points that are created or destroyed) is determined by the form of the nonlinearity.
But we could go further and ask what is in common about these examples that could lead to a definition of the bifurcation of a fixed point for autonomous vector fields? From the common features we give the following definition.
Definition 21 (Bifurcation of a fixed point of a one dimensional autonomous vector field)
We consider a one dimensional autonomous vector field depending on a parameter, \(\mu\). We assume that at a certain parameter value it has a fixed point that is not hyperbolic. We say that a bifurcation occurs at that "nonhyperbolic parameter value" if for \(\mu\) in a neighborhood of that parameter value the number of fixed points and their stability changes.
Finally, we finish the discussion of bifurcations of a fixed point of one dimensional autonomous vector fields with an example showing that a nonhyperbolic fixed point may not bifurcate as a parameter is varied, i.e. non-hyperbolicity is a necessary, but not sufficient, condition for bifurcation.
Example \(\PageIndex{26}\)
We consider the one dimensional autonomous vector field:
\[\dot{x} =\mu-x^3, x \in \mathbb{R}, \label{8.17}\]
where \(\mu\) is a parameter. This vector field has a nonhyperbolic fixed point at x = 0 for \(\mu = 0\). The curve of fixed points in the \(\mu - x\) plane is given by \(\mu = x^3\), and the Jacobian of the vector field is \(-3x^2\), which is strictly negative at all fixed points, except the nonhyperbolic fixed point at the origin.
In fig. 8.9 we plot the fixed points as a function of \(\mu\).
We see that there is no change in the number or stability of the fixed points for \(\mu > 0\) and \(\mu < 0\). Hence, no bifurcation.


5: Early Childhood Last updated Save as PDF Page ID 60359 Lumen Learning Lumen Learning 5.1: Why It Matters- Early Childhood 5.2: Introduction to Physical Development in Early Childhood 5.3: Growth and Nutrition in Early Childhood 5.4: Physical Development in Early Childhood 5.5: Introduction to Cognitive Development in Early Childhood 5.6: Piaget’s Theory of Cognitive Development 5.7: Theory of Mind 5.8: Language Development in Early Childhood 5.9: Introduction to Emotional and Social Development in Early Childhood 5.10: Developing a Concept of Self 5.11: Psychodynamic and Psychosocial Theories of Early Childhood 5.12: Gender and Early Childhood 5.13: Family Life and Parenting Styles 5.14: Learning and Behavior Modification 5.15: Childhood Stress and Development 5.16: Putting It Together- Early Childhood 5.17: Discussion- Parenting Styles 5.18: Assignment- Children’s Toys 5.19: Assignment- Children’s Media 5.20: Assignment- Preschool Journal


5.2: Relating Pressure, Volume, Amount, and Temperature - The Ideal Gas Law
- Identify the mathematical relationships between the various properties of gases
- Use the ideal gas law to compute the values of various gas properties under specified conditions
The Ideal Gas Law
To this point, four separate laws have been discussed that relate pressure, volume, temperature, and the number of moles of the gas:
- Boyle’s law: PV = constant; at constant T and n
- Amontons’s law: \(\dfrac{P}{T}\) = constant; at constant V and n
- Charles’s law: \(\dfrac{V}{T}\) = constant; at constant P and n
- Avogadro’s law: \(\dfrac{V}{n}\) = constant; at constant P and T
Combining these four laws yields the ideal gas law , a relation between the pressure, volume, temperature, and number of moles of a gas:
\[ \frac{P V}{n T}=constant \]
\[PV=nRT \]
where P is the pressure of a gas, V is its volume, n is the number of moles of the gas, T is its temperature on the kelvin scale, and R is a constant called the ideal gas constant or the universal gas constant. The units used to express pressure, volume, and temperature will determine the proper form of the gas constant as required by dimensional analysis, the most commonly encountered values being 0.08206 L atm mol –1 K –1 and 8.3145 kPa L mol –1 K –1 .
The ideal gas law is easy to remember and apply in solving problems , as long as you use the proper values and units for the gas constant, R.
Gases whose properties of P , V , and T are accurately described by the ideal gas law (or the other gas laws) are said to exhibit ideal behavior or to approximate the traits of an ideal gas . An ideal gas is a hypothetical construct that may be used along with kinetic molecular theory to effectively explain the gas laws as will be described in a later module of this chapter. Although all the calculations presented in this module assume ideal behavior, this assumption is only reasonable for gases under conditions of relatively low pressure and high temperature. In the final module of this chapter, a modified gas law will be introduced that accounts for the non-ideal behavior observed for many gases at relatively high pressures and low temperatures.
