on 27-Jan-2017 (Fri)

Instead of using coercion or outright treachery, the perfect courtier got his way through seduction, charm, deception, and subtle strategy, always planning several moves ahead.

Today we face a peculiarly similar paradox to that of the courtier: Everything must appear civilized, decent, democratic, and fair. But if we play by those rules too strictly, if we take them too literally, we are crushed by those around us who are not so foolish.

Niccolò Machiavelli wrote, “Any man who tries to be good all the time is bound to come to ruin among the great number who are not good.”

Napoleon advised: Place your iron hand inside a velvet glove.

If you can master the arts of indirection, learning to seduce, charm, deceive, and subtly outmaneuver your opponents, you will attain the heights of power.

If you can master the arts of indirection, you will be able to make people bend to your will without their realizing what you have done. And if they do not realize what you have done, they will neither resent nor resist you.

Another strategy of the supposed nonplayer is to demand equality in every area of life. Everyone must be treated alike, whatever their status and strength.

To some people the notion of consciously playing power games—no matter how indirect—seems evil, asocial, a relic of the past. They believe they can opt out of the game by behaving in ways that have nothing to do with power. You must beware of such people, for while they express such opinions outwardly, they are often among the most adept players at power.

Yet another way of avoiding the game would be perfect honesty and straightforwardness, since one of the main techniques of those who seek power is deceit and secrecy. But being perfectly honest will inevitably hurt and insult a great many people, some of whom will choose to injure you in return.

The only means to gain one’s ends with people are force and cunning. Love also, they say; but that is to wait for sunshine, and life needs every moment

the better you are at dealing with power, the better friend, lover, husband, wife, and person you become.

Learning the game of power requires a certain way of looking at the world, a shifting of perspective. It takes effort and years of practice, for much of the game may not come naturally. Certain basic skills are required, and once you master these skills you will be able to apply the laws of power more easily.

The most important of these skills, and power’s crucial foundation, is the ability to master your emotions.

An emotional response to a situation is the single greatest barrier to power, a mistake that will cost you a lot more than any temporary satisfaction you might gain by expressing your feelings.

Emotions cloud reason, and if you cannot see the situation clearly, you cannot prepare for and respond to it with any degree of control.

Anger is the most destructive of emotional responses, for it clouds your vision the most.

If you are trying to destroy an enemy who has hurt you, far better to keep him off-guard by feigning friendliness than showing your anger.

Love and affection are potentially destructive, in that they blind you to the often self-serving interests of those whom you least suspect of playing a power game.

You cannot repress anger or love, or avoid feeling them, and you should not try. But you should be careful about how you express them, and most important, they should never influence your plans and strategies in any way.

Related to mastering your emotions is the ability to distance yourself from the present moment and think objectively about the past and future.

You must be able to look in both directions at once, the better to handle danger from wherever it comes. Such is the face you must create for yourself-one face looking continuously to the future and the other to the past.

I thought to myself with what means, with what deceptions, with how many varied arts, with what industry a man sharpens his wits to deceive another, and through these variations the world is made more beautiful.

For the future, the motto is, “No days unalert.” Nothing should catch you by surprise because you are constantly imagining problems before they arise.

Instead of spending your time dreaming of your plan’s happy ending, you must work on calculating every possible permutation and pitfall that might emerge in it.

The further you see, the more steps ahead you plan, the more powerful you become

The other face of Janus looks constantly to the past—though not to remember past hurts or bear grudges.

Half of the game is learning how to forget those events in the past that eat away at you and cloud your reason.

The real purpose of the backward-glancing eye is to educate yourself constantly—you look at the past to learn from those who came before you. (The many historical examples in this book will greatly help that process.)

There are no principles; there are only events.

There is no good and bad, there are only circumstances.

You begin by examining the mistakes you have made in the past, the ones that have most grievously held you back.

You analyze the mistakes in your past in terms of the 48 laws of power, and you extract from them a lesson and an oath: “I shall never repeat such a mistake; I shall never fall into such a trap again.”

