1.49: Unintended consequences of e-cigarette use- a public health epidemic
At the end of this case, students will be able to:
- Describe the social and behavioral influences on e-cigarette use and develop strategies to address those influences
- Discuss the risk factors of e-cigarette use in an adolescent population
- Determine the role of a pharmacist in addressing the e-cigarette public health epidemic
- Identify public health strategies to effectively address the e-cigarette epidemic
Introduction
Electronic cigarettes, also known as e-cigarettes, were first introduced by Herbert A. Gilbert, who patented a “smokeless tobacco cigarette” in August 1965 with a goal of providing a safe and harmless method for smoking. 1 First-generation e-cigarettes looked similar to traditional cigarettes, composed of a battery and a component to house nicotine solution (also called e-liquid or e-juice). 1 Over time, e-cigarette devices have evolved to resemble pens or USB flash drives, with availability of over thousands of e-liquid flavors. 2
Initially marketed by companies as harmless, emerging research has shown that e-cigarettes are anything but. The majority of e-cigarettes contain nicotine in their e-liquid, which remains an addictive substance that can be detrimental to the development of learning, memory, and attention of youth. 3 A commonly sold e-cigarette called JUUL provides its e-liquid in what is known as a pod, which can contain as much nicotine as a regular pack of 20 cigarettes. 4 Unfortunately, an estimated two-thirds of youth choosing to use JUUL are also unaware that JUUL pods always contain nicotine, and therefore, may view it as harmless. 5 Not only can the use of e-cigarettes cause nicotine addiction, but in 2020, the Centers for Disease Control and Prevention reported over 2,800 cases of vaping-associate lung injury across the United States. 6
In the United States, the e-cigarette entered the market in the mid-2000s, and sales have rapidly increased since 2007. E-cigarettes are referred to by many different names, including “e-cigs,” “mods,” “vape pens,” and “vapes,” resulting in the term “vaping” to describe the use of an e-cigarette. Although companies selling e-cigarettes intended their products to be used as a safe way to deliver nicotine to adults who were already using traditional cigarettes, widespread advertisement campaigns of e-cigarettes pushed in both television and print promoting fruity flavors of e-liquid attractive to young people resulted in a very different reality. Accordingly, in 2013, it was estimated that 13.1 million middle school and high school students were aware of e-cigarettes, and by 2018, e-cigarette use among high school students was 20.8%, equating to an estimated 3.6 million youth. 3,7 E-cigarette use has now surpassed that of traditional cigarette use and is the most used tobacco product among youth. 1 E-cigarette use during youth has been shown to progress to traditional cigarette use later in life. 8,9 This can be particularly concerning as research shows this trend holds true for youth who begin e-cigarette use with no intentions of using traditional cigarettes in the future, resulting in a new population of cigarette users. 8
Pharmacists have played an important role in tobacco cessation through motivational patient interviewing and counseling; similarly, there is also a role that the profession can play to address the growing public health epidemic of e-cigarette use, particularly among youth users. As health care professionals, we can emphasize recommendations from the Centers of Disease Control and Prevention to provide education on the potential for e-cigarettes to benefit adult non-pregnant smokers, and urge against the use of e-cigarettes for youth, young adults, pregnant women, and those who do not currently use tobacco products, emphasizing the harm they may cause. 10 Importantly, although patient education and counseling are vital, it is also necessary to recognize education and counseling alone may be insufficient to fully address the current e-cigarette epidemic in the United States. Laws and policies, such as the Tobacco 21 law put into place in 2019, may need stricter enforcement to control the sales of such products to individuals under the age of 21. 11
Case
You are an ambulatory care pharmacist working at a family medicine practice.
CC: “My child keeps getting in trouble for vaping at school and I need them to stop vaping.”
Patient: JL (they/them) is a 17-year-old patient (66 in, 50.1 kg) brought to the clinic by their mother for help in quitting vaping.
HPI: JL has been vaping for the past three years. They initially started vaping occasionally on the weekends when staying at friends’ houses, but now uses throughout the day every day, mostly related to stress and anxiety. They state they go through about 1 gram cartridge in 3-4 days. They use a variety of types, but most commonly JUUL. They either buy it from stores that they know will sell it to them or has their 18-year-old friends purchase it. JL’s mother has also purchased them several pens in the past when they were feeling anxious due to being out.
PMH: GAD
FH:
- Mother: current smoker; asthma
- Father: HTN; hyperlipidemia
- Two younger siblings: both healthy
SH:
- Vapes daily: mainly uses JUUL pods; occasionally borrows other types from friends
- Denies alcohol, cigarette, and illicit substance use
- Denies use of products with THC to their knowledge
Medications:
- Apri (ethinyl estradiol 0.03 mg and desogestrel 0.15 mg) one tablet daily
Allergies: NKDA
Vitals:
- BP 110/76 mmHg
- HR 96bpm
- RR 16/min
- Temp: 97.6°F
- Pain: 0 out of 10
Surgical history: none
Additional context: JL doesn’t understand why they are getting in trouble for vaping at school when they know that there aren’t any harms to using it. They say, “it’s not like smoking cigarettes like my mom does.” They keep the pen in their locker and uses it between classes at school. JL is interested in being able to reduce their use of vaping as they do not want to put their college plans in jeopardy with any more punishments but feels they can still safely vape after school and on weekends. They state by using it, they don’t feel they need to go to their counselor anymore for anxiety, because they use vaping to calm themself down.
