4: Vector Analysis
A vector is a mathematical object that has both a scalar part (i.e., a magnitude and possibly a phase), as well as a direction. Many physical quantities are best described as vectors. For example, the rate of movement through space can be described as speed ; i.e., as a scalar having SI base units of m/s. However, this quantity is more completely described as velocity ; i.e., as a vector whose scalar part is speed and direction indicates the direction of movement. Similarly, force is a vector whose scalar part indicates magnitude (SI base units of N), and direction indicates the direction in which the force is applied. Electric and magnetic fields are also best described as vectors.
-
- 4.1: Vector Arithmetic
- In mathematical notation, a real-valued vector A is said to have a magnitude A=|A| and direction a^ such that A=Aa^(4.1.1) where a^ is a unit vector (i.e., a real-valued vector having magnitude equal to one) having the same direction as A . If a vector is complex-valued, then A is similarly complex-valued
-
- 4.2: Cartesian Coordinates
- Concepts described in that section – i.e., the dot product and cross product – are described in terms of the Cartesian system. In this section, we identify some additional features of this system that are useful in subsequent work and also set the stage for alternative systems; namely the cylindrical and spherical coordinate systems.
-
- 4.3: Cylindrical Coordinates
- The cylindrical system is defined with respect to the Cartesian system. In lieu of x and y , the cylindrical system uses ρ , the distance measured from the closest point on the z axis,1 and ϕ , the angle measured in a plane of constant z , beginning at the +x axis ( ϕ=0 ) with ϕ increasing toward the +y direction.
Contributors and Attributions
-
Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 Licensed with CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0 . Report adoption of this book here . If you are a professor reviewing, adopting, or adapting this textbook please help us understand a little more about your use by filling out this form.


9.7: Summary and Review
Summary
In this chapter, we explored conflict and communication in personal relationships. We learned that conflict is based on interdependent relationships, competing goals, limited resources, and interference from others, and that culture can add a significant dimension to how we perceive and resolve conflict.
It is important to recognize that we approach conflict situations with distinctive styles, including integrating, dominating, avoiding, competing, and obliging preferences. These styles will in turn impact how the conflict is resolved, including win-win, win-lose, and lose-lose outcomes. Conflict can be further complicated by the use of unproductive conflict behaviors, such as gunnysacking, kitchen sinking, blame, beltlining, force, manipulation, personal rejection, counterpunch, labels, and silencers. The use of unproductive conflict behaviors may lead to short-term win-lose outcomes, but over time they can cause conflict to spiral out of control, hurt the relationship, and hurt the self-concepts of the people on the receiving end.
To close the chapter, we explored specific communication tools to help enhance successful conflict management, as well as an integrative model of conflict negotiation. There are a variety of communication tools that can encourage the peaceful resolution of conflict, including mindfulness, being unconditionally constructive, practicing the 5 to 1 Rule, bracketing, perception checking, active listening and mirroring, assertiveness, empathy, and emotional intelligence. For conflicts that are ongoing and have a significant impact, practicing the integrative conflict resolution model may help to solve the conflict. Last, it is important to recognize when conflict may benefit from the help of a third party, such as a conflict mediator or therapist.
Interpersonal conflict, when managed with empathy and grace, can lead to a variety of benefits. It is important to remember to commit to honoring yourself, the other person, and the relationship. In doing so, you may experience greater personal awareness, understanding of the other person, and relationship growth.
Discussion Questions
- Think of a recent situation in which you experienced a conflict or a problem in a relationship with a friend, family member, romantic partner, or co-worker. What type of conflict goal is at the heart of the problem? If you could go back and renegotiate the conflict, what are three conflict management skills you learned in this chapter that you would apply for peaceful problem resolution? How might the context of the situation influence your choice of conflict management style?
- Reflect on your own style of conflict management (avoidance, dominating, integrating, obliging, compromising). Do you try to jump right in and solve the problem, or would you rather not deal with issues? Do you try and protect the relationship first, or do you try to win and get your way? What is the benefit of your approach? What is the downside to your approach? Based on what you have learned in this chapter, how might you adapt your conflict management approach in the future?
- In your family, what are the expectations for handling conflict? (Consider cultural factors such as race and ethnicity, gender, age, individualism/collectivism, and sexual orientation.) How do these expectations impact your relationships with others?


14.5: The Bhagavad Gita
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The Bhagavad Gita , or “Song of the Blessed One,” is a fascinating story, with great religious significance if one accepts it at face value (Mitchell, 2000). It consists primarily of a conversation between Krishna , a great Avatar or divine incarnation, and Arjuna, a great warrior, on the eve of a battle. The battle is a civil war, with noble warriors and relatives split between both sides. Arjuna decides that no good can come of killing so many people in this battle, and he decides not to fight. Krishna, who is driving Arjuna’s chariot, instructs Arjuna in Yoga as he discusses what is right both for Arjuna and for all people. Many of the principles of Yoga derive from what Krishna told Arjuna, thus it is believed by many that these Yoga principles come from the mouth of God. This is similar to what Jews, Christians, and Muslims believe about the Torah, the Bible, and the Koran: that they are divinely inspired texts.
My favorite quote from the Bhagavad Gita refers directly to the self:
The Self is a friend for him who masters himself by the Self;
but for him who is not self-mastered, the self is the cruelest foe.
(pg. 89; Mitchell, 2000)
This quote suggests that we can be our own best friend, or our own worst enemy. Indeed, Krishna tells Arjuna that what he must do is to be himself. It is only through his own actions that Arjuna can fulfill his potential. However, Arjuna must not remain attached to the consequences of his actions; he must simply act and allow the universe to move forward as it will. Only by truly understanding the nature of the universe, and the nature of ourselves, can we properly make this choice. The practice of Yoga helps us to see this reality, and the Bhagavad Gita helps to describe the essential practices.
Figure \(\PageIndex{1}\)
The altar prepared for the ceremony when the author was initiated into Kriya Yoga.
Together, the Bhagavad Gita and the Yoga Sutras contain all of the basic information on Yoga we will explore in this chapter. There is actually a fair amount of overlap between the books, but it is unclear which one may have been written first. Most scholars believe that the Bhagavad Gita was written between 500 B.C. and 100 A.D. (Mitchell, 2000), which falls right in the middle of when Patanjali is believed to have written down the Yoga Sutras . Since both philosophies seem to come from much older sources, it may well be that they owe their commonalities to some older tradition that can no longer be specifically identified.
Placing Yoga in Context: An Ancient Plan for Self Development
Yoga is much older than any other theory described in this book, with the exception of those parts of other theories that were borrowed from Yoga and Buddhism. The ancient Vedas , which provide much of the mythological and philosophical basis for Hinduism, are 4,000 to 5,000 years old (placing them amongst the oldest recorded literature in the world). The Yoga Sutras of Patanjali and the Bhagavad Gita , which provide the basic teachings of traditional Yoga, were written as early as 600 B.C. (though there is little consensus on exactly when). Kriya-Yoga, the yoga believed by many to be the original Yoga of Patanjali, was lost to the world for many centuries, until it was reintroduced by Lahiri Mahasaya in 1861. Yoga continues to evolve today, with many different styles being introduced and revised, both in the East and the West.
Although this ancient philosophy may not seem relevant to modern personality theory, it has actually been part of psychology from the very beginning. Most notably, Jung and Rogers were clearly influenced by their travels to India and China, respectively. The knowledge of Yoga and Buddhism they developed as a result of those and other experiences helped to shape their personality theories. Fromm also examined how psychoanalysis and Buddhist meditation compare to each other. Today, as positive psychology examines topics such as happiness and well being, and as spiritual psychotherapists examine the important role that spirituality plays in the lives of many people, those practices that Yoga, Buddhism, and other spiritual disciplines have in common are being examined more closely by psychologists. In the next chapter we will examine similar spiritual disciplines that exist within the Jewish, Christian, and Muslim traditions.
Spirit, Nature, and Consciousness
In the metaphysics of Yoga there is believed to be a duality between spirit and nature. Spirit is pure consciousness; nature is the opposite of spirit.  The body and mind come from nature, but the transcendental self comes from spirit. A key point here is that it is the transcendental self that comes from spirit, not the self we are aware of every day. Our thoughts, feelings, sensations, indeed our mind itself, are all the result of neural activity in our brain, and our brain is part of our body. Thus, our awareness is not our consciousness.
Our true self, the transcendental self, is a temporary manifestation of Spirit in essence. The great mistake in our lives is to confuse our body and mind with who we really are, to believe that this body and this mind are our self. The practice of Yoga, however, teaches us to still our minds, to eliminate all thought and sensation, so that we might be in union with our transcendental self and the universal spirit. Once we have accomplished this task, by subjugating our natural tendency to think and restraining our mind itself, we will know who and what we really are ( Yoga Sutras I:2,3 [Bailey, 1927]). This is not an easy task, but it has a great reward. As Sri Yukteswar told Yogananda:
The soul expanded into Spirit remains alone in the region of lightless light, darkless dark, thoughtless thought, intoxicated with its ecstasy of joy in God’s dream of cosmic creation (Yogananda, 1946; pgs. 489-490).
William James, America’s foremost psychologist, is best known for his theory on the stream of consciousness. According to James, it is the continuity of consciousness that defines our self. This is in direct contradiction to Eastern philosophies, which consider the conscious mind to be derived from the natural world, and therefore only an illusion. Eastern philosophies consider the transcendental self to be real, but obscured from us by the distraction of the so-called conscious mind. Can we find some compromise between these points of view? No! However, such a contradiction would probably not bother James. James (1892) did talk about the soul, the transcendental ego, and spirit as possible sources for the conscious self, but considered that possibility to be outside the realm of psychological study
discussion Question \(\PageIndex{1}\)
Do you believe in a transcendental self (whether you call it self, spirit, soul…whatever)? What does this make you feel about your physical body? As for all of nature can you really believe it is just an illusion?