The ideal gas equation contains five terms, the gas constant R and the variable properties P , V , n , and T . Specifying any four of these terms will permit use of the ideal gas law to calculate the fifth term as demonstrated in the following example exercises.
Methane, CH 4 , is being considered for use as an alternative automotive fuel to replace gasoline. One gallon of gasoline could be replaced by 655 g of CH 4 . What is the volume of this much methane at 25 °C and 745 torr?
Solution
The first step is to convert everything to the necessary units
- \( 655 \mathrm{~g} \; \mathrm{CH}_4 \times\left(\frac{1 \; \mathrm{Mole} \; \mathrm{CH}_4}{16.0 \; \mathrm{~g} \; \mathrm{CH}_4}\right)=40.9 \; \mathrm{mol} \; \mathrm{CH}_4 \nonumber \)
- \( 25^{\circ} \mathrm{C}+273=298 \mathrm{~K} \nonumber \)
- \( 745 \;torr \left(\frac{1 \; \mathrm{ATM}}{760 \text { Torr }}\right)=0.980 \; ATM \nonumber \)
Substitute in these values for the ideal gas equation as:
\[ V = \frac{\left(40.9 \operatorname{mol} \mathrm{CH}_4\right)\left(0.0821 \frac{\text {L ATM }}{\text {mol K }}\right)(298 \mathrm{~K})}{0.980 \text { ATM }} \]
V = 1021 L
Calculate the pressure in bar of 2520 moles of hydrogen gas stored at 27 °C in the 180-L storage tank of a modern hydrogen-powered car.
- Answer
-
350 bar
Combined Gas Law
If the number of moles of an ideal gas are kept constant while other conditions are changed, we can simplify the ideal gas equation to easily solve many common questions. This mathematical relationship is called the combined gas law where P 1 , V 1 , and T 1 describe the initial set of conditions while P 2 , V 2 , and T 2 describe the final set of conditions.
\[\dfrac{P_1V_1}{T_1}=\dfrac{P_2V_2}{T_2}\]
One convenient feature of this equation is that the units for pressure and volume cancel - so as long as you are consistent with both the initial and final conditions you can use this equation without requiring unit conversions. NOTE: The temperature must still use an absolute scale like kelvin.
When filled with air, a typical scuba tank with a volume of 13.2 L has a pressure of 153 atm (Figure \(\PageIndex{1}\)). If the water temperature is 27 °C, how many liters of air will such a tank provide to a diver’s lungs at a depth of approximately 70 feet in the ocean where the pressure is 3.13 atm?
Letting 1 represent the air in the scuba tank and 2 represent the air in the lungs, and noting that body temperature (the temperature the air will be in the lungs) is 37 °C, we have:
\[\dfrac{P_1V_1}{T_1}=\dfrac{P_2V_2}{T_2}⟶\mathrm{\dfrac{(153\:atm)(13.2\:L)}{(300\:K)}=\dfrac{(3.13\:atm)(\mathit{V}_2)}{(310\:K)}} \nonumber \]
Solving for V 2 :
\[V_2=\mathrm{\dfrac{(153\cancel{atm})(13.2\:L)(310\cancel{K})}{(300\cancel{K})(3.13\cancel{atm})}=667\:L} \nonumber \]
(Note: Be advised that this particular example is one in which the assumption of ideal gas behavior is not very reasonable, since it involves gases at relatively high pressures and low temperatures. Despite this limitation, the calculated volume can be viewed as a good “ballpark” estimate.)
A sample of ammonia is found to occupy 0.250 L under laboratory conditions of 27 °C and 0.850 atm. Find the volume of this sample at 0 °C and 1.00 atm.
- Answer
-
0.193 L
Whether scuba diving at the Great Barrier Reef in Australia (shown in Figure \(\PageIndex{2}\)) or in the Caribbean, divers must understand how pressure affects a number of issues related to their comfort and safety.