If you can evaluate and observe your past self with the 48 laws, you can learn to break the patterns of the past—an immensely valuable skill.

Power requires the ability to play with appearances. To this end you must learn to wear many masks and keep a bag full of deceptive tricks.

Deception and masquerade should not be seen as ugly or immoral. All human interaction requires deception on many levels

Deception is a developed art of civilization and the most potent weapon in the game of power.

You cannot succeed at deception unless you take a somewhat distanced approach to yourself— unless you can be many different people, wearing the mask that the day and the moment require.

If deception is the most potent weapon in your arsenal, then patience in all things is your crucial shield.

Like mastering your emotions, patience is a skill—it does not come naturally.

nothing about power is natural; power is more godlike than anything in the natural world.

patience is the supreme virtue of the gods, who have nothing but time.

Impatience, on the other hand, only makes you look weak. It is a principal impediment to power.

Power is essentially amoral and one of the most important skills to acquire is the ability to see circumstances rather than good or evil

Power is a game—this cannot be repeated too often—and in games you do not judge your opponents by their intentions but by the effect of their actions.

It is only natural for people to cover up their actions with all kinds of justifications, always assuming that they have acted out of goodness. You must learn to inwardly laugh each time you hear this and never get caught up in gauging someone’s intentions and actions through a set of moral judgments that are really an excuse for the accumulation of power.

It is a game. Both of you behave as gentlemen. observing the rules of the game and taking nothing personally.

Half of your mastery of power comes from what you do not do, what you do not allow yourself to get dragged into.

Half of your mastery of power comes from what you do not do, what you do not allow yourself to get dragged into. For this skill you must learn to judge all things by what they cost you. As Nietzsche wrote, “The value of a thing sometimes lies not in what one attains with it, but in what one pays for it —what it costs us.”

Perhaps you will attain your goal, and a worthy goal at that, but at what price? Apply this standard to everything, including whether to collaborate with other people or come to their aid.

Never waste valuable time, or mental peace of mind, on the affairs of others—that is too high a price to pay.

Power is a social game. To learn and master it, you must develop the ability to study and understand people.

To be a master player you must also be a master psychologist. You must recognize motivations and see through the cloud of dust with which people surround their actions.

An understanding of people’s hidden motives is the single greatest piece of knowledge you can have in acquiring power. It opens up endless possibilities of deception, seduction, and manipulation.

People are of infinite complexity and you can spend a lifetime watching them without ever fully understanding them. So it is all the more important, then, to begin your education now.

Never discriminate as to whom you study and whom you trust.

Never trust anyone completely and study everyone, including friends and loved ones.

Finally, you must learn always to take the indirect route to power.

Disguise your cunning. Like a billiard ball that caroms several times before it hits its target, your moves must be planned and developed in the least obvious way.

By training yourself to be indirect, you can thrive in the modern court, appearing the paragon of decency while being the consummate manipulator.

The laws have a simple premise: Certain actions almost always increase one’s power (the observance of the law), while others decrease it and even ruin us (the transgression of the law).

The gods of power frown on the frivolous; they give ultimate satisfaction only to those who study and reflect, and punish those who skim the surfaces looking for a good time.

Any man who tries to be good all the time is bound to come to ruin among the great number who are not good. Hence a prince who wants to keep his authority must learn how not to be good, and use that knowledge, or refrain from using it, as necessity requires.

LAW 1 NEVER OUTSHINE THE MASTER

Always make those above you feel comfortably superior.

In your desire to please and impress your superiors, do not go too far in displaying your talents or you might accomplish the opposite—inspire fear and insecurity.

Make your masters appear more brilliant than they are and you will attain the heights of power.

Nicolas Fouquet, Louis XIV’s finance minister in the first years of his reign, loved lavish parties and money. Fouquet was clever and indispensable to the king, so when the prime minister died, he expected to be named the successor. Instead, the king abolish the position. This made Fouquet suspect that he was falling out of favor, and so he decided to ingratiate himself by staging a the best party ever. The party’s ostensible purpose would be to commemorate the completion of Fouquet’s château, but its real function was to pay tribute to the king.