Karma
Karma is a difficult concept to grasp. We generally think of karma as the consequences of things we have done wrong, but karma does not apply simply to our misbehavior, it applies to all of our actions. An easy to understand discussion of karma has been written by Goldstein and Kornfield (2001). The law of karma can be understood on two levels. First, karma refers to cause and effect. Whenever we perform an action, we experience some consequence at a later time. The second level of karma may be more important, as it refers to our state of mind at the time when we performed the action in question. Our intentions, or the motives behind an action, determine the nature of the consequences we experience. The importance of this point is that we control the nature of our karma. This, of course, has important implications for personality development. Once we understand the karmic law, it is only natural that we should begin to plant the seeds of healthy karma. In other words, we should be inclined to act only in ways that are healthy and socially beneficial, so that the consequences we then experience will lead to greater well-being for ourselves.
The second level of karma, that it is our intentions and motivation that affect the outcome of our lives, seems quite similar to cognitive theories in psychology. Cognitive psychology focuses on the nature of our thought, and problems often arise when we are trapped in a series of automatic thoughts that create problems for us. In other words, when we view the world negatively, we react in negative and maladaptive ways. Similarly, our past karma influences the karma we create for the future. If we think and act in negative ways, we create negative karma, but it is also true that if we think and act in positive ways we create positive karma. Cognitive therapy resembles much of what is written in the East about recognizing the cause-and-effect pattern that our karma traps us within. Successful cognitive therapy is something like enlightenment: when we realize the truth of what we are doing we have a chance to break that pattern and move in a healthy direction.
discussion question \(\PageIndex{2}\)
Karma refers to the cosmic law of cause and effect, the idea that our past actions will someday affect our current and future lives. Do you believe this, and can you provide any examples of this happening to you?
The Three Principles of Creation
Would it be possible for us to avoid or circumvent the law of karma by remaining inactive? Sometimes many of us do just want to get away from everything and everyone. However, social withdrawal is often an early sign of psychological distress or mental illness, suggesting that this is generally not a healthy action. According to Yoga, all nature is composed of three aspects called the gunas: rajas, tamas, and sattva. According to Krishna, as recorded in the Bhagavad Gita , the three gunas “bind to the mortal body the deathless embodied Self” (pg. 158; Mitchell, 2000). Rajas (born of craving) binds us to action, tamas (born of ignorance) binds us to dullness, and sattva (untainted and luminous) binds us to knowledge and joy. Only through faithful Yoga practice can an individual transcend the gunas and achieve unity with God. Only then will the individual be free from any attachment to action and free from the law of karma.
Another important and practical question that arises from an understanding of the three gunas is: what should we eat? In America we generally associate being vegetarian with practicing Yoga or being a Buddhist. But is there a good reason for this? Everything is made from the three gunas, but in different proportions. So we might say that a hot, spicy dish like Gang Garee from Thailand (a personal favorite) is predominantly rajas. Heavy food, with a thick sauce or gravy is predominantly tamas. Fruits and vegetables, however, are lighter and more refreshing; they are predominantly sattva. So a vegetarian diet should increase the relative amount of sattva in our body, thus making us a better person. This is just like the old saying: “You are what you eat!” This goes beyond diet, however, since diet only directly affects the body. What about the mind? Everything we take in is comprised of the three gunas: words and ideas that we hear, music, the emotions expressed by people we spend time with, and so on (Vivekananda, 1955a). It is important not only that we eat well, but also that we spend time in relaxing and healthy environments, associate with good people, and generally try to cultivate a life that moves more toward sattva than the other gunas.
The Guru or Teacher
Spirit vs. nature, consciousness, karma, the three gunas, it can all seem very strange to those of us who did not grow up with these concepts. Another common source of confusion is the distinction between the terms yogi and guru . A yogi is anyone who practices Yoga. A guru is a teacher, someone advanced in Yoga and capable of leading others on the path. A true guru is typically revered in the East, and in English they are often referred to as prophets, or saints.
Many believe that a guru is essential to the practice of Yoga. The practice of Yoga can be difficult, and the principles can be confusing. Some argue that only through initiation by a guru can an individual truly and correctly use mantras such as Om. Otherwise, the would-be student of Yoga will not know the correct frequency or nature of the mantra. Perhaps most important, though, is the spiritual consciousness provided by the guru. As described by Yogananda:
…If I entered the hermitage in a worried or indifferent frame of mind, my attitude imperceptibly changed. A healing calm descended at the mere sight of my guru. Each day with him was a new experience in joy, peace, and wisdom. Never did I find him deluded or emotionally intoxicated with greed, anger, or human attachment. (pg. 137; Yogananda, 1946)
Another aspect involved in the importance of a guru is his or her lineage. In Western culture we are very individualistic, so it is common for each new teacher or leader to try establishing a new beginning. In the collectivist cultures of the East, however, they pay more attention to the earlier teachers of current teachers. Establishing this lineage is very important, not to suggest that one line of Yoga is better than another, but to connect the past to the present and, presumably, the present to the future. Recent research in the field of positive psychology has suggested that such a balanced time perspective, emphasizing the past, present, and future, is an important facet of psychological well-being (Boniwell, 2005; Boniwell & Zimbardo, 2003).
discussion question \(\PageIndex{3}\)
In Eastern cultures they talk about gurus, in Western cultures we often hear about mentors. Do the words “guru” and “mentor” mean the same thing to you? Have you ever been inspired by someone you thought of as a guru or mentor?


10.4: Addition of Signed Numbers
- be able to add numbers with like signs and with unlike signs
- be able to use the calculator for addition of signed numbers
Addition of Numbers with Like Signs
The addition of the two positive numbers 2 and 3 is performed on the number line as follows.
Begin at 0, the origin.
Since 2 is positive, move 2 units to the right.
Since 3 is positive, move 3 more units to the right.
We are now located at 5.
Thus, \(2 + 3 = 5\).
Summarizing, we have
\(\text{(2 positive units)} + \text{(3 positive units)} = \text{(5 positive units)}\)
The addition of the two negative numbers -2 and -3 is performed on the number line as follows.
Begin at 0, the origin.
Since -2 is negative, move 2 units to the left.
Since -3 is negative, move 3 more units to the left.
We are now located at -5.
Thus, \((-2) + (-3) = -5\)
Summarizing, we have
\(\text{(2 negative units)} + \text{(3 negative units)} = \text{(5 negative units)}\)
Observing these two examples, we can suggest these relationships:
\(\text{(postitive number)} + \text{(positive number)} = \text{(positive number)}\)
\(\text{(negative number)} + \text{(negative number)} = \text{(negative number)}\)
Adding Numbers with the Same
Sign
Addition of numbers with like sign:
To add two real numbers that have the
same
sign, add the absolute values of the numbers and associate with the sum the common sign.
Find the sums.
3 + 7
Solution
\(\begin{array} {l} {|3| = 3} \\ {|7| = 7} \end{array} \big \}\) Add these absolute values.
3 + 7 = 10
The common sign is “+.”
Thus, \(3 + 7 = +10\), or \(3 + 7 = 10\).
Find the sums.
(-4) + (-9)
Solution
\(\begin{array} {l} {|-4| = 4} \\ {|-9| = 9} \end{array} \big \}\) Add these absolute values.
4 + 9 = 13
The common sign is “-.”
Thus, \((-4) + (-9) = -13\).
Practice Set A
Find the sums.
8 + 6
- Answer
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14
Practice Set A
41 + 11
- Answer
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52
Practice Set A
(-4) + (-8)
- Answer
-
-12
Practice Set A
(-36) + (-9)
- Answer
-
-45
Practice Set A
-14 + (-20)
- Answer
-
-34
Practice Set A
\(-\dfrac{2}{3} + (-\dfrac{5}{3})\)
- Answer
-
\(-\dfrac{7}{3}\)
Practice Set A
-2.8 + (-4.6)
- Answer
-
-7.4
Practice Set A
0 + (-16)
- Answer
-
-16
Addition With Zero
Addition with Zero
Notice that
\((0) + \text{(a positive number)} = \text{(that same positive number)}\).
\((0) + \text{(a negative number)} = \text{(that same negative number)}\).
Since adding zero to a real number leaves that number unchanged, zero is called the additive identity .
Addition of Numbers with Unlike Signs
The addition \(2 + (-6)\),
two numbers with unlike signs , can also be illustrated using the number line.
Begin at 0, the origin.
Since 2 is positive, move 2 units to the right.
Since -6 is negative, move, from 2, 6 units to the left.
We are now located at -4.
We can suggest a rule for adding two numbers that have unlike signs by noting that if the signs are disregarded, 4 can be obtained by subtracting 2 from 6. But 2 and 6 are precisely the absolute values of 2 and -6. Also, notice that the sign of the number with the larger absolute value is negative and that the sign of the resulting sum is negative.
Adding Numbers with Unlike Signs
Addition of numbers with unlike signs: To add two real numbers that have
unlike signs
, subtract the smaller absolute value from the larger absolute value and associate with this difference the sign of the number with the larger absolute value.
Find the following sums.
7 + (-2)
Solution
\(\underbrace{|7| = 7}_{\begin{array} {c} {\text{Larger absolute}} \\ {\text{value. Sign is positive.}}\end{array}}\) \(\underbrace{|-2| = 2}_{\begin{array} {c} {\text{Larger absolute}} \\ {\text{value. Sign is positive.}}\end{array}}\)
Subtract absolute values: 7 - 2 = 5.
Attach the proper sign: "+."
Thus, \(7 + (-2) = +5\) or \(7 + (-2) = 5\).
3 + (-11)
Solution
\(\underbrace{|3| = 3}_{\begin{array} {c} {\text{Smaller absolute}} \\ {\text{value}}\end{array}}\) \(\underbrace{|-11| = 11}_{\begin{array} {c} {\text{Larger absolute}} \\ {\text{value. Sign is negative.}}\end{array}}\)
Subtract absolute values: 11 - 3 = 8.
Attach the proper sign: "-."
Thus, \(3 + (-11) = -8\).