Case Questions
- What are the risks of using e-cigarettes in a teenager?
- Does JL qualify for use of nicotine replacement therapy, Chantix, or Zyban?
- What are the social and behavioral factors that may make quitting use of e-cigarettes difficult for JL? How might you help them to address them?
- What are the risks of using e-cigarettes and their current medications together?
- How does the marketing and availability of e-cigarette products affect their use by teenagers?
- What are some action steps that pharmacists can take to help address this public health issue? Consider a variety of practice settings: community, hospital, outpatient, public health department, research.
Author Commentary
Addressing the use of tobacco and tobacco products is a role that pharmacists already play. By utilizing the Transtheoretical Model for Change or the 5 A’s approach, pharmacists can help a patient understand where they are at in their desire to make the change and meet them there to best help them achieve success. Understanding the importance of tobacco cessation and the negative consequences of nicotine addiction, pharmacists can also address rising e-cigarette use in youth and adults. It is important for pharmacists to stay abreast of emerging data regarding e-cigarette use. The CDC provides updates on e-cigarette, or vaping, product use-associated lung injury (EVALI) that are useful for pharmacists to be aware of. Equally important is staying up to date on the long-term consequences of vaping and its impact on comorbid health conditions.
Patient Approaches and Opportunities
Pharmacists have tools (Transtheoretical Model for Change or the 5 A’s approach) available to provide education to youth and adults in multiple settings regarding the dangers of e-cigarettes. Pharmacists may collaborate with schools to provide classroom education on the potential consequences of vaping. These education sessions may help in addressing some of the myths about vaping (i.e., JUUL pods not containing nicotine) and lead youth to make smarter decisions regarding vaping. Education should be unbiased, and evidence based. Pharmacists should address the influences of peer pressure and include tips on how to combat it and should include motivational messages to discourage youth from picking up potentially addictive and detrimental behaviors. With each encounter, either individual or in group settings, pharmacists can assess the use of vaping devices and if a patient desires to quit. As the scope of pharmacists continue to expand to prescribing pharmacotherapy for tobacco and vaping cessation, pharmacists will play an even larger role in serving patients. Under current regulations, pharmacists can still be valuable in assess and recommending OTC products as well as connecting patients with prescribers and resources.
As scopes of practice expand, pharmacists prescribing appropriate pharmacotherapy to tobacco users trying to quit will also be key. Additionally, if and when possible, pharmacists can also be part of the larger movement to enact policies that increase enforcement on the sale of e-cigarettes to minors. The e-cigarette pandemic must be tackled from multiple perspectives to really be able to find a sustainable solution.
Related chapters of interest:
External resources:
-
Websites:
- Centers for Disease Control and Prevention. Electronic cigarettes. https://www.cdc.gov/tobacco/basic_information/e-cigarettes/index.htm
- US Department of Health ang Human Services. Know the risks: e-cigarettes & young people https://e-cigarettes.surgeongeneral.gov/
- Surgeon General advisory on e-cigarette use among youth. https://e-cigarettes.surgeongeneral.gov/documents/surgeon-generals-advisory-on-e-cigarette-use-among-youth-2018.pdf
- Smokefreeteen.gov. How to quit vaping. https://teen.smokefree.gov/vaping-quit-plan
-
Journal articles:
- St Helen G, Eaton DL. Public health consequences of e-cigarette use. JAMA Intern Med 2018;178(7):984-986.
- Leventhal AM, Goldenson NI, Cho J, Kirkpatrick MG, McConnell RS, Stone MD, Pang RD, Audrain-McGovern J, Barrington-Trimis JL. Flavored e-cigarette use and progression of vaping in adolescents. Pediatrics 2019;144(5):e20190789.
- Phone applications:
References
- Centers for Disease Control and Prevention. Introduction, conclusions, and historical background relative to e-cigarettes. https://www.cdc.gov/tobacco/data_statistics/sgr/e-cigarettes/pdfs/2016_SGR_Chap_1_508.pdf . Accessed November 9, 2020.
- Zhu SH, Sun JY, Bonnevie E, Cummins SE, Gamst A, Yin L, Lee M. Four hundred and sixty brands of e-cigarettes and counting: implications for product regulation. Tob Control 2014;23(Suppl 3):iii3-9.
- Office of the Surgeon General. Surgeon General’s advisory on E-cigarette use among youth. https://e-cigarettes.surgeongeneral.gov/documents/surgeon-generals-advisory-on-e-cigarette-use-among-youth-2018.pdf . Accessed November 9, 2020.
- Willett JG, Bennett M, Hair EC, et al. Recognition, use and perceptions of JUUL among youth and young adults. Tob Control 2019;28(1):115-6.