Everyone was invited and everything was fancy as fuck.

He personally escorted the king through tours and shit, you know, brown nosing cheeks deep.

The party ran well into the night and everyone agreed it was the most amazing party they had ever attended.

The next day, Fouquet was arrested by the king. Three months later he went on trial for stealing from the country’s treasury. (Actually, most of the stealing he was accused of he had done on the king’s behalf and with the king’s permission.) Fouquet was found guilty and sent to prison.

Louis XIV, the Sun King, was a proud and arrogant man who wanted to be the center of attention at all times; he could not countenance being outdone in lavishness by anyone, and certainly not his finance minister. To succeed Fouquet, Louis chose Jean-Baptiste Colbert, a man famous for his parsimony and for giving the dullest parties in Paris. Colbert made sure that any money liberated from the treasury went straight into Louis’s hands. With the money, Louis built a palace even more magnificent than Fouquet’s—the glorious palace of Versailles. He used the same architects, decorators, and garden designer. And at Versailles, Louis hosted parties even more extravagant than the one that cost Fouquet his freedom.

Let us examine the situation. The evening of the party, as Fouquet presented spectacle on spectacle to Louis, each more magnificent than the one before, he imagined the affair as demonstrating his loyalty and devotion to the king. Not only did he think the party would put him back in the king’s favor, he thought it would show his good taste, his connections, and his popularity, making him indispensable to the king and demonstrating that he would make an excellent prime minister. Instead, however, each new spectacle, each appreciative smile bestowed by the guests on Fouquet, made it seem to Louis that his own friends and subjects were more charmed by the finance minister than by the king himself, and that Fouquet was actually flaunting his wealth and power. Rather than flattering Louis XIV, Fouquet’s elaborate party offended the king’s vanity. Louis would not admit this to anyone, of course—instead, he found a convenient excuse to rid himself of a man who had inadvertently made him feel insecure.

Such is the fate, in some form or other, of all those who unbalance the master’s sense of self, poke holes in his vanity, or make him doubt his pre-eminence.

When the evening began, Fouquet was at the top of the world. By the time it had ended, he was at the bottom. Voltaire, 1694-1778

To succeed Fouquet, Louis chose Jean-Baptiste Colbert, a man famous for his parsimony and for giving the dullest parties in Paris.

Colbert (new finance minister) made sure that any money liberated from the treasury went straight into Louis’s hands.

With it, Louis built a palace even more magnificent than Fouquet’s—the glorious palace of Versailles.

There, Louis hosted parties even more extravagant than the one that cost Fouquet his freedom.

The evening of the party, as Fouquet presented spectacle on spectacle to Louis, each more magnificent, thinking he was demonstrating his loyalty and devotion to the king. He was basically showing of connections, taste etc. and thought this would make him indispensable.

This made it seem to Louis that his own friends and subjects were more charmed by the finance minister than by the king himself, and that Fouquet was actually flaunting his wealth and power.

Rather than flattering Louis XIV, the party offended the king’s vanity.

Louis would not admit this —instead, he found a convenient excuse to rid himself of a man who had inadvertently made him feel insecure.

As Fouquet, in some form or other, of all those who unbalance the master’s sense of self, poke holes in his vanity, or make him doubt his pre-eminence.

Galileo depended on the generosity of rulers to support his research so he would sometimes make gifts of his inventions to the leading patrons of the time but his patrons usually paid him with gifts, not cash. This made for a life of constant insecurity and dependence.

He hit on a new strategy, when he discovered the moons of Jupiter. Instead of dividing the discovery among his patrons, as he had done in the past, he decided to focus exclusively on the Medicis.

He chose the Medicis because they had made Jupiter, the mightiest of the gods, the Medici symbol—a symbol of a power that went beyond politics and banking.