The morning temperature on a winter's day in Lake Tahoe was -12 degrees. The afternoon temperature was 25 degrees warmer. What was the afternoon temperature?
Solution
We need to find \(-12 + 25\).
\(\underbrace{|-12| = 12}_{\begin{array} {c} {\text{Smaller absolute}} \\ {\text{value}}\end{array}}\) \(\underbrace{|25| = 25}_{\begin{array} {c} {\text{Larger absolute}} \\ {\text{value. Sign is positive.}}\end{array}}\)
Subtract absolute values: 25 - 12 = 16.
Attach the proper sign: "+."
Thus, \(-12 + 25 = 13\).
Practice Set B
Find the sums.
4 + (-3)
- Answer
-
1
Practice Set B
-3 + 5
- Answer
-
2
Practice Set B
15 + (-18)
- Answer
-
-3
Practice Set B
0 + (-6)
- Answer
-
-6
Practice Set B
-26 + 12
- Answer
-
-14
Practice Set B
35 + (-78)
- Answer
-
-43
Practice Set B
15 + (-10)
- Answer
-
5
Practice Set B
1.5 + (-2)
- Answer
-
-0.5
Practice Set B
-8 + 0
- Answer
-
-8
Practice Set B
0 + (0.57)
- Answer
-
0.57
Practice Set B
-879 + 454
- Answer
-
-425
Calculators
Calculators having the
key can be used for finding sums of signed numbers.
Use a calculator to find the sum of -147 and 84.
| Display Reads | |||
| Type | 147 | 147 | |
| Press | +/- | -147 | This key changes the sign of a number. It is different than -. |
| Press | + | -147 | |
| Type | 84 | 84 | |
| Press | = | -63 |
Practice Set C
Use a calculator to find each sum.
673 + (-721)
- Answer
-
-48
Practice Set C
-8,261 + 2,206
- Answer
-
-6,085
Practice Set C
-1,345.6 + (-6,648.1)
- Answer
-
-7,993.7
Exercises
Find the sums in the following 27 problems. If possible, use a calculator to check each result.
Exercise \(\PageIndex{1}\)
4 + 12
- Answer
-
16
Exercise \(\PageIndex{2}\)
8 + 6
Exercise \(\PageIndex{3}\)
(-3) + (-12)
- Answer
-
-15
Exercise \(\PageIndex{4}\)
(-6) + (-20)
Exercise \(\PageIndex{5}\)
10 + (-2)
- Answer
-
8
Exercise \(\PageIndex{6}\)
8 + (-15)
Exercise \(\PageIndex{7}\)
-16 + (-9)
- Answer
-
-25
Exercise \(\PageIndex{8}\)
-22 + (-1)
Exercise \(\PageIndex{9}\)
0 + (-12)
- Answer
-
-12
Exercise \(\PageIndex{10}\)
0 + (-4)
Exercise \(\PageIndex{11}\)
0 + (24)
- Answer
-
24
Exercise \(\PageIndex{12}\)
-6 + 1 + (-7)
Exercise \(\PageIndex{13}\)
-5 + (-12) + (-4)
- Answer
-
-21
Exercise \(\PageIndex{14}\)
-5 + 5
Exercise \(\PageIndex{15}\)
-7 + 7
- Answer
-
0
Exercise \(\PageIndex{16}\)
-14 + 14
Exercise \(\PageIndex{17}\)
4 + (-4)
- Answer
-
0
Exercise \(\PageIndex{18}\)
9 + (-9)
Exercise \(\PageIndex{19}\)
84 + (-61)
- Answer
-
23
Exercise \(\PageIndex{20}\)
13 + (-56)
Exercise \(\PageIndex{21}\)
452 + (-124)
- Answer
-
328
Exercise \(\PageIndex{22}\)
636 + (-989)
Exercise \(\PageIndex{23}\)
1,811 + (-935)
- Answer
-
876
Exercise \(\PageIndex{24}\)
-373 + (-14)
Exercise \(\PageIndex{25}\)
-1,211 + (-44)
- Answer
-
-1,255
Exercise \(\PageIndex{26}\)
-47.03 + (-22.71)
Exercise \(\PageIndex{27}\)
-1.998 + (-4.086)
- Answer
-
-6.084
Exercise \(\PageIndex{28}\)
In order for a small business to break even on a project, it must have sales of $21,000. If the amount of sales was $15,000, by how much money did this company fall short?
Exercise \(\PageIndex{29}\)
Suppose a person has $56 in his checking account. He deposits $100 into his checking account by using the automatic teller machine. He then writes a check for $84.50. If an error causes the deposit not to be listed into this person’s account, what is this person’s checking balance?
- Answer
-
-$28.50
Exercise \(\PageIndex{30}\)
A person borrows $7 on Monday and then $12 on Tuesday. How much has this person borrowed?
Exercise \(\PageIndex{31}\)
A person borrows $11 on Monday and then pays back $8 on Tuesday. How much does this person owe?
- Answer
-
$3.00
Exercises for Review
Exercise \(\PageIndex{32}\)
Find the reciprocal of \(8 \dfrac{5}{6}\).
Exercise \(\PageIndex{33}\)
Find the value of \(\dfrac{5}{12} + \dfrac{7}{18} - \dfrac{1}{3}\).
- Answer
-
\(\dfrac{17}{36}\)
Exercise \(\PageIndex{34}\)
Round 0.01628 to the nearest tenth.
Exercise \(\PageIndex{35}\)
Convert 62% to a fraction.
- Answer
-
\(\dfrac{62}{100} = \dfrac{31}{50}\)
Exercise \(\PageIndex{36}\)
Find the value of |-12|.


1.31: Galaxy Formation
After recombination, baryons fall into the gravitational potential wells provided by dark matter halos. As hydrogen gas falls into halos, it gains kinetic energy and heats up to a temperate roughly determined by the halo mass; for a certain range of halo masses, cooling mechanisms allow the gas to cool, condense near the center of the halo, and begin the process of star and galaxy formation. The first generations of stars and galaxies flood the universe with high-energy radiation, reionizing neutral hydrogen in the intergalactic medium. A rich set of feedback processes then shape the evolution of galaxy populations over subsequent epochs.
Like nonlinear dark matter structure formation, galaxy formation and evolution are complex processes that generally require numerical simulations to model accurately. This Chapter provides a broad overview of galaxy formation theory and the connection between galaxies and dark matter halos, which are important tools for cosmological analyses based on galaxy populations.
How do Galaxies Form?
After recombination, most of the baryons in the universe are in the form of neutral hydrogen. During this epoch, dark matter overdensities are growing (\(\delta\propto a\)) and collapsing into dark matter halos. Consider a region in which hydrogen gas is gravitationally attracted towards the center of a halo; as it falls in, it heats up to a virial temperature
\[ T_{\mathrm{vir}}\approx 10^5\ \mathrm{K}~\left(\frac{V_{\mathrm{vir}}}{100\ \mathrm{km\ s}^{-1}}\right)^2 \label{eqn:tvir}, \]
where \(V_{\mathrm{vir}}\) is the virial velocity of the halo, which is set by its mass and density profile. For example, a halo of mass \(10^{12}\ M_{\mathrm{\odot}}\) (\(10^{9}\ M_{\mathrm{\odot}}\)) roughly has \(V_{\mathrm{vir}} \approx 200\ \mathrm{km\ s}^{-1} \) (\(V_{\mathrm{vir}} \approx 20\ \mathrm{km\ s}^{-1} \)). In this way, the halo's gravitational potential sets the temperature of the infalling gas.
Gravitational acceleration isn't the only process that sets the gas temperature. In particular, hydrogen undergoes electromagnetic interactions including collisional excitation , ionization , recombination , and bremsstrahlung with photons and free electrons, which largely act to cool the gas at these early times. The net effect of these processes is captured by the cooling function \(\Lambda(T)\), which is defined to set the characteristic cooling time via
\[ t_{\mathrm{cool}} = \frac{3nk_B T}{2 n_\mathrm{H}^2 \Lambda(T)}, \label{eqn:tcool} \]
where \(n\) is the total number density of gas particles and \(n_{\mathrm{H}}\) is the number density of neutral hydrogen atoms in the gas. If the cooling time is shorter than the free-fall time, the gas cools more quickly than it can gravitationally equilibrate and condenses to the center of the halo.
Let's estimate the minimum virial velocity necessary to efficiently cool atomic hydrogen. Other than bremsstrahlung, the cooling processes mentioned above all involve electron transitions between hydrogen energy levels, the lowest of which are separated by \(13.6\ \mathrm{eV}\). Thus, the gas must be warmer than roughly \(10\ \mathrm{eV}\) to initiate these processes, corresponding to \(T_{\mathrm{vir,min}}\approx 10^4\ \mathrm{K}\) and \(V_{\mathrm{vir,min}}\approx 10\ \mathrm{km\ s}^{-1}\). This threshold is known as the atomic cooling limit , and it corresponds to a halo mass of roughly \(10^{8}\ M_{\mathrm{\odot}}\) today. Problem 32.1 explores how the cooling time depends on redshift, gas density, and gas composition.
Box \(\PageIndex{1}\)
Exercise 32.1.1: Estimate the minimum halo mass necessary to efficiently cool molecular hydrogen (\(\mathrm{H}_2\)). To convert from virial velocity to halo mass, you can combine the estimates of this relation above with \(V_{\mathrm{vir}}\propto M^{1/3}\).
As a result of the cooling processes described above, sufficiently dense regions of cold gas clump into giant molecular clouds (GMCs) that host star formation. The dynamic range of the star formation process is remarkable: gas accretes into halos on scales of hundreds of kiloparsecs (roughly corresponding to the size of a typical halo), condenses into GMCs with characteristic sizes of tens of parsecs, and forms into stars with sizes of \(\approx 10^{-7}\) parsecs. Importantly, star formation in GMCs is highly inefficient, with typical star formation timescales that are orders of magnitude larger than the corresponding free-fall times.