- Truth Initiative. JUUL e-cigarettes gain popularity among youth, but awareness of nicotine presence remains low. https://truthinitiative.org/news/juul-e-cigarettes-gain-popularity-among-youth . Accessed November 9, 2020.
- Centers for Disease Control and Prevention. Outbreak of lung injury associated with e-cigarette use, or vaping. Updated February 25, 2020. www.cdc.gov/tobacco/basic_information/e-Cigarettes/severe-Lung-Disease.html . Accessed November 9, 2020.
- Wang B, King BA, Corey CG, Arrazola RA, Johnson SE. Awareness and use of non-conventional tobacco products among U.S. students, 2012. Am J Prevent Med 2014;47(2 Suppl 1):S36-S52.
- Primack BA, Soneji S, Stoolmiller M, Fine MJ, Sargent JD. Progression to traditional cigarette smoking after electronic cigarette use among US adolescents and young adults. JAMA Pediatr 2015;169(11):1018-23.
- Primack BA, Shensa A, Sidani JE, et al. Initiation of traditional cigarette smoking after electronic cigarette use among tobacco-naïve US young adults. Am J Med 2018;131(4):443.e1-9.
- Centers for Disease Control and Prevention. Electronic cigarettes. Reviewed September 2020. https://www.cdc.gov/tobacco/basic_information/e-cigarettes/index.htm . Accessed February 15, 2021.
- U.S. Food and Drug Administration. Tobacco 21. https://www.fda.gov/tobacco-products/retail-sales-tobacco-products/tobacco-21 . Accessed February 15, 2021.


10.6: J. S. Bach- His Works
Works
In 1950, a thematic catalogue called Bach-Werke-Verzeichnis (Bach Works Catalogue) was compiled by Wolfgang Schmieder. Schmieder largely followed the Bach-Gesellschaft-Ausgabe , a comprehensive edition of the composer’s works that was produced between 1850 and 1900: BWV 1–224 are cantatas; BWV 225–249, large-scale choral works including his Passions; BWV 250–524, chorales and sacred songs; BWV 525–748, organ works; BWV 772–994, other keyboard works; BWV 995–1000, lute music; BWV 1001–40, chamber music; BWV 1041–71, orchestral music; and BWV 1072–1126, canons and fugues.
Organ Works
Bach was best known during his lifetime as an organist, organ consultant, and composer of organ works in both the traditional German free genres—such as preludes, fantasias, and toccatas—and stricter forms, such as chorale preludes and fugues. At a young age, he established a reputation for his great creativity and ability to integrate foreign styles into his organ works. A decidedly North German influence was exerted by Georg Böhm, with whom Bach came into contact in Lüneburg, and Dieterich Buxtehude, whom the young organist visited in Lübeck in 1704 on an extended leave of absence from his job in Arnstadt. Around this time, Bach copied the works of numerous French and Italian composers to gain insights into their compositional languages, and later arranged violin concertos by Vivaldi and others for organ and harpsichord. During his most productive period (1708–14) he composed about a dozen pairs of preludes and fugues, five toccatas and fugues, and the Little Organ Book , an unfinished collection of forty-six short chorale preludes that demonstrates compositional techniques in the setting of chorale tunes. After leaving Weimar, Bach wrote less for organ, although some of his best-known works (the six trio sonatas, the German Organ Mass in Clavier-Übung III from 1739, and the Great Eighteen chorales, revised late in his life) were composed after his leaving Weimar. Bach was extensively engaged later in his life in consulting on organ projects, testing newly built organs, and dedicating organs in afternoon recitals.
Other Keyboard Works
- The Well-Tempered Clavier , Books 1 and 2 (BWV 846–893). Each book consists of a prelude and fugue in each of the 24 major and minor keys in chromatic order from C major to B minor (thus, the whole collection is often referred to as “the 48”). “Well-tempered” in the title refers to the temperament (system of tuning); many temperaments before Bach’s time were not flexible enough to allow compositions to utilize more than just a few keys.
- The Inventions and Sinfonias (BWV 772–801). These short two- and three-part contrapuntal works are arranged in the same chromatic order as The Well-Tempered Clavier , omitting some of the rarer keys. These pieces were intended by Bach for instructional purposes.
- Three collections of dance suites: the English Suites (BWV 806–811), the French Suites (BWV 812–817), and the Partitas for keyboard ( Clavier-Übung I , BWV 825–830). Each collection contains six suites built on the standard model ( Allemande – Courante – Sarabande –(optional movement)– Gigue ). The English Suites closely follow the traditional model, adding a prelude before the allemande and including a single movement between the sarabande and the gigue . The French Suites omit preludes, but have multiple movements between the sarabande and the gigue . The partitas expand the model further with elaborate introductory movements and miscellaneous movements between the basic elements of the model.
- The Goldberg Variations (BWV 988), an aria with thirty variations. The collection has a complex and unconventional structure: the variations build on the bass line of the aria, rather than its melody, and musical canons are interpolated according to a grand plan. There are nine canons within the thirty variations, every third variation is a canon. These variations move in order from canon at the unison to canon at the ninth. The first eight are in pairs (unison and octave, second and seventh, third and sixth, fourth and fifth). The ninth canon stands on its own due to compositional dissimilarities. The final variation, instead of being the expected canon at the tenth, is a quodlibet.