Galileo announced that "the moons of Jupiter offered themselves in the heavens” to his telescope at the same time as Cosimo II’s enthronement. He said that the four moons harmonized with the number of the Medicis.

After he dedicated the discovery to the Medicis, Galileo commissioned an emblem representing Jupiter sitting on a cloud with the four stars circling about him, and presented this to Cosimo II as a symbol of his link to the stars.

Cosimo II made Galileo his official court philosopher and mathematician, with a full salary.

For a scientist this was the shit. The days of begging for patronage were over.

Galileo gained more with his new strategy than he had in years of begging. The reason is simple: All masters want to appear more brilliant than other people.

They do not care about science or empirical truth or the latest invention ; they care about their name and glory. Galileo gave the Medicis infinitely more glory by linking their name with cosmic forces than he had by making them the patrons of some new scientific gadget or discovery.

Scientists too must serve masters. And their intellectual powers can make the master feel insecure, as if he were only there to supply the funds—an ugly, ignoble job.

The producer of a great work wants to feel he is more than just the provider of the financing.

He wants to appear creative and powerful, and also more important than the work produced in his name.

Galileo did not challenge the intellectual authority of the Medicis with his discovery, or make them feel inferior in any way; by literally aligning them with the stars, he made them shine brilliantly among the courts of Italy. He did not outshine the master, he made the master outshine all others.

Cells of the nervous system ● These fall broadly into two categories ○ Neurons: the excitable cells of the nervous system. ○ Glia: intimately associated with neurons, support neuronal function.

Others: endothelial and ependymal cells, fibroblasts ○ Endothelial cells form the blood vessels within the brain. ○ Ependymal cells line the ventricles ○ Fibroblast cells are in the meninges

All of the neurons possess specialised regions, some may have: ○ Cell body, also known as perikaryon or soma ○ An axon, which may branch, and may be very long ○ Dendrites (or sensory nerve endings, for ​sensory neurones which don’t have dendrites

The producer of a great work wants to feel he is more than just the provider of the financing.

Everyone has insecurities. When you show yourself in the world and display your talents, you naturally stir up all kinds of resentment, envy, and other manifestations of insecurity. This is to be expected. You cannot spend your life worrying about the petty feelings of others. With those above you, however, you must take a different approach: When it comes to power, outshining the master is perhaps the worst mistake of all.

Do not fool yourself into thinking that life has changed much since the days of Louis XIV and the Medicis. Those who attain high standing in life are like kings and queens: They want to feel secure in their positions, and superior to those around them in intelligence, wit, and charm. It is a deadly but common misperception to believe that by displaying and vaunting your gifts and talents, you are winning the master’s affection. He may feign appreciation, but at his first opportunity he will replace you with someone less intelligent, less attractive, less threatening, just as Louis XIV replaced the sparkling Fouquet with the bland Colbert. And as with Louis, he will not admit the truth, but will find an excuse to rid himself of your presence.

This Law involves two rules that you must realize. First, you can inadvertently outshine a master simply by being yourself. There are masters who are more insecure than others, monstrously insecure; you may naturally outshine them by your charm and grace.

No one had more natural talents than Astorre Manfredi, prince of Faenza. The most handsome of all the young princes of Italy, he captivated his subjects with his generosity and open spirit.

In the year 1500, Cesare Borgia laid siege to Faenza. When the city surrendered, the citizens expected the worst from the cruel Borgia, who, however, decided to spare the town: He simply occupied its fortress, executed none of its citizens, and allowed Prince Manfredi, eighteen at the time, to remain with his court, in complete freedom.

A few weeks later, though, soldiers hauled Astorre Manfredi away to a Roman prison. A year after that, his body was fished out of the River Tiber, a stone tied around his neck. Borgia justified the horrible deed with some sort of trumped-up charge of treason and conspiracy, but the real problem was that he was notoriously vain and insecure. The young man was outshining him without even trying. Given Manfredi’s natural talents, the prince’s mere presence made Borgia seem less attractive and charismatic. The lesson is simple: If you cannot help being charming and superior, you must learn to avoid such monsters of vanity. Either that, or find a way to mute your good qualities when in the company of a Cesare Borgia.