The first generations of stars form in dark matter halos about \(100\) million years after the Big Bang, transitioning the universe from post-recombination "dark ages" to the epoch of "cosmic dawn." These stars eventually die, exploding in supernovae and flooding the universe with high-energy radiation, which heats the gas in halos, inhibiting star formation. This radiation also reionizes hydrogen in the intergalactic medium, transitioning the universe from a dark, opaque neutral hydrogen gas to a transparent, ionized plasma. Several independent lines of evidence, including the optical depth of the cosmic microwave background and the absorption spectra of high-redshift quasars, indicate that the universe is fully reionized by \(z\approx 6\), roughly 1 billion years after the Big Bang.
| Simulation of the end of the "cosmic dark ages," the formation of the first stars and galaxies during "cosmic dawn," and the epoch of reionization. Dark regions consist of neutral hydrogen, which fills the universe after recombination. Bright regions show ionized bubbles emanating from the first generations of stars as they explode in supernovae. These ionized bubbles expand to fill the entire simulation volume by the time reionization is complete, about 1 billion years after the Big Bang. |
How do Galaxies Evolve?
Early models and simulations of galaxy formation drastically overpredicted the number of galaxies as a function of galaxy luminosity, or the galaxy luminosity function , compared to observations. This issue, known as the "overcooling problem," is overcome by feedback--- the effects of galaxy formation on the process itself. The supernova and photoionization heating mechanisms discussed above constitute one form of feedback. These mechanisms most strongly affect galaxies that inhabit halos with masses below \(\approx 10^{12}\ M_{\mathrm{\odot}}\) by heating gas to a significant fraction of these halos' virial temperatures. For halos with masses below \(\approx 10^{10}\ M_{\mathrm{\odot}}\), photoionization heating partially or completely shuts down subsequent star formation; we observe the relic of ancient star formation in these small halos as dwarf galaxies today.
The overcooling problem also applied to halos with masses above \(\approx 10^{12}\ M_{\mathrm{\odot}}\), which were predicted to form stars more efficiently than inferred from observations. Supernova feedback and photoionization heating do not significantly affect star formation in these halos due to their large virial temperatures. Instead, high-energy radiation from the matter orbiting supermassive black holes (SMBHs) at the centers of massive galaxies heats the surrounding gas. This mechanism, known as active galactic nucleus (AGN) feedback , helps bring the predicted luminosity function for the brightest galaxies into agreement with observations. Note that AGN feedback is ineffective in lower-mass systems because SMBH mass (and thus the amount of energy AGN feedback releases) decreases rapidly in smaller halos. The left panel of the figure below illustrates the resulting galaxy--halo connection ; specifically, the relation between stellar mass and total halo mass, and the halo mass ranges in which various forms of feedback affect galaxy formation.
Galaxy properties, including (but not limited to) stellar mass, star formation rate, size, shape, color, and visual appearance (or morphology ) evolve over cosmic time. The interplay between cooling, star formation, and feedback causes the global star formation rate in the universe to peak roughly 10 billion years after the Big Bang at \(z\approx 2\), or "cosmic noon," as shown in the right panel below. Note that the rise and fall of AGN activity in the universe roughly coincides with the rise and fall of the global star formation rate.
| Wechsler & Tinker 2018. Right panel : The average cosmic star formation rate per unit volume as a function of redshift (bottom axis) and lookback time (top axis). Points with errorbars show observational measurements, and the line shows a best-fitting star formation rate density model. Adapted from Madau & Dickinson 2014 . |
Galaxy morphology is historically classified according to the Hubble sequence , which delineates four major types of galaxies based on their visual appearance: elliptical, lenticular, spiral, and irregular. These classes of galaxies are arranged in a "tuning fork" sequence, which does not correspond to how galaxies typically evolve. In particular, although ellipticals and spirals are respectively referred to as "early-type" and "late-type" galaxies, modern observations indicate that the first generations of active, star-forming galaxies are predominantly spirals and irregulars. Mergers between these classes of galaxies yield ellipticals; this process often coincides with a shutdown in star formation, such that ellipticals are generally redder and less actively star-forming compared to spirals. Problem 32.2 explores how different galaxy morphologies arise and the relation between galaxy and halo evolution.
| An updated version of the original Hubble sequence, which delineates four major types of galaxies: elliptical (E), lenticular (S0), spiral (S), and irregular (I). Note that galaxies do not generally evolve from left to right along the Hubble sequence. Instead, the universe is dominated by spiral and irregular galaxies at early times, which merge to form elliptical galaxies. Adapted from Kormendy & Bender 1996 . |
HOMEWORK Problems
Problem \(\PageIndex{1}\): Behavior of the cooling time.
Consider the behavior of the cooling time as a function of density, redshift, and gas composition:
a) Starting from Equation \ref{eqn:tcool}, argue that \(t_{\mathrm{cool}}\propto \rho_g^{-1}\), where \(\rho_g\) is the gas density. Qualitatively explain why an inverse dependence on gas density is reasonable. What does this dependence imply about the temperature profile of the gas as a function of radius from the center of a halo?
b) Use the result from part a) to show that \(t_{\mathrm{cool}}\propto a^3\). Qualitatively describe what consequences more efficient cooling at early times might have for galaxy formation.
c) Qualitatively describe how the presence of heavier elements affects the cooling time and minimum virial temperature necessary for efficient cooling. This is an important consideration because stars deposit heavier elements into the interstellar medium when they explode in supernovae.
Problem \(\PageIndex{2}\): Galaxy morphology and evolution.
This problem walks through a few additional aspects of galaxy morphology and evolution.
a) Using conservation of angular momentum, qualitatively describe why thin galactic disks form out of gas that falls into halos from the intergalactic medium.
b) When two spiral galaxies merge, qualitatively describe why the resulting stellar distribution is extended and elliptical.
c) Considering that small halos form first and merge together to build larger halos and how galaxy properties evolve as a result of mergers, qualitatively describe how you expect the stellar mass--halo mass relation to evolve over cosmic time.


2.2.4: Expressing Relations in a Structure
One main use formulas can be put to is to express properties and relations in a structure \(\Struct M\) in terms of the primitives of the language \(\Lang L\) of \(\Struct M\) . By this we mean the following: the domain of \(\Struct M\) is a set of objects. The constant symbols, function symbols, and predicate symbols are interpreted in \(\Struct M\) by some objects in \(\Domain M\) , functions on \(\Domain M\) , and relations on \(\Domain M\) . For instance, if \(\Obj A^2_0\) is in \(\Lang L\) , then \(\Struct M\) assigns to it a relation \(R = \Assign{ {\Obj A^2_0} }{M}\) . Then the formula \(\Atom{\Obj A^2_0}{\Obj v_1, \Obj v_2}\) expresses that very relation, in the following sense: if a variable assignment \(s\) maps \(\Obj v_1\) to \(a \in \Domain{M}\) and \(\Obj v_2\) to \(b \in \Domain M\) , then \[Rab \quad\text{ iff}\quad \Sat[,s]{M}{\Atom{\Obj A^2_0}{\Obj v_1, \Obj v_2}}.\nonumber\] Note that we have to involve variable assignments here: we can’t just say “ \(Rab\) iff \(\Sat{M}{\Atom{\Obj A^2_0}{a, b}}\) ” because \(a\) and \(b\) are not symbols of our language: they are elements of \(\Domain{M}\) .
Since we don’t just have atomic formulas, but can combine them using the logical connectives and the quantifiers, more complex formulas can define other relations which aren’t directly built into \(\Struct M\) . We’re interested in how to do that, and specifically, which relations we can define in a structure.
Let \(A(\Obj v_1,\dots, \Obj v_n)\) be a formula of \(\Lang L\) in which only \(\Obj v_1\) ,…, \(\Obj v_n\) occur free, and let \(\Struct M\) be a structure for \(\Lang L\) . \(A(\Obj v_1,\dots, \Obj v_n)\) expresses the relation \(R \subseteq \Domain M^n\) iff \[Ra_1\dots a_n \quad\text{ iff}\quad \Sat[,s]{M}{\Atom{A}{\Obj v_1,\dots, \Obj v_n}}\nonumber\] for any variable assignment \(s\) with \(s(\Obj v_i) = a_i\) ( \(i = 1, \dots, n\) ).
In the standard model of arithmetic \(\Struct N\) , the formula \(\Obj v_1 < \Obj v_2 \lor \eq[\Obj v_1][\Obj v_2]\) expresses the \(\le\) relation on \(\Nat\) . The formula \(\eq[\Obj v_2][\Obj v_1']\) expresses the successor relation, i.e., the relation \(R \subseteq \Nat^2\) where \(Rnm\) holds if \(m\) is the successor of \(n\) . The formula \(\eq[\Obj v_1][\Obj v_2']\) expresses the predecessor relation. The formulas \(\lexists{\Obj v_3}{(\eqN[\Obj v_3][\Obj 0] \land \eq[\Obj v_2][(\Obj v_1 + \Obj v_3)])}\) and \(\lexists{\Obj v_3}{\eq[(\Obj v_1 + {\Obj v_3}')][v_2]}\) both express the \(\Obj <\) relation. This means that the predicate symbol \(<\) is actually superfluous in the language of arithmetic; it can be defined.
This idea is not just interesting in specific structures, but generally whenever we use a language to describe an intended model or models, i.e., when we consider theories. These theories often only contain a few predicate symbols as basic symbols, but in the domain they are used to describe often many other relations play an important role. If these other relations can be systematically expressed by the relations that interpret the basic predicate symbols of the language, we say we can define them in the language.
Find formulas in \(\Lang L_A\) which define the following relations:
- \(n\) is between \(i\) and \(j\) ;
- \(n\) evenly divides \(m\) (i.e., \(m\) is a multiple of \(n\) );
- \(n\) is a prime number (i.e., no number other than \(1\) and \(n\) evenly divides \(n\) ).
Suppose the formula \(A(\Obj v_1, \Obj v_2)\) expresses the relation \(R \subseteq \Domain M^2\) in a structure \(\Struct M\) . Find formulas that express the following relations:
- the inverse \(R^{-1}\) of \(R\) ;
- the relative product \(R \mid R\) ;
Can you find a way to express \(R^+\) , the transitive closure of \(R\) ?