- Miscellaneous pieces such as the Overture in the French Style ( French Overture , BWV 831) and the Italian Concerto (BWV 971) (published together as Clavier-Übung II ), and the Chromatic Fantasia and Fugue (BWV 903).
Among Bach’s lesser known keyboard works are seven toccatas (BWV 910–916), four duets (BWV 802–805), sonatas for keyboard (BWV 963–967), the Six Little Preludes (BWV 933–938), and the Aria variata alla maniera italiana (BWV 989).
Orchestral and Chamber Music
Bach wrote for single instruments, duets, and small ensembles. Many of his solo works, such as his six sonatas and partitas for violin (BWV 1001–1006), six cello suites (BWV 1007–1012), and partita for solo flute (BWV 1013), are widely considered among the most profound works in the repertoire. Bach composed a suite and several other works which have been claimed (since 1900) for the solo lute, but there is no evidence that he wrote for this instrument. He wrote trio sonatas; solo sonatas (accompanied by continuo) for the flute and for the viola da gamba; and a large number of canons and ricercars, mostly with unspecified instrumentation. The most significant examples of the latter are contained in The Art of Fugue and The Musical Offering .
Bach’s best-known orchestral works are the Brandenburg Concertos , so named because he submitted them in the hope of gaining employment fromMargrave Christian Ludwig of Brandenburg-Schwedt in 1721; his application was unsuccessful. These works are examples of the concerto grosso genre. Other surviving works in the concerto form include two violin concertos (BWV 1041 and BWV 1042); a concerto for two violins in D minor (BWV 1043), often referred to as Bach’s “double” concerto; and concertos for one to four harpsichords. It is widely accepted that many of the harpsichord concertos were not original works, but arrangements of his concertos for other instruments now lost. A number of violin, oboe, and flute concertos have been reconstructed from these. In addition to concertos, Bach wrote four orchestral suites, and a series of stylized dances for orchestra, each preceded by a French overture.
Vocal and Choral Works
Cantatas
As the Thomaskantor , beginning mid of 1723, Bach performed a cantata each Sunday and feast day that corresponded to the lectionary readings of the week. Although Bach performed cantatas by other composers, he composed at least three entire annual cycles of cantatas at Leipzig, in addition to those composed at Mühlhausen and Weimar. In total he wrote more than three hundred sacred cantatas, of which nearly two hundred survive.
His cantatas vary greatly in form and instrumentation, including those for solo singers, single choruses, small instrumental groups, and grand orchestras. Many consist of a large opening chorus followed by one or more recitative-aria pairs for soloists (or duets) and a concluding chorale. The recitative is part of the corresponding Bible reading for the week and the aria is a contemporary reflection on it. The melody of the concluding chorale often appears as a cantus firmus in the opening movement. Among his best known cantatas are:
- Christ lag in Todes Banden , BWV 4
- Ich hatte viel Bekümmernis , BWV 21
- Ein feste Burg ist unser Gott , BWV 80
- Gottes Zeit ist die allerbeste Zeit , BWV 106 ( Actus Tragicus )
- Wachet auf, ruft uns die Stimme , BWV 140
- Herz und Mund und Tat und Leben , BWV 147
In addition, Bach wrote a number of secular cantatas, usually for civic events such as council inaugurations. These include wedding cantatas, the Wedding Quodlibet , the Peasant Cantata , and the Coffee Cantata .
Motets
Bach’s motets (BWV 225–231) are pieces on sacred themes for choir and basso continuo, with instruments playing colla parte. Several of them were composed for funerals. The six motets certainly composed by Bach are Singet dem Herrn ein neues Lied , Der Geist hilft unser Schwachheit auf , Jesu, meine Freude , Fürchte dich nicht , Komm, Jesu, komm , and Lobet den Herrn, alle Heiden . The motet Sei Lob und Preis mit Ehren (BWV 231) is part of the composite motet Jauchzet dem Herrn, alle Welt (BWV Anh. 160), other parts of which may be based on work by Telemann.
Passions, Oratorios, Magnificat
Bach’s large choral-orchestral works include the grand scale St Matthew Passion and St John Passion , both written for Good Friday vesper services at the Thomaskirche and the Nikolaikirche in alternate years, and the Christmas Oratorio (a set of six cantatas for use in the liturgical season of Christmas). Shorter works are the Easter Oratorio , the Ascension Oratorio , and the Magnificat.
Mass in B Minor
Bach assembled his last large work, the Mass in B minor, near the end of his life, between 1748 and 1749. The mass was never performed in full during Bach’s lifetime. He incorporated the Sanctus of 1724 and the Missa in B minor, composed in 1733. He derived many movements from his cantatas, such as Weinen, Klagen, Sorgen, Zagen , BWV 12 , written in 1714, and composed some new movements. All of these movements have substantial solo parts as well as choruses. It is not known what direction of development Bach had intended for his last Mass to take. As Stauffer states,
If Bach had lived longer, it is likely that he would have created a definitive fair copy of the Mass, similar to those of the St. John and St. Matthew Passions… As Otto Bettmann once remarked, Bach’s ‘music sets in order what life cannot.’