Pressure increases with ocean depth, and the pressure changes most rapidly as divers reach the surface. The pressure a diver experiences is the sum of all pressures above the diver (from the water and the air). Most pressure measurements are given in units of atmospheres, expressed as “atmospheres absolute” or ATA in the diving community: Every 33 feet of salt water represents 1 ATA of pressure in addition to 1 ATA of pressure from the atmosphere at sea level. As a diver descends, the increase in pressure causes the body’s air pockets in the ears and lungs to compress; on the ascent, the decrease in pressure causes these air pockets to expand, potentially rupturing eardrums or bursting the lungs. Divers must therefore undergo equalization by adding air to body airspaces on the descent by breathing normally and adding air to the mask by breathing out of the nose or adding air to the ears and sinuses by equalization techniques; the corollary is also true on ascent, divers must release air from the body to maintain equalization. Buoyancy, or the ability to control whether a diver sinks or floats, is controlled by the buoyancy compensator (BCD). If a diver is ascending, the air in his BCD expands because of lower pressure according to Boyle’s law (decreasing the pressure of gases increases the volume). The expanding air increases the buoyancy of the diver, and she or he begins to ascend. The diver must vent air from the BCD or risk an uncontrolled ascent that could rupture the lungs. In descending, the increased pressure causes the air in the BCD to compress and the diver sinks much more quickly; the diver must add air to the BCD or risk an uncontrolled descent, facing much higher pressures near the ocean floor. The pressure also impacts how long a diver can stay underwater before ascending. The deeper a diver dives, the more compressed the air that is breathed because of increased pressure: If a diver dives 33 feet, the pressure is 2 ATA and the air would be compressed to one-half of its original volume. The diver uses up available air twice as fast as at the surface.
Standard Conditions of Temperature and Pressure
We have seen that the volume of a given quantity of gas and the number of molecules (moles) in a given volume of gas vary with changes in pressure and temperature. Chemists sometimes make comparisons against a standard temperature and pressure (STP) for reporting properties of gases: 273.15 K (0.00 °C) and 1 bar pressure (100 kPa). Be aware that prior to 1982 STP was defined at a pressure of 1 atm (101.325 kPa) and this definition is still in common use. At STP , an ideal gas has a volume of about 22.4 L—this is referred to as the standard molar volume (Figure \(\PageIndex{3}\)).
Summary
The behavior of gases can be described by several laws based on experimental observations of their properties. The pressure of a given amount of gas is directly proportional to its absolute temperature, provided that the volume does not change (Amontons’s law). The volume of a given gas sample is directly proportional to its absolute temperature at constant pressure (Charles’s law). The volume of a given amount of gas is inversely proportional to its pressure when temperature is held constant (Boyle’s law). Under the same conditions of temperature and pressure, equal volumes of all gases contain the same number of molecules (Avogadro’s law).
The equations describing these laws are special cases of the ideal gas law, PV = nRT , where P is the pressure of the gas, V is its volume, n is the number of moles of the gas, T is its kelvin temperature, and R is the ideal (universal) gas constant.
Key Equations
- PV = nRT
Summary
- absolute zero
- temperature at which the volume of a gas would be zero according to Charles’s law.
- Amontons’s law
- (also, Gay-Lussac’s law) pressure of a given number of moles of gas is directly proportional to its kelvin temperature when the volume is held constant
- Avogadro’s law
- volume of a gas at constant temperature and pressure is proportional to the number of gas molecules
- Boyle’s law
- volume of a given number of moles of gas held at constant temperature is inversely proportional to the pressure under which it is measured
- Charles’s law
- volume of a given number of moles of gas is directly proportional to its kelvin temperature when the pressure is held constant
- ideal gas
- hypothetical gas whose physical properties are perfectly described by the gas laws
- ideal gas constant ( R )
- constant derived from the ideal gas equation R = 0.08226 L atm mol –1 K –1 or 8.314 L kPa mol –1 K –1
- ideal gas law
- relation between the pressure, volume, amount, and temperature of a gas under conditions derived by combination of the simple gas laws
- standard conditions of temperature and pressure (STP)
- 273.15 K (0 °C) and 1 bar (100 kPa)
- standard molar volume
- volume of 1 mole of gas at STP, approximately 22.4 L for gases behaving ideally


8.7F: Deltaproteobacteria
- Review the Deltaproteobacteria class of Proteobacteria
Deltaproteobacteria is a class of Proteobacteria. All species of this group are, like all Proteobacteria, Gram-negative.