Let \(\Lang{L}\) be the language containing a 2-place predicate symbol \(<\) only (no other constant symbols, function symbols or predicate symbols— except of course \(\eq[][]\) ). Let \(\Struct{N}\) be the structure such that \(\Domain{N} = \Nat\) , and \(\Assign{<}{N} = \Setabs{\tuple{n,m}}{n < m}\) . Prove the following:
- \(\{ 0 \}\) is definable in \(\Struct{N}\) ;
- \(\{ 1 \}\) is definable in \(\Struct{N}\) ;
- \(\{ 2 \}\) is definable in \(\Struct{N}\) ;
- for each \(n \in \Nat\) , the set \(\{ n \}\) is definable in \(\Struct{N}\) ;
- every finite subset of \(\Domain{N}\) is definable in \(\Struct{N}\) ;
- every co-finite subset of \(\Domain{N}\) is definable in \(\Struct{N}\) (where \(X \subseteq \Nat\) is co-finite iff \(\Nat \setminus X\) is finite).


5.3: Areas, lengths and angles
Problem 174 A rectangular piece of fruitcake has a layer of icing on top and down one side to form a larger rectangular slab of cake (as shown in Figure 3 ).
Figure 3: Icing on the cake
Describe how to make a single straight cut so as to divide both the fruitcake and the icing exactly in half. (The thickness of the icing on top is not necessarily the same as the thickness down the side.)
Problem 175
(a) What is the angle between the two hands of a clock at 1:35? Can you find another time when the angle between the two hands is the same as this?
(b) How many times each day do the two hands of a clock ‘coincide’? And at what times do they coincide?
(c) If we add a second hand, how many times each day do the three hands coincide?
Problem 176 The twelve hour marks for a clock are marked on the circumference of a unit circle to form the vertices of a regular dodecagon ABCDEFGHIJKL. Calculate exactly (i.e. using Pythagoras’ Theorem rather than trigonometry) the lengths of all the possible line segments joining two vertices of the dodecagon.
Problem 177 Consider the lattice of all points (k,m,n) in 3-dimensions with integer coordinates k , m , n . Which of the following distances can be realised between lattice points?
Problem 178
(a) Five vertices A, B, C, D , E are arranged in cyclic order. However instead of joining each vertex to its two immediate neighbours to form a convex pentagon, we join each vertex to the next but one vertex to form a pentagonal star, or pentagram ACEBD. Calculate the sum of the five “angles” in any such pentagonal star.
(b) There are two types of 7-gonal stars. Calculate the sum of the angles at the seven vertices for each type.
(c) Try to extend the previous two results (and the proofs) to arbitrary n -gonal stars.
Problem 179
(a) A regular pentagon ABCDE with edges of length 1 is surrounded in the plane by five new regular pentagons - ABLMN joined to AB , BCOPQ joined to BC , and so on.
(i) Prove that M, N , X , Y lie on a line.
(ii) Prove that MPSVY is a regular pentagon.
(iii) Find the edge length of this larger surrounding regular pentagon.
(b) Given a regular pentagon MPSVY , with edge length 1, draw the five diagonals to form the pentagram MSYPV . Let PY meet MV at A, and MS at B ; let PV meet MS at C and SY at D; and let SY meet VM at E .
(i) Prove that ABCDE is a regular pentagon.
(ii) Prove that A , B , and M are three vertices of a regular pentagon ABLMN , where L lies on MP and N lies on MY .
(iii) Find the edge length of the regular pentagon ABCDE .


18.8: Summary
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- Jennifer Hasty, David G. Lewis, & Marjorie M. Snipes
- OpenStax
Animals play essential roles in many areas of human life. While it may be difficult to define an animal, and sometimes controversial to speak the scientific truth that human are animals, too, the continuum between us and them is incontrovertible. In describing animals, anthropologist Claude Lévi-Strauss said that they are “good to think” of (1963, 89) because they show up prolifically in our cultures. Human-animal scholars often use a research approach known as multispecies ethnography as a way of understanding the symbiosis between humans and animals.
Of all animal species, the dog has played the most transformative role in human cultures historically. An early domesticate, dogs have served as guards, hunters, herders, transport, food, and (most commonly) companions in many different societies. Many human subsistence systems depend on animals; hunting, herding, fishing, and factory farming are the primary ways in which humans access meat. Indigenous hunters practice empathy and appreciation as ways of connecting as predators to prey, and many pastoralists have a symbiotic relationship with their herd animals, migrating periodically to provide pasture for their herds. Animals are also symbols. In totemic societies, animal species and relationships are used as ways of ordering human society; human groups have relationships of respect with their totemic emblem and identify with some of the qualities of the animal. Animals also play important roles in oral tradition and religious systems as teachers, messengers, and sacrificial tokens. Many religious systems reflect the awareness that life is not the exclusive domain of the human species and that our world is a shared community.
Animals are also pets and cultural artifacts. Domesticated animals have been genetically reconfigured to meet the needs of human societies. This includes selectively breeding for neotony , a tendency for an animal to maintain both physical and behavioral juvenile characteristics. While many Indigenous societies practice pet keeping as companionship and sometimes also as a way to teach young children about animal behaviors, in modern Western societies, pet keeping has become an industry.
There are also animal trades in Western societies, from zoos, aquariums, and circuses to wild animal reserves where ecotourism generates funds to preserve wild animal habitats. Often, these industries have both negative and positive attributes. In the medical industry, animals have long served as human stand-ins for research. Increasingly today, there are laws and regulations to improve the plight of animals in medical labs, but this continues to be a challenge, and the improvements are rarely adequate. Still, the contributions that animals have made to human health and welfare have been substantial, whether in labs, on farms, in forests, or in our homes. Animals have always mattered to human beings.


9.2: Maximization By The Simplex Method
In this section, you will learn to solve linear programming maximization problems using the Simplex Method:
- Identify and set up a linear program in standard maximization form
- Convert inequality constraints to equations using slack variables
- Set up the initial simplex tableau using the objective function and slack equations
- Find the optimal simplex tableau by performing pivoting operations.
- Identify the optimal solution from the optimal simplex tableau.
In the last chapter, we used the geometrical method to solve linear programming problems, but the geometrical approach will not work for problems that have more than two variables. In real-life situations, linear programming problems consist of literally thousands of variables and are solved by computers. We can solve these problems algebraically, but that will not be very efficient. Suppose we were given a problem with, say, 5 variables and 10 constraints. By choosing all combinations of five equations with five unknowns, we could find all the corner points, test them for feasibility, and come up with the solution, if it exists. But the trouble is that even for a problem with so few variables, we will get more than 250 corner points, and testing each point will be very tedious. So we need a method that has a systematic algorithm and can be programmed for a computer. The method has to be efficient enough so we wouldn't have to evaluate the objective function at each corner point. We have just such a method, and it is called the simplex method .
The simplex method was developed during the Second World War by Dr. George Dantzig. His linear programming models helped the Allied forces with transportation and scheduling problems. In 1979, a Soviet scientist named Leonid Khachian developed a method called the ellipsoid algorithm which was supposed to be revolutionary, but as it turned out it is not any better than the simplex method. In 1984, Narendra Karmarkar, a research scientist at AT&T Bell Laboratories developed Karmarkar's algorithm which has been proven to be four times faster than the simplex method for certain problems. But the simplex method still works the best for most problems.
The simplex method uses an approach that is very efficient. It does not compute the value of the objective function at every point; instead, it begins with a corner point of the feasibility region where all the main variables are zero and then systematically moves from corner point to corner point, while improving the value of the objective function at each stage. The process continues until the optimal solution is found.
To learn the simplex method, we try a rather unconventional approach. We first list the algorithm, and then work a problem. We justify the reasoning behind each step during the process. A thorough justification is beyond the scope of this course.
We start out with an example we solved in the last chapter by the graphical method. This will provide us with some insight into the simplex method and at the same time give us the chance to compare a few of the feasible solutions we obtained previously by the graphical method. But first, we list the algorithm for the simplex method.
- Set up the problem. That is, write the objective function and the inequality constraints.
- Convert the inequalities into equations. This is done by adding one slack variable for each inequality.
- Construct the initial simplex tableau. Write the objective function as the bottom row.
- The most negative entry in the bottom row identifies the pivot column.
- Calculate the quotients. The smallest quotient identifies a row. The element in the intersection of the column identified in step 4 and the row identified in this step is identified as the pivot element. The quotients are computed by dividing the far right column by the identified column in step 4. A quotient that is a zero, or a negative number, or that has a zero in the denominator, is ignored.
- Perform pivoting to make all other entries in this column zero. This is done the same way as we did with the Gauss-Jordan method.
- When there are no more negative entries in the bottom row, we are finished; otherwise, we start again from step 4.
- Read off your answers. Get the variables using the columns with 1 and 0s. All other variables are zero. The maximum value you are looking for appears in the bottom right-hand corner.
Now, we use the simplex method to solve Example 8.1.1 solved geometrically in section 8.1.
Niki holds two part-time jobs, Job I and Job II. She never wants to work more than a total of 12 hours a week. She has determined that for every hour she works at Job I, she needs 2 hours of preparation time, and for every hour she works at Job II, she needs one hour of preparation time, and she cannot spend more than 16 hours for preparation. If she makes $40 an hour at Job I, and $30 an hour at Job II, how many hours should she work per week at each job to maximize her income?
Solution
In solving this problem, we will follow the algorithm listed above.
STEP 1. Set up the problem. Write the objective function and the constraints.
Since the simplex method is used for problems that consist of many variables, it is not practical to use the variables \(x\), \(y\), \(z\) etc. We use symbols \(x_1\), \(x_2\), \(x_3\), and so on.
Let
- \(x_1\) = The number of hours per week Niki will work at Job I and
- \(x_2\) = The number of hours per week Niki will work at Job II.