Second, never imagine that because the master loves you, you can do anything you want. Entire books could be written about favorites who fell out of favor by taking their status for granted, for daring to outshine. In late-sixteenth-century Japan, the favorite of Emperor Hideyoshi was a man called Sen no Rikyu. The premier artist of the tea ceremony, which had become an obsession with the nobility, he was one of Hideyoshi’s most trusted advisers, had his own apartment in the palace, and was honored throughout Japan. Yet in 1591, Hideyoshi had him arrested and sentenced to death. Rikyu took his own life, instead. The cause for his sudden change of fortune was discovered later: It seems that Rikyu, former peasant and later court favorite, had had a wooden statue made of himself wearing sandals (a sign of nobility) and posing loftily. He had had this statue placed in the most important temple inside the palace gates, in clear sight of the royalty who often would pass by. To Hideyoshi this signified that Rikyu had no sense of limits. Presuming that he had the same rights as those of the highest nobility, he had forgotten that his position depended on the emperor, and had come to believe that he had earned it on his own. This was an unforgivable miscalculation of his own importance and he paid for it with his life. Remember the...

When you show yourself in the world and display your talents, you naturally stir up all kinds of resentment, envy, and other manifestations of insecurity.

You cannot spend your life worrying about the petty feelings of others. With those above you, however, you must take a different approach: When it comes to power, outshining the master is perhaps the worst mistake of all.

The levels argument of the factor function must refer to those exact terms for reading in the data. But for the resulting output, the levels can be relabeled to whatever might be more meaningful for the application, such as “Bottom SES,” “Middle SES,” and “Top SES.” We accomplish the relabeling in R with the labels argument

A matrix is simply a two-dimensional array of values of the same type. A matrix can be created in R using the matrix command. The first argument specifies the contents of the matrix, in order. Other arguments specify the size of the matrix, the order in which the matrix should be filled, and, optionally, the names of the dimensions and rows and columns.

As you may have inferred by now, the indices are ordered such that the first index refers to the row and the second index refers to the column

An entire row or column of a matrix can be accessed by specifying its entire range or by leaving its range unspecified.

An array is a generalization of a matrix to multiple dimensions. There is really no need for a separate matrix function because it is merely the two-dimensional case of the array function. In the array function, the first argument specifies the ordered contents, the second argument specifies the size of each dimension, and an optional third argument specifies the names of the dimensions and levels within dimensions

the array function fills the array by incrementing the first index (row) first, then incrementing the second index (column) next, then incrementing the third index (layer) next, and so forth. Unlike the matrix function, there is no built-in way to load the contents into the array in a different ordering of dimensions

The list structure is a generic vector in which components can be of different types, and named.

We can get element i from a list, including its name, by putting i inside single square brackets. We can get the contents of element i by putting i inside double square brackets.

A data frame is much like a matrix, insofar as it has several columns of equal length. But each column can be of a different type, and, in particular, columns can be factors. A data frameisreallyatypeof list in which each component is thought of as a named column of a matrix, with different columns possibly of different types. The elements of a data frame can be accessed as if it were a list or as if it were a matrix.

The elements of the data frame can be accessed as for a list, by using names or single brackets or double brackets

This book uses a typical convention for arranging data in which every row of a data file contains one measurement instance, which might be one person in a survey, or one trial in a response-time experiment. Each row contains the value of the key measurement to be explained or predicted, and each row also contains values that indicate predictors or explanatory variables for that instance.

CSV files are easily loaded into R’s memory using the read.csv function

Thus, the columns of HGNdf are vectors or factors, named according to the words in the first row of the CSV file, and all of length equal to the number of data rows in the CSV file. It is important to note that columns with any character (non-numeric) entries are turned into factors

One way to save a data table is with the write.csv function. For example, the command

If you want to save data frames with all the factor information intact, then you can use the save command. The resulting file is in a special R format, not generic text, and the standard filename extension for this type of file is “.Rdata.”