Musical Style
During the Baroque period, many composers only wrote the framework, and performers embellished this framework with ornaments and other elaboration. This practice varied considerably between the schools of European music; Bach notated most or all of the details of his melodic lines, leaving little for performers to interpolate. This accounted for his control over the dense contrapuntal textures that he favored, and decreased leeway for spontaneous variation of musical lines. At the same time, Bach left the instrumentation of major works including The Art of Fugue open.
Bach’s devout relationship with the Christian God in the Lutheran tradition and the high demand for religious music of his times placed sacred music at the centre of his repertory. He taught Luther’s Small Catechism as the Thomaskantor in Leipzig, and some of his pieces represent it;the Lutheran chorale hymn tune was the basis of much of his work. He wrote more cogent, tightly integrated chorale preludes than most. The large-scale structure of some of Bach’s sacred works is evidence of subtle, elaborate planning. For example, the St Matthew Passion illustrates the Passion with Bible text reflected in recitatives, arias, choruses, and chorales.
Bach’s drive to display musical achievements was evident in his composition. He wrote much for the keyboard and led its elevation from continuo to solo instrument with harpsichord concertos and keyboard obbligato. Bach produced collections of movements that explored the range of artistic and technical possibilities inherent in various genres. The most famous example is The Well-Tempered Clavier , in which each book presents a prelude and fugue in every major and minor key. Each fugue displays a variety of contrapuntal and fugal techniques.
Performances
Present-day Bach performers usually pursue one of two traditions: so-called “authentic performance practice”, utilizing historical techniques; or the use of modern instruments and playing techniques, often with larger ensembles. In Bach’s time orchestras and choirs were usually smaller than those of later composers, and even Bach’s most ambitious choral works, such as his Mass in B minor and Passions, were composed for relatively modest forces. Some of Bach’s important chamber music does not indicate instrumentation, which allows for a greater variety of ensembles.
Modern adaptations of Bach’s music contributed greatly to Bach’s popularization in the second half of the 20th century. Among these were the Swingle Singers’ versions of Bach pieces (for instance, the “ Air ” from Orchestral Suite No. 3, or the Wachet Auf… chorale prelude) and Wendy Carlos’s 1968 Switched-On Bach , which used the Moog electronic synthesizer. Jazz musicians have adopted Bach’s music, with Jacques Loussier, Ian Anderson,Uri Caine, and the Modern Jazz Quartet among those creating jazz versions of Bach works.
Contributors and Attributions
- Johann Sebastian Bach. Located at : https://en.Wikipedia.org/wiki/Johann_Sebastian_Bach . License : CC BY-SA: Attribution-ShareAlike
- Bach Brandenburg Concerto no 6. Authored by : Jason Weinberger. Located at : https://www.flickr.com/photos/jasonweinberger/6483948851/ . License : CC BY: Attribution


13.1: Gustation (Taste)
Gustation is the special sense associated with the tongue. The surface of the tongue, along with the rest of the oral cavity, is lined by a stratified squamous epithelium. Raised bumps called papillae (singular = papilla) contain the structures for gustatory transduction. There are four types of papillae, based on their appearance (Figure \(\PageIndex{1}\)): circumvallate, foliate, filiform, and fungiform. Within the structure of the papillae are taste buds that contain specialized gustatory receptor cells for the transduction of taste stimuli. These receptor cells are sensitive to the chemicals contained within foods that are ingested, and they release neurotransmitters based on the amount of the chemical in the food.


3.1: Broad and Narrow Transcription
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Learning to use the IPA to transcribe speech can be very challenging, for many reasons.  Another challenge is simply remembering which symbols correspond to which sounds. The tables in Units 2.4 and 3.2 may seem quite daunting, but the more you practice, the better you’ll get at remembering the IPA symbols.
A challenge that many beginner linguists face is deciding exactly how much detail to include in their IPA transcriptions. For example, if you know that Canadian English speakers tend to diphthongize the mid-tense vowels [e] and [o] in words like say and show , should you transcribe them as the diphthongs [eɪ] and [oʊ]? And the segment [p] in the word apple doesn’t sound quite like the [p] in pear ; how should one indicate that? Does the word manager really begin with the same syllable that the word human ends with?
Part of learning to transcribe involves making a decision about exactly how much detail to include in your transcription. If your transcription includes enough information to identify the place and manner of articulation of consonants, the voicing of stops and fricatives, and the tongue and lip position for vowels, this is usually enough information for someone reading your transcription to be able to recognize the words you’ve transcribed. A transcription at this level is called a broad transcription .
But it’s possible to include a great deal more detail in your transcription, to more accurately represent the particulars of accent and dialect and the variations in certain segments. A transcription that includes a lot of phonetic detail is called a narrow transcription . The rest of this chapter discusses the most salient details that would be included in a narrow transcription of the most widespread variety of Canadian English.