It is customary to choose the variable that is to be maximized as \(Z\).
The problem is formulated the same way as we did in the last chapter.
\[\begin{array}{ll}
\textbf { Maximize } & \mathrm{Z}=40 \mathrm{x}_{1}+30 \mathrm{x}_{2} \\
\textbf { Subject to: } & \mathrm{x}_{1}+\mathrm{x}_{2} \leq 12 \\
& 2 \mathrm{x}_{1}+\mathrm{x}_{2} \leq 16 \\
& \mathrm{x}_{1} \geq 0 ; \mathrm{x}_{2} \geq 0
\end{array}\nonumber \]
STEP 2. Convert the inequalities into equations. This is done by adding one slack variable for each inequality.
For example to convert the inequality \(x_1 + x_2 ≤ 12\) into an equation, we add a non-negative variable \(s_1\), and we get
\[x_1 + x_2 + s_1 = 12 \nonumber \]
Here the variable \(s_1\) picks up the slack, and it represents the amount by which \(x_1 + x_2\) falls short of 12. In this problem, if Niki works fewer than 12 hours, say 10, then \(s_1\) is 2. Later when we read off the final solution from the simplex table, the values of the slack variables will identify the unused amounts.
We rewrite the objective function \(Z = 40x_1 + 30x_2\) as \(- 40x_1 - 30x_2 + Z = 0\).
After adding the slack variables, our problem reads
\[\begin{array}{ll}
\text { Objective function } & - 40x_1 - 30x_2 + Z = 0 \\
\text { Subject to constraints: } & x_1 + x_2 + s_1 = 12 \\
& 2x_1 + x_2 + s_2 = 16 \\
& x1 ≥ 0; x2 ≥ 0
\end{array} \nonumber \]
STEP 3. Construct the initial simplex tableau . Each inequality constraint appears in its own row. (The non-negativity constraints do not appear as rows in the simplex tableau.) Write the objective function as the bottom row.
Now that the inequalities are converted into equations, we can represent the problem into an augmented matrix called the initial simplex tableau as follows.
Here the vertical line separates the left-hand side of the equations from the right side. The horizontal line separates the constraints from the objective function. The right side of the equation is represented by the column C.
The reader needs to observe that the last four columns of this matrix look like the final matrix for the solution of a system of equations. If we arbitrarily choose \(x_1 = 0\) and \(x_2 = 0\), we get
\[\left[\begin{array}{cccc}
s_{1} & s_{2} & Z & | & C \\
1 & 0 & 0 & | & 12 \\
0 & 1 & 0 & | & 16 \\
0 & 0 & 1 & | & 0
\end{array}\right]\nonumber \]
which reads
\[s_1 = 12 \quad s_2 = 16 \quad Z = 0 \nonumber \]
The solution obtained by arbitrarily assigning values to some variables and then solving for the remaining variables is called the basic solution associated with the tableau. So the above solution is the basic solution associated with the initial simplex tableau. We can label the basic solution variable in the right of the last column as shown in the table below.
STEP 4. The most negative entry in the bottom row identifies the pivot column.
The most negative entry in the bottom row is -40; therefore column 1 is identified.
Question Why do we choose the most negative entry in the bottom row?
Answer The most negative entry in the bottom row represents the largest coefficient in the objective function - the coefficient whose entry will increase the value of the objective function the quickest.
The simplex method begins at a corner point where all the main variables, the variables that have symbols such as \(x_1\), \(x_2\), \(x_3\) etc., are zero. It then moves from a corner point to the adjacent corner point always increasing the value of the objective function. In the case of the objective function \(Z = 40x_1+ 30x_2\), it will make more sense to increase the value of \(x_1\) rather than \(x_2\). The variable \(x_1\) represents the number of hours per week Niki works at Job I. Since Job I pays $40 per hour as opposed to Job II which pays only $30, the variable \(x_1\) will increase the objective function by $40 for a unit of increase in the variable \(x_1\).
STEP 5. Calculate the quotients. The smallest quotient identifies a row. The element in the intersection of the column identified in step 4 and the row identified in this step is identified as the pivot element.
Following the algorithm, in order to calculate the quotient, we divide the entries in the far right column by the entries in column 1, excluding the entry in the bottom row.
The smallest of the two quotients, 12 and 8, is 8. Therefore row 2 is identified. The intersection of column 1 and row 2 is the entry 2, which has been highlighted. This is our pivot element.
Question Why do we find quotients, and why does the smallest quotient identify a row?
Answer When we choose the most negative entry in the bottom row, we are trying to increase the value of the objective function by bringing in the variable \(x_1\). But we cannot choose any value for \(x_1\). Can we let \(x_1 = 100\)? Definitely not! That is because Niki never wants to work for more than 12 hours at both jobs combined: \(x_1 + x_2 ≤ 12\). Can we let \(x_1 = 12\)? Again, the answer is no because the preparation time for Job I is two times the time spent on the job. Since Niki never wants to spend more than 16 hours for preparation, the maximum time she can work is 16 ÷ 2 = 8.
Now you see the purpose of computing the quotients; using the quotients to identify the pivot element guarantees that we do not violate the constraints.
Question Why do we identify the pivot element?
Answer As we have mentioned earlier, the simplex method begins with a corner point and then moves to the next corner point always improving the value of the objective function. The value of the objective function is improved by changing the number of units of the variables. We may add the number of units of one variable, while throwing away the units of another. Pivoting allows us to do just that.
The variable whose units are being added is called the entering variable , and the variable whose units are being replaced is called the departing variable . The entering variable in the above table is \(x_1\), and it was identified by the most negative entry in the bottom row. The departing variable \(s_2\) was identified by the lowest of all quotients.
STEP 6. Perform pivoting to make all other entries in this column zero.
In chapter 7, we used pivoting to obtain the row echelon form of an augmented matrix. Pivoting is a process of obtaining a 1 in the location of the pivot element, and then making all other entries zeros in that column. So now our job is to make our pivot element a 1 by dividing the entire second row by 2 (notation: (1/2)\(R_2\) →\(R_2\)). The result follows.
To obtain a zero in the entry first above the pivot element, we multiply the second row by -1 and add it to row 1(notation: \(-1R_2\) + \(R_1\) → \(R_1\)). We get
To obtain a zero in the element below the pivot, we multiply the second row by 40 and add it to the last row(notation: \(40R_2\) + \(R_3\) → \(R_3\)).
We now determine the basic solution associated with this tableau. By arbitrarily choosing \(x_2 = 0\) and \(s_2 = 0\), we obtain \(x_1 = 8\), \(s_1 = 4\), and \(z = 320\). If we write the augmented matrix, whose left side is a matrix with columns that have one 1 and all other entries zeros, we get the following matrix stating the same thing.
\[\left[\begin{array}{ccccc}
\mathrm{x}_{1} & \mathrm{s}_1 & \mathrm{Z} & | & \mathrm{C} \\
0 & 1 & 0 & | & 4 \\
1 & 0 & 0 & | & 8 \\
0 & 0 & 1 & | & 320
\end{array}\right] \nonumber \]
We can restate the solution associated with this matrix as \(x_1 = 8\), \(x_2 = 0\), \(s_1 = 4\), \(s_2 = 0\) and \(z = 320\). At this stage of the game, it reads that if Niki works 8 hours at Job I, and no hours at Job II, her profit Z will be $320. Recall from Example 8.1.1 in section 8.1 that (8, 0) was one of our corner points. Here \(s_1 = 4\) and \(s_2 = 0\) mean that she will be left with 4 hours of working time and no preparation time.
STEP 7. When there are no more negative entries in the bottom row, we are finished; otherwise, we start again from step 4.
Since there is still a negative entry, -10, in the bottom row, we need to begin, again, from step 4. This time we will not repeat the details of every step, instead, we will identify the column and row that give us the pivot element, and highlight the pivot element. The result is as follows.
1
We make the pivot element 1 by multiplying row 1 by 2 (notation: 2\(R_1\) → \(R_1\)), and we get
Now to make all other entries as zeros in this column, we first multiply row 1 by -1/2 and add it to row 2(notation: (-1/2)\(R_1\) + \(R_2\) → \(R_2\)), and then multiply row 1 by 10 (notation: 10\(R_1\) + \(R_3\) → \(R_3\)) and add it to the bottom row.
We no longer have negative entries in the bottom row, therefore we are finished.
Question Why are we finished when there are no negative entries in the bottom row?
Answer The answer lies in the bottom row. The bottom row corresponds to the equation:
\[\begin{array}{l}
0 x_{1}+0 x_{2}+20 s_{1}+10 s_{2}+Z=400 \quad \text { or } \\
z=400-20 s 1-10 s 2
\end{array}\nonumber \]
Since all variables are non-negative, the highest value \(Z\) can ever achieve is 400, and that will happen only when \(s_1\) and \(s_2\) are zero.
STEP 8. Read off your answers.
We now read off our answers, that is, we determine the basic solution associated with the final simplex tableau. Again, we look at the columns that have a 1 and all other entries zeros. Since the columns labeled \(s_1\) and \(s_2\) are not such columns, we arbitrarily choose \(s_1 = 0\), and \(s_2 = 0\), and we get
\[\left[\begin{array}{ccccc}
\mathrm{x}_{1} & \mathrm{x}_{2} & \mathrm{Z} & | & \mathrm{C} \\
0 & 1 & 0 & | & 8 \\
1 & 0 & 0 & | & 4 \\
0 & 0 & 1 & | & 400
\end{array}\right] \nonumber \]
The matrix reads \(x_1 = 4\), \(x_2= 8\) and \(z = 400\).
The final solution says that if Niki works 4 hours at Job I and 8 hours at Job II, she will maximize her income to $400. Since both slack variables are zero, it means that she would have used up all the working time, as well as the preparation time, and none will be left.


14.2: What Is Stress?