The Deltaproteobacteria comprise a branch of predominantly aerobic genera, the fruiting-body-forming Myxobacteria that release myxospores in unfavorable environments. It is a branch of strictly anaerobic genera, which contains most of the known sulfate- (Desulfovibrio, Desulfobacter, Desulfococcus, Desulfonema, etc. ) and sulfur-reducing bacteria (e.g. Desulfuromonas spp.) alongside several other anaerobic bacteria with different physiology (e.g. ferric iron-reducing Geobacter spp. and syntrophic Pelobacter and Syntrophus spp.). A pathogenic intracellular Deltaproteobacteria has recently been identified.
The myxobacteria (“slime bacteria”) are a group of bacteria that predominantly live in the soil and feed on insoluble organic substances. The myxobacteria have very large genomes, relative to other bacteria, e.g. 9–10 million nucleotides. Sorangium cellulosum has the largest known (as of 2008) bacterial genome, at 13.0 million nucleotides. Myxobacteria are included among the delta group of proteobacteria, a large taxon of Gram-negative forms.
Myxobacteria can move actively by gliding. They typically travel in swarms (also known as wolf packs), containing many cells kept together by intercellular molecular signals. Individuals benefit from aggregation as it allows accumulation of extracellular enzymes which are used to digest food. This in turn increases feeding efficiency. Myxobacteria produce a number of biomedically and industrially useful chemicals, such as antibiotics. They export those chemicals outside of the cell.
The currently accepted taxonomy is based on the List of Prokaryotic Names with Standing in Nomenclature (LPSN) and National Center for Biotechnology Information (NCBI) and the phylogeny is based on 16S rRNA-based LTP release 106 by ‘The All-Species Living Tree’ Project.
Key Points
- The Deltaproteobacteria comprise a branch of predominantly aerobic genera.
- Deltaproteobacteria include the fruiting-body-forming Myxobacteria which release myxospores in unfavorable environments.
- Deltaproteobacteria include a branch of strictly anaerobic genera, which contains most of the known sulfate- (Desulfovibrio, Desulfobacter, Desulfococcus, Desulfonema, etc. ) and sulfur-reducing bacteria (e.g. Desulfuromonas spp. ).
Key Terms
- aerobic : Living or occurring only in the presence of oxygen.
- Deltaproteobacteria : Deltaproteobacteria is a class of Proteobacteria. All species of this group are, like all Proteobacteria, Gram-negative.
- Gram-negative : Gram-negative bacteria are bacteria that do not retain crystal violet dye in the Gram staining protocol. In a Gram stain test, a counterstain (commonly safranin) is added after the crystal violet, coloring all Gram-negative bacteria with a red or pink color.


19.3: Implications of Climate Change
Although we’ve all experienced the effects of climate change over the past decade it’s not straightforward for climatologists to make the connection between a warming climate and specific weather events, and most are justifiably reluctant to ascribe any specific event to climate change. In this respect, the best measures of climate change are those that we can detect over several decades, such as the temperature changes shown in Figure \(\PageIndex{2}\), or the sea-level rise shown in Figure \(\PageIndex{1}\). As already stated, sea level has risen approximately 20 cm since 1750, and that rise is attributed to both warming (and therefore expanding) seawater and melting glaciers and other land-based snow and ice (melting of sea ice does not contribute directly to sea-level rise as it is already floating in the ocean).
Projections for sea-level rise to the end of this century vary widely. This is in large part because we do not know which of the possible climate change scenarios we will most closely follow, but many are in the range from 0.5 m to 2.0 m. One of the problems in predicting sea-level rise is that we do not have a strong understanding of how large ice sheets—such as Greenland and Antarctica—will respond to future warming. Another issue is that the oceans don’t respond immediately to warming. For example, with the current amount of warming, we are already committed to a future sea-level rise of between 1.3 m and 1.9 m, even if we could stop climate change today. This is because it takes decades to centuries for the existing warming of the atmosphere to be transmitted to depth within the oceans and to exert its full impact on large glaciers. Most of that committed rise would take place over the next century, but some would be delayed longer. And for every decade that the current rates of climate change continue, that number increases by another 0.3 m. In other words, if we don’t make changes quickly, by the end of this century we’ll be locked into about 3 m of future sea-level rise.