9.5: Behavioristic Approach – Parsing a Sentence
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Behavioristic experiments about how human beings parse a sentence often use syntactically ambiguous sentences. Because it is easier to realize that sentence-analysing mechanisms called parsing take place when using sentences in which we cannot automatically constitute the meaning of the sentence. There are two different theories about how humans parse sentences. The syntax-first approach claims that syntax plays the main part whereas semantics has only a supporting role, whereas the interactionist approach states that both syntax and semantics work together to determine the meaning of a sentence. Both theories will be explained below in more detail.
The Syntax-First Approach of Parsing The syntax-first approach concentrates on the role of syntax when parsing a sentence. That humans infer the meaning of a sentence with help of its syntactical structure (Kako and Wagner 2001) can easily be seen when considering Lewis Carroll´s poem ‘Jabberwocky’:
"Twas brillig, and the slithy toves Did gyre and gimble in the wabe: All mimsy were the borogoves, And the mome raths outgrabe."
Although most of the words in the poems have no meaning one may ascribe at least some sense to the poem because of its syntactical structure.
There are many different syntactic rules that are used when parsing a sentence. One important rule is the principle of late closure which means that a person assumes that a new word he perceives is part of the current phrase. That this principle is used for parsing sentences can be seen very good with help of a so called garden-path sentence. Experiments with garden-path sentences have been done by Frazier and Fayner 1982. One example of a garden-path sentence is: “Because he always jogs a mile seems a short distance to him.” When reading this sentence one first wants to continue the phrase “Because he always jogs” by adding “a mile” to the phrase, but when reading further one realizes that the words “a mile” are the beginning of a new phrase. This shows that we parse a sentence by trying to add new words to a phrase as long as possible. Garden-path sentences show that we use the principle of late closure as long it makes syntactically sense to add a word to the current phrase but when the sentence starts to get incorrect semantics are often used to rearrange the sentence. The syntax-first approach does not disregard semantics. According to this approach we use syntax first to parse a sentence and semantics is later on used to make sense of the sentence.
Apart from experiments which show how syntax is used for parsing sentences there were also experimens on how semantics can influence the sentence processing. One important experiment about that issue has been done by Daniel Slobin in 1966. He showed that passive sentences are understood faster if the semantics of the words allow only one subject to be the actor. Sentences like “The horse was kicked by the cow.” and “The fence was kicked by the cow.” are grammatically equal and in both cases only one syntactical parsing is possible. Nevertheless the first sentence semantically provides two subjects as possible actors and therefore it needs longer to parse this sentence. By measuring this significant difference Daniel Slobin showed that semantics play an important role in parsing a sentence, too.


8.6: Medical uses of radioisotopes
The use of nuclear chemistry in medical technologies is increasing over time. The medical uses can be divided into two categories :
- medical imaging of organs or diagnosing any malfunction and
- therapeutic use, mainly for killing cancerous cells.
Radioisotopes in medical imaging
- those that emit \(\ce{\gamma}\)-rays, directly or indirectly, during their decay,
- can be delivered to the organ of interest in pure form or tagged in a compound,
- are short-lived or can be excreted from the body after use.
Table 8.6.1 lists some radioisotopes commonly used in medical imaging. A low dose of the radioisotope is administered to a patient. The \(\ce{\gamma}\)-rays cross over the body and are recorded like X-rays. A computer finally converts the recording into a useful image. The image is compared with an image of a healthy organ to diagnose any malfunction.
| Radioisotope | Symbol | Mode of decay | Half-life | Use in medical imaging |
|---|---|---|---|---|
| Carbon-11 | \(\ce{_6^11C}\) | β + , g | 20.3 m | Brain scan to trace glucose |
| Fluorine-19 | \(\ce{_9^18F}\) | β + , g | 109m | Brain scan to trace glucose |
| Chromium-51 | \(\ce{_24^51Cr}\) | E.C., g | 27.3 d | Diagnose albinism; image the spleen and gastrointestinal track |
| Gallium-67 | \(\ce{_31^67Ga}\) | E.C., g | 78.3 h | Whole-body scan for tumors |
| Selenium-75 | \(\ce{_34^75Se}\) | E.C., g | 118 d | Pancreas scan |
| Krypton-81m | \(\ce{_36^{81m}Kr}\) | g | 13.3 s | Lung ventilation scan |
| Xenon-133 | \(\ce{_54^133Xe}\) | β | 5.24 d | Lung ventilation scan |
| Strontium-81 | \(\ce{_38^81Sr}\) | β | 22.2 m | Scan for bone cancer and bone diseases |
| Mercury-197 | \(\ce{_80^197Hg}\) | E.C., g | 64.1 h | Kidney scan |
| Iron-59 | \(\ce{_26^59Fe}\) | β, g | 44.5 d | Bone marrow function and anemias |
| Iodine-131 | \(\ce{_53^131I}\) | β, g | 8.04 d | Diagnosis of thyroid malfunction |
| Iodine-123 | \(\ce{_53^123I}\) | E.C., g | 13.2 h | Diagnosis of thyroid malfunction |
| Technetium-99m | \(\ce{_43^{99m}Te}\) | g | 6.01 h | Brain, liver, kidney, bone scans, and diagnosis of the damaged heart muscle |
| Phosphorous-32 | \(\ce{_15^32P}\) | β | 14.3 d | Detect eye tumors |
| Thallium-201 | \(\ce{_81^201Tl}\) | E.C., g | 3.05 d | Heart scan and exercise stress test |
An example of medical imaging is the thyroid gland in the neck that produces the hormone thyroxin, which controls the overall rate of metabolism in the body. Each thyroxin molecule contains four iodine atoms. Administration of radioactive Na 131 I or Na 123 I salt accumulates the iodine in the thyroid gland in a few hours. Decay of 131 I and 123 I involves \(\ce{\gamma}\)-emission.