Learning Objectives
- Differentiate between stimulus-based and response-based definitions of stress
- Define stress as a process
- Differentiate between good stress and bad stress
- Describe the early contributions of Walter Cannon and Hans Selye to the stress research field
- Understand the physiological basis of stress and describe the general adaptation syndrome
The term stress as it relates to the human condition first emerged in scientific literature in the 1930s, but it did not enter the popular vernacular until the 1970s (Lyon, 2012). Today, we often use the term loosely in describing a variety of unpleasant feeling states; for example, we often say we are stressed out when we feel frustrated, angry, conflicted, overwhelmed, or fatigued. Despite the widespread use of the term, stress is a fairly vague concept that is difficult to define with precision.
Researchers have had a difficult time agreeing on an acceptable definition of stress. Some have conceptualized stress as a demanding or threatening event or situation (e.g., a high-stress job, overcrowding, and long commutes to work). Such conceptualizations are known as stimulus-based definitions because they characterize stress as a stimulus that causes certain reactions. Stimulus-based definitions of stress are problematic, however, because they fail to recognize that people differ in how they view and react to challenging life events and situations. For example, a conscientious student who has studied diligently all semester would likely experience less stress during final exams week than would a less responsible, unprepared student.
Others have conceptualized stress in ways that emphasize the physiological responses that occur when faced with demanding or threatening situations (e.g., increased arousal). These conceptualizations are referred to as response-based definitions because they describe stress as a response to environmental conditions. For example, the endocrinologist Hans Selye , a famous stress researcher, once defined stress as the “response of the body to any demand, whether it is caused by, or results in, pleasant or unpleasant conditions” (Selye, 1976, p. 74). Selye’s definition of stress is response-based in that it conceptualizes stress chiefly in terms of the body’s physiological reaction to any demand that is placed on it. Neither stimulus-based nor response-based definitions provide a complete definition of stress. Many of the physiological reactions that occur when faced with demanding situations (e.g., accelerated heart rate) can also occur in response to things that most people would not consider to be genuinely stressful, such as receiving unanticipated good news: an unexpected promotion or raise.
A useful way to conceptualize stress is to view it as a process whereby an individual perceives and responds to events that he appraises as overwhelming or threatening to his well-being (Lazarus & Folkman, 1984). A critical element of this definition is that it emphasizes the importance of how we appraise—that is, judge—demanding or threatening events (often referred to as stressors ); these appraisals, in turn, influence our reactions to such events. Two kinds of appraisals of a stressor are especially important in this regard: primary and secondary appraisals. A primary appraisal involves judgment about the degree of potential harm or threat to well-being that a stressor might entail. A stressor would likely be appraised as a threat if one anticipates that it could lead to some kind of harm, loss, or other negative consequence; conversely, a stressor would likely be appraised as a challenge if one believes that it carries the potential for gain or personal growth. For example, an employee who is promoted to a leadership position would likely perceive the promotion as a much greater threat if she believed the promotion would lead to excessive work demands than if she viewed it as an opportunity to gain new skills and grow professionally. Similarly, a college student on the cusp of graduation may face the change as a threat or a challenge (See figure \(\PageIndex{1}\) below).
The perception of a threat triggers a secondary appraisal : judgment of the options available to cope with a stressor, as well as perceptions of how effective such options will be (Lyon, 2012) (See figure \(\PageIndex{2}\)). As you may recall from what you learned about self-efficacy, an individual’s belief in his ability to complete a task is important (Bandura, 1994). A threat tends to be viewed as less catastrophic if one believes something can be done about it (Lazarus & Folkman, 1984). Imagine that two middle-aged women, Robin and Maria, perform breast self-examinations one morning and each woman notices a lump on the lower region of her left breast. Although both women view the breast lump as a potential threat (primary appraisal), their secondary appraisals differ considerably. In considering the breast lump, some of the thoughts racing through Robin’s mind are, “Oh my God, I could have breast cancer! What if the cancer has spread to the rest of my body and I cannot recover? What if I have to go through chemotherapy? I’ve heard that experience is awful! What if I have to quit my job? My husband and I won’t have enough money to pay the mortgage. Oh, this is just horrible…I can’t deal with it!” On the other hand, Maria thinks, “Hmm, this may not be good. Although most times these things turn out to be benign, I need to have it checked out. If it turns out to be breast cancer, there are doctors who can take care of it because the medical technology today is quite advanced. I’ll have a lot of different options, and I’ll be just fine.” Clearly, Robin and Maria have different outlooks on what might turn out to be a very serious situation: Robin seems to think that little could be done about it, whereas Maria believes that, worst case scenario, a number of options that are likely to be effective would be available. As such, Robin would clearly experience greater stress than would Maria.
To be sure, some stressors are inherently more stressful than others in that they are more threatening and leave less potential for variation in cognitive appraisals (e.g., objective threats to one’s health or safety). Nevertheless, appraisal will still play a role in augmenting or diminishing our reactions to such events (Everly & Lating, 2002).
If a person appraises an event as harmful and believes that the demands imposed by the event exceed the available resources to manage or adapt to it, the person will subjectively experience a state of stress. In contrast, if one does not appraise the same event as harmful or threatening, she is unlikely to experience stress. According to this definition, environmental events trigger stress reactions by the way they are interpreted and the meanings they are assigned. In short, stress is largely in the eye of the beholder: it’s not so much what happens to you as it is how you respond (Selye, 1976).
Good Stress?
Although stress carries a negative connotation, at times it may be of some benefit. Stress can motivate us to do things in our best interests, such as study for exams, visit the doctor regularly, exercise, and perform to the best of our ability at work. Indeed, Selye (1974) pointed out that not all stress is harmful. He argued that stress can sometimes be a positive, motivating force that can improve the quality of our lives. This kind of stress, which Selye called eustress (from the Greek eu = “good”), is a good kind of stress associated with positive feelings, optimal health, and performance. A moderate amount of stress can be beneficial in challenging situations. For example, athletes may be motivated and energized by pregame stress, and students may experience similar beneficial stress before a major exam. Indeed, research shows that moderate stress can enhance both immediate and delayed recall of educational material. Male participants in one study who memorized a scientific text passage showed improved memory of the passage immediately after exposure to a mild stressor as well as one day following exposure to the stressor (Hupbach & Fieman, 2012).
Increasing one’s level of stress will cause performance to change in a predictable way. As shown in figure \(\PageIndex{3}\), as stress increases, so do performance and general well-being (eustress); when stress levels reach an optimal level (the highest point of the curve), performance reaches its peak. A person at this stress level is colloquially at the top of his game, meaning he feels fully energized, focused, and can work with minimal effort and maximum efficiency. But when stress exceeds this optimal level, it is no longer a positive force—it becomes excessive and debilitating, or what Selye termed distress (from the Latin dis = “bad”). People who reach this level of stress feel burned out; they are fatigued, exhausted, and their performance begins to decline. If the stress remains excessive, health may begin to erode as well (Everly & Lating, 2002).
The Prevalence of Stress
Stress is everywhere and, as shown in figure \(\PageIndex{4}\), it has been on the rise over the last several years. Each of us is acquainted with stress—some are more familiar than others. In many ways, stress feels like a load you just can’t carry—a feeling you experience when, for example, you have to drive somewhere in a crippling blizzard, when you wake up late the morning of an important job interview, when you run out of money before the next pay period, and before taking an important exam for which you realize you are not fully prepared.
Stress is an experience that evokes a variety of responses, including those that are physiological (e.g., accelerated heart rate, headaches, or gastrointestinal problems), cognitive (e.g., difficulty concentrating or making decisions), and behavioral (e.g., drinking alcohol, smoking, or taking actions directed at eliminating the cause of the stress). Although stress can be positive at times, it can have deleterious health implications, contributing to the onset and progression of a variety of physical illnesses and diseases (Cohen & Herbert, 1996).
The scientific study of how stress and other psychological factors impact health falls within the realm of health psychology , a subfield of psychology devoted to understanding the importance of psychological influences on health, illness, and how people respond when they become ill (Taylor, 1999). Health psychology emerged as a discipline in the 1970s, a time during which there was increasing awareness of the role behavioral and lifestyle factors play in the development of illnesses and diseases (Straub, 2007). In addition to studying the connection between stress and illness, health psychologists investigate issues such as why people make certain lifestyle choices (e.g., smoking or eating unhealthy food despite knowing the potential adverse health implications of such behaviors). Health psychologists also design and investigate the effectiveness of interventions aimed at changing unhealthy behaviors. Perhaps one of the more fundamental tasks of health psychologists is to identify which groups of people are especially at risk for negative health outcomes, based on psychological or behavioral factors. For example, measuring differences in stress levels among demographic groups and how these levels change over time can help identify populations who may have an increased risk for illness or disease.
Figure \(\PageIndex{5}\) depicts the results of three national surveys in which several thousand individuals from different demographic groups completed a brief stress questionnaire; the surveys were administered in 1983, 2006, and 2009 (Cohen & Janicki-Deverts, 2012). All three surveys demonstrated higher stress in women than in men. Unemployed individuals reported high levels of stress in all three surveys, as did those with less education and income; retired persons reported the lowest stress levels. However, from 2006 to 2009 the greatest increase in stress levels occurred among men, Whites, people aged \(45-64\), college graduates, and those with full-time employment. One interpretation of these findings is that concerns surrounding the 2008–2009 economic downturn (e.g., threat of or actual job loss and substantial loss of retirement savings) may have been especially stressful to White, college-educated, employed men with limited time remaining in their working careers.
Early contributions to the Study of Stress
As previously stated, scientific interest in stress goes back nearly a century. One of the early pioneers in the study of stress was Walter Cannon , an eminent American physiologist at Harvard Medical School (See figure \(\PageIndex{6}\)). In the early part of the 20 th century, Cannon was the first to identify the body’s physiological reactions to stress.