In a 2008 report, the Organisation for Economic Co-operation and Development (OECD) estimated that by 2070 approximately 150 million people living in coastal areas could be at risk of flooding due to the combined effects of sea-level rise, increased storm intensity, and land subsidence. The assets at risk (buildings, roads, bridges, ports, etc.) are in the order of $35 trillion ($35,000,000,000,000). Countries with the greatest exposure of population to flooding are China, India, Bangladesh, Vietnam, U.S.A., Japan, and Thailand. Some of the major cities at risk include Shanghai, Guangzhou, Mumbai, Kolkata, Dhaka, Ho Chi Minh City, Tokyo, Miami, and New York.
One of the other risks for coastal populations, besides sea-level rise, is that climate warming is also associated with an increase in the intensity of tropical storms (e.g., hurricanes or typhoons), which almost always bring serious flooding from intense rain and storm surges. Some recent examples are New Orleans in 2005 with Hurricane Katrina, and New Jersey and New York in 2012 with Hurricane Sandy (Figure \(\PageIndex{2}\)).
Tropical storms get their energy from the evaporation of warm seawater in tropical regions. In the Atlantic Ocean, this takes place between 8° and 20° N in the summer. Figure \(\PageIndex{3}\) shows the variations in the sea-surface temperature (SST) of the tropical Atlantic Ocean (in blue) versus the amount of power represented by Atlantic hurricanes between 1950 and 2008 (in red). Not only has the overall intensity of Atlantic hurricanes increased with the warming since 1975, but the correlation between hurricanes and sea-surface temperatures is very strong over that time period.
Because warm air is able to hold more water than cold air, the general global trend over the past century has been one of increasing precipitation (Figure \(\PageIndex{4}\)).
A similar trend is evident for British Columbia based on weather data from 1945 to 2005 for 29 stations distributed around the province (Figure \(\PageIndex{5}\)). Of those stations, 19 show an increase in precipitation and 10 show a decrease. While the decreases are all less than 12%, some of the increases are greater than 48%. Based on the data from these stations, it is estimated that approximately 60 mm/year more precipitation fell on British Columbia in 2005 compared with 1945. That is equivalent to about six months of the average flow of the Fraser River.
While the overall amount of precipitation (total volume of rain plus snow) increased at 19 out of 29 stations between 1945 and 2005, the amount of snowfall decreased at every single station. This is a disturbing trend for many of us, including: operators and users of winter resorts and hydroelectric dams, the Wildfire Management Branch, people who drink water from reservoirs that are replenished by snow, and people who eat food that is grown across western Canada and is irrigated with water derived from melting snow.
Figure \(\PageIndex{6}\) shows the monthly precipitation data for Penticton, BC from 1950 to 2005 (solid line) along with the ENSO (El Niño Southern Oscillation) index values (dotted line). High ENSO index values correspond to strong El Niño events, such as 1983 and 1998. Describe the relationship between ENSO and precipitation in B.C.’s southern interior.
It’s not necessarily a consistent relationship.
See Appendix 3 for Exercise 19.4 answers.
The geographical ranges of diseases and pests, especially those caused or transmitted by insects, have been shown to extend toward temperate regions because of climate change. West Nile virus and Lyme disease are two examples that already directly affect Canadians, while dengue fever could be an issue in the future. Canadians are also indirectly affected by the increase in populations of pests such as the mountain pine beetle (Figure \(\PageIndex{7}\)).
A summary of the impacts of climate change on natural disasters is given in Figure \(\PageIndex{8}\). The major types of disasters related to climate are floods and storms, but the health implications of extreme temperatures are also becoming a great concern. In the decade 1971 to 1980, extreme temperatures were the fifth most common natural disasters; by 2001 to 2010, they were the third most common.
For several days in both June and July of 2019, many parts of Europe experienced massive heat waves with all-time national record temperatures set in several countries (Belgium, Finland, France, Germany, Luxembourg, Netherlands, and United Kingdom) (Figure \(\PageIndex{9}\)). At the time of writing (August 2019) the death toll from these events is not known. A similar event in Russia in 2010 is estimated to have resulted in over 55,000 deaths.
If you look back to Figure \(\PageIndex{3}\) and the related text, you can easily see what aspects of our way of life are the most responsible for climate change. Think about how you could make changes to your own lifestyle to reduce your impact on the climate. It may depend on where you live, and the degree to which fossil fuels are used to generate the electricity that you use, but it’s most likely to include how, how far, how fast, and how frequently you move around.