\[\ce{_53^131I -> _54^131Xe + _{-1}^{0}{e}} + \gamma\nonumber\]
\[\ce{_53^123I + _{-1}^{0}{e} -> _52^123Xe} + \gamma\nonumber\]
The \(\ce{\gamma}\)-emission from the iodine localized in the thyroid gland is recorded, as shown in Fig. 8.6.1. An overactive thyroid (hyperthyroidism) cumulates more and underactive thyroid (hypothyroidism) cumulates less iodine than a healthy thyroid.
Another example is positron emission tomography (PET). Positron emitters like carbon-11 and fluorine-18 incorporated in a suitable compound like glucose allow following the metabolic path of the compound. For example, 18-fluorodeoxyglucose )18-FDG) is a glucose molecule in which one of the oxygen is replaced with 18 F. Intravenous injection of the 18-FDG ultimately results in the cumulation of 18-FDG in the brain and other body organs where glucose is used in the metabolic process. The 18 F emits a positron, which, being an anti-particle of the electron, reacts with the electron and produces two g \(\ce{\gamma}\)-rays.
\[\ce{_9^18F -> _8^18O + _{1}^{0}{e}} + \gamma\nonumber\]
\[\ce{_{1}^{0}{e} + _{1}^{0}{e} ->} 2\gamma\nonumber\]
The \(\ce{\gamma}\)-rays are used to obtain an image of the organ. The image reveals problem areas in the form of an abnormal concentration of glucose in the part of the organ. For example, Fig. 8.6.2 compares the PET image of a healthy brain versus a brain with Alzheimer's disease.
Since glucose metabolism happens in all organs, whole-body PET scans can be used to diagnose lung, colorectal, head and neck, and esophageal cancers as well as other diseases that involve abnormal glucose metabolism. For example, tumors have high metabolic rates; the PET scans using 18-FDG are used to detect them, as shown in Fig. 8.6.3.
Medical imaging without using radioisotopes
Major examples of medical imaging using external radiation sources include the following.
- X-rays, which are external radiation source, is commonly used for medical imaging of organs and bones.
- Computed tomography (CT) scan uses computer processing of many X-ray measurements, from different angles, to produce a cross-sectional view (virtual slices) of the organ.
- Magnetic resonance image (MRI) is another powerful medical imaging technique that is based on the fact that hydrogen atoms splint into two two energy states when placed in a strong magnetic field. When illuminated with infrared (IR) radiation of the energy matching with the energy gap between the two groups, the hydrogen atoms are excited from the lower to higher energy state. The decay of the excited state emits the IR radiations that are recorded to obtain the image of soft tissues that contain many hydrogen atoms in the form of water molecules. Fig. 8.6.4 shows some examples.
Radiation therapy
The purpose of radiation therapy is to selective kill the diseased cells or tissues by exposing them to radiation. Higher radiation doses are required for therapy than for imaging. The radiation source can be external or internal.
External radiation therapy
In the external irradiation, the radiation from the source, such as coba which are often used, other radiation sources are being developed, such as proton beam from the cyclotron, which have been used to kill inoperable tumors near or in the eye, skull base, or spine. Proton therapy uses a beam of proton to deliver radiation directly to the tumor.
Internal radiation therapy
Internal radiation therapy is used when a short-lived radioisotope can be made to selectively concentrate in the organ or tissue of interest. For example, iodine-131 is \(\ce{\beta}\) and \(\ce{\gamma}\)-emitter, is administered to a patient, is picked up by the thyroid gland, and is used to treat hyperthyroidism. Another example is actinium-225, which is an \(\ce{\alpha}\)-emitter with a half-life of 10 days. Actinium-225 installs in a monoclonal antibody that is attached to a prostate-specific antigen and delivered selectively to treat prostate tumors.
Brachytherapy
Brachytherapy or seed implantation is a form of internal radiation treatment that delivers a high dose for a short period compared to the external radiation treatment. Fig. 8.6.6 shows the sites in the body where brachytherapy can be used to treat cancer. For example, 40 or more titanium capsules, about the size of a rice grain (Fig. 8.6.7), are implanted to treat prostate cancer. The seed contains \(\ce{\gamma}\)-emitter like iodine-125 (half-life 60 days), palladium-103 (half-life 17 days), or cesium-131 (half-life 10 days). The seed may be left in the body because, due to the short half-life, they are no more significantly radioactive after the treatment.