Cannon and the Fight-or-Flight Response
Imagine that you are hiking in the beautiful mountains of Colorado on a warm and sunny spring day. At one point during your hike, a large, frightening-looking black bear appears from behind a stand of trees and sits about 50 yards from you. The bear notices you, sits up, and begins to lumber in your direction. In addition to thinking, “This is definitely not good,” a constellation of physiological reactions begins to take place inside you. Prompted by a deluge of epinephrine (adrenaline) and norepinephrine (noradrenaline) from your adrenal glands, your pupils begin to dilate. Your heart starts to pound and speeds up, you begin to breathe heavily and perspire, you get butterflies in your stomach, and your muscles become tense, preparing you to take some kind of direct action. Cannon proposed that this reaction, which he called the fight-or-flight response , occurs when a person experiences very strong emotions—especially those associated with a perceived threat (Cannon, 1932). During the fight-or-flight response, the body is rapidly aroused by activation of both the sympathetic nervous system and the endocrine system (See figure \(\PageIndex{7}\)). This arousal helps prepare the person to either fight or flee from a perceived threat.
According to Cannon, the fight-or-flight response is a built-in mechanism that assists in maintaining homeostasis—an internal environment in which physiological variables such as blood pressure, respiration, digestion, and temperature are stabilized at levels optimal for survival. Thus, Cannon viewed the fight-or-flight response as adaptive because it enables us to adjust internally and externally to changes in our surroundings, which is helpful in species survival.
Selye and the General Adaptation Syndrome
Another important early contributor to the stress field was Hans Selye , mentioned earlier. He would eventually become one of the world’s foremost experts in the study of stress (See figure \(\PageIndex{8}\)). As a young assistant in the biochemistry department at McGill University in the 1930s, Selye was engaged in research involving sex hormones in rats. Although he was unable to find an answer for what he was initially researching, he incidentally discovered that when exposed to prolonged negative stimulation (stressors)—such as extreme cold, surgical injury, excessive muscular exercise, and shock—the rats showed signs of adrenal enlargement, thymus and lymph node shrinkage, and stomach ulceration. Selye realized that these responses were triggered by a coordinated series of physiological reactions that unfold over time during continued exposure to a stressor. These physiological reactions were nonspecific, which means that regardless of the type of stressor, the same pattern of reactions would occur. What Selye discovered was the general adaptation syndrome , the body’s nonspecific physiological response to stress.
The general adaptation syndrome, shown in figure \(\PageIndex{9}\), consists of three stages:
- alarm reaction
- stage of resistance
- stage of exhaustion (Selye, 1936; 1976).
Alarm reaction describes the body’s immediate reaction upon facing a threatening situation or emergency, and it is roughly analogous to the fight-or-flight response described by Cannon. During an alarm reaction, you are alerted to a stressor, and your body alarms you with a cascade of physiological reactions that provide you with the energy to manage the situation. A person who wakes up in the middle of the night to discover her house is on fire, for example, is experiencing an alarm reaction.
If exposure to a stressor is prolonged, the organism will enter the stage of resistance . During this stage, the initial shock of alarm reaction has worn off and the body has adapted to the stressor. Nevertheless, the body also remains on alert and is prepared to respond as it did during the alarm reaction, although with less intensity. For example, suppose a child who went missing is still missing \(72\) hours later. Although the parents would obviously remain extremely disturbed, the magnitude of physiological reactions would likely have diminished over the \(72\) intervening hours due to some adaptation to this event.
If exposure to a stressor continues over a longer period of time, the stage of exhaustion ensues. At this stage, the person is no longer able to adapt to the stressor: the body’s ability to resist becomes depleted as physical wear takes its toll on the body’s tissues and organs. As a result, illness, disease, and other permanent damage to the body—even death—may occur. If a missing child still remained missing after three months, the long-term stress associated with this situation may cause a parent to literally faint with exhaustion at some point or even to develop a serious and irreversible illness.
In short, Selye’s general adaptation syndrome suggests that stressors tax the body via a three-phase process—an initial jolt, subsequent readjustment, and a later depletion of all physical resources—that ultimately lays the groundwork for serious health problems and even death. It should be pointed out, however, that this model is a response-based conceptualization of stress, focusing exclusively on the body’s physical responses while largely ignoring psychological factors such as appraisal and interpretation of threats. Nevertheless, Selye’s model has had an enormous impact on the field of stress because it offers a general explanation for how stress can lead to physical damage and, thus, disease. As we shall discuss later, prolonged or repeated stress has been implicated in development of a number of disorders such as hypertension and coronary artery disease.
The Physiological Basis of Stress
What goes on inside our bodies when we experience stress? The physiological mechanisms of stress are extremely complex, but they generally involve the work of two systems—the sympathetic nervous system and the hypothalamic-pituitary-adrenal (HPA) axis . When a person first perceives something as stressful (Selye’s alarm reaction), the sympathetic nervous system triggers arousal via the release of adrenaline from the adrenal glands. Release of these hormones activates the fight-or-flight responses to stress, such as accelerated heart rate and respiration. At the same time, the HPA axis, which is primarily endocrine in nature, becomes especially active, although it works much more slowly than the sympathetic nervous system. In response to stress, the hypothalamus (one of the limbic structures in the brain) releases corticotrophin-releasing factor, a hormone that causes the pituitary gland to release adrenocorticotropic hormone (ACTH) (See figure \(\PageIndex{10}\)). The ACTH then activates the adrenal glands to secrete a number of hormones into the bloodstream; an important one is cortisol, which can affect virtually every organ within the body. Cortisol is commonly known as a stress hormone and helps provide that boost of energy when we first encounter a stressor, preparing us to run away or fight. However, sustained elevated levels of cortisol weaken the immune system.
In short bursts, this process can have some favorable effects, such as providing extra energy, improving immune system functioning temporarily, and decreasing pain sensitivity. However, extended release of cortisol—as would happen with prolonged or chronic stress—often comes at a high price. High levels of cortisol have been shown to produce a number of harmful effects. For example, increases in cortisol can significantly weaken our immune system (Glaser & Kiecolt-Glaser, 2005), and high levels are frequently observed among depressed individuals (Geoffroy, Hertzman, Li, & Power, 2013). In summary, a stressful event causes a variety of physiological reactions that activate the adrenal glands, which in turn release epinephrine, norepinephrine, and cortisol. These hormones affect a number of bodily processes in ways that prepare the stressed person to take direct action, but also in ways that may heighten the potential for illness.
When stress is extreme or chronic, it can have profoundly negative consequences. For example, stress often contributes to the development of certain psychological disorders, including post-traumatic stress disorder, major depressive disorder, and other serious psychiatric conditions. Additionally, we noted earlier that stress is linked to the development and progression of a variety of physical illnesses and diseases. For example, researchers in one study found that people injured during the September 11, 2001, World Trade Center disaster or who developed post-traumatic stress symptoms afterward later suffered significantly elevated rates of heart disease (Jordan, Miller-Archie, Cone, Morabia, & Stellman, 2011). Another investigation yielded that self-reported stress symptoms among aging and retired Finnish food industry workers were associated with morbidity 11 years later. This study also predicted the onset of musculoskeletal, nervous system, and endocrine and metabolic disorders (Salonen, Arola, Nygård, & Huhtala, 2008). Another study reported that male South Korean manufacturing employees who reported high levels of work-related stress were more likely to catch the common cold over the next several months than were those employees who reported lower work-related stress levels (Park et al., 2011). Later, you will explore the mechanisms through which stress can produce physical illness and disease.
Summary
Stress is a process whereby an individual perceives and responds to events appraised as overwhelming or threatening to one’s well-being. The scientific study of how stress and emotional factors impact health and well-being is called health psychology, a field devoted to studying the general impact of psychological factors on health. The body’s primary physiological response during stress, the fight-or-flight response, was first identified in the early \(20^{th}\) century by Walter Cannon. The fight-or-flight response involves the coordinated activity of both the sympathetic nervous system and the hypothalamic-pituitary-adrenal (HPA) axis. Hans Selye, a noted endocrinologist, referred to these physiological reactions to stress as part of general adaptation syndrome, which occurs in three stages: alarm reaction (fight-or-flight reactions begin), resistance (the body begins to adapt to continuing stress), and exhaustion (adaptive energy is depleted, and stress begins to take a physical toll).
Glossary
- alarm reaction
- first stage of the general adaptation syndrome; characterized as the body’s immediate physiological reaction to a threatening situation or some other emergency; analogous to the fight-or-flight response
- cortisol
- stress hormone released by the adrenal glands when encountering a stressor; helps to provide a boost of energy, thereby preparing the individual to take action
- distress
- bad form of stress; usually high in intensity; often leads to exhaustion, fatigue, feeling burned out; associated with erosions in performance and health
- eustress
- good form of stress; low to moderate in intensity; associated with positive feelings, as well as optimal health and performance
- fight-or-flight response
- set of physiological reactions (increases in blood pressure, heart rate, respiration rate, and sweat) that occur when an individual encounters a perceived threat; these reactions are produced by activation of the sympathetic nervous system and the endocrine system
- general adaptation syndrome
- Hans Selye’s three-stage model of the body’s physiological reactions to stress and the process of stress adaptation: alarm reaction, stage of resistance, and stage of exhaustion
- health psychology
- subfield of psychology devoted to studying psychological influences on health, illness, and how people respond when they become ill
- hypothalamic-pituitary-adrenal (HPA) axis
- set of structures found in both the limbic system (hypothalamus) and the endocrine system (pituitary gland and adrenal glands) that regulate many of the body’s physiological reactions to stress through the release of hormones
- primary appraisal
- judgment about the degree of potential harm or threat to well-being that a stressor might entail
- secondary appraisal
- judgment of options available to cope with a stressor and their potential effectiveness
- stage of exhaustion
- third stage of the general adaptation syndrome; the body’s ability to resist stress becomes depleted; illness, disease, and even death may occur
- stage of resistance
- second stage of the general adaptation syndrome; the body adapts to a stressor for a period of time
- stress
- process whereby an individual perceives and responds to events that one appraises as overwhelming or threatening to one’s well-being
- stressors
- environmental events that may be judged as threatening or demanding; stimuli that initiate the stress process
Contributors and Attributions
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Rose M. Spielman with many significant contributors. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact<EMAIL_ADDRESS>.Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.