If desired, several different variables can all be saved in a single Rdata file, simply by specifying them as the initial comma- separated arguments in the save command. All the variables are saved along with their names

The summary function detects the type, or “class,” of the argument provided to it, and returns a summary appropriate for that class of object.

if we construct a vector consisting of numerical values, summary provides the minimum value in the vector, median value, etc., as shown by this example

However, if we convert the vector to a factor, then the summary function provides a table with the frequency of each level

Thehead function returns the first few components of the variable put in its argument. The str function returns a compact display of the structure of its argument.

The aggregate function is very useful for summarizing data according to factor characteristics.

Another way to generate a table of counts is with the table function

The apply function is handy for collapsing arrays across specified dimensions, and applying a function to the data within the collapsed dimensions

Notice that the array is three-dimensional, with a datum in every one of the 24 cells. We would like to rearrange the data so that there is one datum per row, with each row also specifying the levels of the array from which the datum came. This is done by the melt command

In R’s basic command window, set the working directory by selecting menu items File → Change dir. In RStudio, set the working directory by selecting menu items Session → Set Working Directory.

In general, a function in R in defined by code of the form: functionName = function( arguments ) { commands } The commands inside the curly braces can extend over many lines of code.

You invoke the function by commanding R thus: functionName( arguments=argumentValues )

Notice the arrangement of curly braces across lines in the if-else structure above. In particular, the line containing “else” begins with a closing curly brace, which tells R that the “else” clause is continuing the preceding “if.” A line that begins with “else” causes an error

A simple way to measure processing time is with the proc.time function, which returns the current computer-system time. To measure the duration of a process, use proc.time at the beginning and end of the process, and compute the difference between the times

In general, for loops are slow relative to vectorized operations

Uncertainty is measured in terms of probability

Whenever we ask about how likely an outcome is, we always ask with a set of possible outcomes in mind. This set exhausts all possible outcomes, and the outcomes are all mutually exclusive. This set is called the sample space

The sample space is determined by the measurement operation we use to make an observation of the world.

Many coins minted by governments have the picture of an important person’s head on one side. This side is called “heads” or, technically, the “obverse.”

The degree of belief about a parameter can be denoted p(θ )

Both “probability” of head or tail outcome and “degree of belief ” in biases refer to sample spaces.

When we flip a given coin, we are sampling from the space of head or tail. When we grab a coin at random from a sack of coins, in which each coin may have a different bias, we are sampling from the space of possible biases

If you want values displayed in scientific notation (floating-point form) whatever their size, enter the command format short e

Use format bank for financial calculations; you get fixed point with two dec- imal digits (for cents)

Use format hex to get hexadecimal display.

Use format rat to display a number as a rational approximation (ratio of two integers)

he number of times the loop is executed may be calculated from the fol- lowing equation: floor last −first increment +1 where the MATLAB function floor(x) rounds x down toward −∞. This value is called the iteration or trip count.

A more general form of for is for index = v where v is any vector. The index moves through each element of the vector in turn, providing a neat way of processing each item in a list.

The MATLAB function clock returns a six-element vector with the current date and time in the format year, month, day, hour, minute, seconds. Thus, t0 records when the calculation starts.

The function etime returns the time in seconds elapsed between its two ar- guments, which must be vectors as returned by clock.

here is a neater way of monitoring the time taken to interpret MATLAB state- ments: the tic and toc function.

The MATLAB function rand generates a random number in the range 0–1

The simplest form of if in a single line is if condition; statements; end

In general, the elseif clause is used: if condition1 statementsA elseif condition2 statementsB

switch executes certain statements based on the value of a variable or expres- sion.