If you hold the opinion that there isn’t much point in making changes to your lifestyle because others won’t or because your contribution is only a tiny fraction of the problem, bear in mind that all of us have the opportunity to set an example that others can follow. And remember the words of the American anthropologist Margaret Mead:
Never doubt that a small group of thoughtful, committed citizens can change the world. Indeed, it is the only thing that ever has.
Image Descriptions
| Negative 12% to 0% | 0% to 12% | 12% to 24% | 24% to 36% | 26% to 48% | Greater than 48% |
|---|---|---|---|---|---|
|


1: Vectors in Euclidean Space
In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually \(x, y\) or \(x, y, z\), respectively). The graph of a function of two variables, say, \(z = f(x,y)\), lies in Euclidean space , which in the Cartesian coordinate system consists of all ordered triples of real numbers \((a, b, c)\). Since Euclidean space is 3-dimensional, we denote it by \(\mathbb{R}^{3}\). The graph of \(f\) consists of the points \((x, y, z) = (x, y, f(x, y))\).
-
- 1.1: Introduction
- In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually \(x, y\) or \(x, y, z\), respectively). The graph of a function of two variables, say, \(z = f(x,y)\), lies in Euclidean space, which in the Cartesian coordinate system consists of all ordered triples of real numbers \((a, b, c)\). Since Euclidean space is 3-dimensional, we denote it by \(\mathbb{R}^{3}\). The graph of \(f\) consists of the points \((x, y, z) = (x, y, f(x, y))\).
-
- 1.2: Vector Algebra
- Now that we know what vectors are, we can start to perform some of the usual algebraic operations on them (e.g. addition, subtraction). Before doing that, we will introduce the notion of a scalar. The term scalar was invented to convey the sense of something that could be represented by a point on a scale or graduated ruler. The word vector comes from Latin, where it means "carrier''. Examples of scalar quantities are mass, electric charge, and speed (not velocity).
-
- 1.4: Cross Product
- In Section 1.3 we defined the dot product, which gave a way of multiplying two vectors. The resulting product, however, was a scalar, not a vector. In this section we will define a product of two vectors that does result in another vector. This product, called the cross product, is only defined for vectors in \(\mathbb{R}^{3}\). The definition may appear strange and lacking motivation, but we will see the geometric basis for it shortly.
-
- 1.5: Lines and Planes
- Now that we know how to perform some operations on vectors, we can start to deal with some familiar geometric objects, like lines and planes, in the language of vectors. The reason for doing this is simple: using vectors makes it easier to study objects in 3-dimensional Euclidean space. We will first consider lines.
-
- 1.6: Surfaces
- A plane in Euclidean space is an example of a surface, which we will define informally as the solution set of the equation F(x,y,z)=0 in R3, for some real-valued function F. For example, a plane given by ax+by+cz+d=0 is the solution set of F(x,y,z)=0 for the function F(x,y,z)=ax+by+cz+d. Surfaces are 2-dimensional. The plane is the simplest surface, since it is "flat''. In this section we will look at some surfaces that are more complex, the most important of which are spheres and the cylinders
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- 1.7: Curvilinear Coordinates
- The two types of curvilinear coordinates which we will consider are cylindrical and spherical coordinates. Instead of referencing a point in terms of sides of a rectangular parallelepiped, as with Cartesian coordinates, we will think of the point as lying on a cylinder or sphere. Cylindrical coordinates are often used when there is symmetry around the \(z\)-axis; spherical coordinates are useful when there is symmetry about the origin.
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- 1.8: Vector-Valued Functions
- A vector-valued function of a real variable is a rule that associates a vector \(\textbf{f}(t)\) with a real number \(t\), where \(t\) is in some subset \(D\) of \(\mathbb{R}^1\) (called the domain of \(f\)). We write f: \(D → \)\(\mathbb{R}^ 3\) to denote that f is a mapping of \(D\) into \(\mathbb{R}^ 3\) .
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- 1.9: Arc Length
- A curve can have many parametrizations, with different speeds, so which one is the best to use? In some situations the arc length parametrization can be useful. The idea behind this is to replace the parameter \(t\), for any given smooth parametrization \(\textbf{f}(t)\) defined on \([a,b]\), by a new parameter \(s\).
Thumbnail: Illustration of the Cartesian coordinate system for 3D. (Public Domain; Jorge Stolfi).