The symbol i may be used to assign complex values, for example, z=2+3*

If z is a complex number, real(z), imag(z), conj(z), and abs(z) all have the obvious meanings

A complex number may be represented in polar coordinates: z =re iθ angle(z) returns θ between −π and π ; that is, atan2(imag(z), real(z)). abs(z) returns the magnitude r

If y is complex, the statement plot(y) is equivalent to plot(real(y), imag(y)) The statement axis(’equal’) is necessary to make circles look round; it changes what is known as the aspect ratio of the monitor. axis(’normal’) gives the default aspect ratio.

For complex matrices, the operations ’ and .’ behave differently. The ’ op- erator is the complex conjugate transpose, meaning rows and columns are inter- changed and signs of imaginary parts are changed. The .’ operator, on the other hand, does a pure transpose without taking the complex conjugate.

In the simplest case of a uniform vector field ~ A and a surface S per- pendicular to the direction of the field, the flux U is defined as the product of the field magnitude and the area of the surface

If the vector field is uniform but is not perpendicular to the surface, as in Figure 1.6(b), the flux may be determined simply by finding the component of ~ A perpendicular to the surface and then multiplying that value by the surface area

A vector is the mathematical representation of a physical entity that may be characterized by size (or “magnitude”) and direction

The word “vector” comes from the Latin vehere meaning “to carry;” it was first used by eighteenth-century astronomers investigating the mechanism by which a planet is “carried” around the Sun

In text, the vector nature of an object is often indicated by placing a small arrow over the variable representing the object (such as F), or by using a bold font (such as F), or by underlining (such as F or F ∼ ).

In other words, if you were to move the vector shown in Figure 1.1(a) to a different location without varying its length or its pointing direction, would it still be the same vector? In some applications, the answer is “yes,” and those vectors are called free vectors.

You can move a free vector anywhere you’d like as long as you don’t change its length or direction, and it remains the same vector. But in many physics and engineering problems, you’ll be dealing with vectors that apply at a given location; such vectors are called “bound” or “anchored” vectors,

You may see the term “sliding” vectors used for vectors that are free to move along their length but are not free to change length or direction; such vectors are useful for problems involving torque and angular motion

You can understand the usefulness of bound vectors if you think about an application such as representing the velocity of the wind at various points in the atmosphere. To do that, you could choose to draw a bound vector at each point of interest, and each of those vectors would show the speed and direction of the wind at that location (most people draw the vector with its tail – the end without the arrow – at the point to which the vector is bound). A collection of such vectors is called a vector field

If you think about the ways in which you might represent a bound vector, you may realize that the vector can be defined simply by specifying the start and end points of the arrow. So in a three-dimensional Cartesian coordinate system, you only need to know the values of x, y, and z for each end of the vector

Now consider the special case in which the vector is anchored to the origin of the coordinate system (that is, the end without the arrowhead is at the point of intersection of the coordinate axes, as shown in Figure 1.2(b). 3 Such vectors may be completely specified simply by listing the three numbers that represent the x-, y-, and z-coordinates of the vector’s end point.

In this representation, the values that represent the vector are called the “components” of the vector, and the number of components it takes to define a vector is equal to the number of dimensions in the space in which the vector exists

A scalar is the mathematical representation of a physical entity that may be characterized by magnitude only

you might suspect that there are other entities involving magnitude and directions that are more complex than vectors (that is, requiring more numbers than the number of spatial dimensions). Indeed there are, and such entities are called “tensors.”

A tensor is the mathematical representation of a physical entity that may be characterized by magnitude and multiple directions

An example of a tensor is the inertia that relates the angular velocity of a rotating object to its angular momentum. Since the angular velocity vector has a direction and the angular momentum vector has a (potentially different) direction, the inertia tensor involves multiple directions

So in 3-dimensional space, a second-rank tensor is represented by 3 2 = 9 numbers. In N-dimensional space, scalars still require only one number, vectors require N numbers, and tensors require N R numbers

a tensor may be represented by an array of 3 R numbers in 3-dimensional space. In this expression, “R” represents the rank of the tensor.

Recognizing scalars, vectors, and tensors is easy once you realize that a scalar can be represented by a single number, a vector by an ordered set of numbers, and a tensor by an array of numbers.