Another option is temporary brachytherapy, e.g., iridium-192 needles that deliver higher radiation doses are inserted to treat prostate cancer and removed after 5 to 10 min.  The iridium-192 wires are inserted through a catheter implanted in the space from where the tumor was removed. The wires are removed after delivering the required radiation dose. The process is repeated twice a day for five days. The catheter is removed, and no radioactive material is left in the body after the treatment.


7.14: Inductance of a Coaxial Structure
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Let us now determine the inductance of coaxial structure, shown in Figure \(\PageIndex{1}\). The inductance of this structure is of interest for a number of reasons – in particular, for determining the characteristic impedance of coaxial transmission line, as addressed in Section 3.10.
Figure \(\PageIndex{1}\): Determining the inductance of coaxial line.For our present purpose, we may model the structure as shown in Figure \(\PageIndex{1}\). This model consists of two concentric perfectly-conducting cylinders of radii \(a\) and \(b\), separated by a homogeneous material having permeability \(\mu\). To facilitate analysis, let us place the \(+z\) axis along the common axis of the concentric cylinders, so that the cylinders may be described as the surfaces \(\rho=a\) and \(\rho=b\).
Below we shall find the inductance by assuming a current \(I\) on the inner conductor and integrating over the resulting magnetic field to obtain the magnetic flux \(\Phi\) between the conductors. Then, inductance can be determined as the ratio of the response flux to the source current.
Before we get started, note the derivation we are about to do is similar to the derivation of the capacitance of a coaxial structure, addressed in Section 5.24. The reader may benefit from a review of that section before attempting this derivation.
The first step is to find the magnetic field inside the structure. This is relatively simple if we may neglect fringing fields, since then the internal field may be assumed to be constant with respect to \(z\). This analysis will also apply to the case where the length \(l\) pertains to one short section of a much longer structure; in this case we will obtain the inductance per length as opposed to the total inductance for the structure. Note that the latter is exactly what we need for the transmission line lumped-element equivalent circuit model (Section 3.4).
To determine the inductance, we invoke the definition: \[L \triangleq \frac{\Phi}{I} \label{m0125_eIndDef} \] A current \(I\) flowing in the \(+z\) direction on the inner conductor gives rise to a magnetic field inside the coaxial structure. The magnetic field intensity for this scenario was determined in Section 7.5 where we found \[{\bf H} = \hat{\phi} \frac{I}{2\pi \rho} ~~, ~~ a \le\rho\le b \nonumber \] The reader should note that in that section we were considering merely a line of current; not a coaxial structure. So, on what basis do we claim the field for inside the coaxial structure is the same? This is a consequence of Ampere’s Law (Section 7.14): \[\oint_{\mathcal C} {\bf H}\cdot d{\bf l} = I_{encl} \nonumber \] If in this new problem we specify the same circular path \(\mathcal{C}\) with radius greater than \(a\) and less than \(b\), then the enclosed current is simply \(I\). The presence of the outer conductor does not change the radial symmetry of the problem, and nothing else remains that can change the outcome. This is worth noting for future reference:
The magnetic field inside a coaxial structure comprised of concentric conductors bearing current \(I\) is identical to the magnetic field of the line current \(I\) in free space.
We’re going to need magnetic flux density (\({\bf B}\)) as opposed to \({\bf H}\) in order to get the magnetic flux. This is simple since they are related by the permeability of the medium; i.e., \({\bf B}=\mu{\bf H}\). Thus: \[{\bf B} = \hat{\phi} \frac{\mu I}{2\pi \rho} ~~, ~~ a \le\rho\le b \nonumber \]
Next, we get \(\Phi\) by integrating over the magnetic flux density \[\Phi = \int_{\mathcal S}{ {\bf B} \cdot d{\bf s} } \nonumber \] where \(\mathcal{S}\) is any open surface through which all magnetic field lines within the structure must pass. Since this can be any such surface, we may as well choose the simplest one. The simplest such surface is a plane of constant \(\phi\), since such a plane is a constant-coordinate surface and perpendicular to the magnetic field lines. This surface is shown as the shaded area in Figure \(\PageIndex{1}\). Using this surface we find:
\begin{aligned}
\Phi &=\int_{\rho=a}^{b} \int_{z=0}^{l}\left(\hat{\phi} \frac{\mu I}{2 \pi \rho}\right) \cdot(\hat{\phi} d \rho d z) \\
&=\frac{\mu I}{2 \pi}\left(\int_{z=0}^{l} d z\right)\left(\int_{\rho=a}^{b} \frac{d \rho}{\rho}\right) \\
&=\frac{\mu I l}{2 \pi} \ln \left(\frac{b}{a}\right)
\end{aligned}
Wrapping up: \[L \triangleq \frac{\Phi}{I} = \frac{\left(\mu Il/2\pi\right)\ln\left(b/a\right) }{ I } \nonumber \] Note that factors of \(I\) in the numerator and denominator cancel out, leaving: \[\boxed{ L = \frac{\mu l}{2\pi} \ln\left(\frac{b}{a}\right) } \nonumber \] Note that this is dimensionally correct, having units of H. Also note that this is expression depends only on materials (through \(\mu\)) and geometry (through \(l\), \(a\), and \(b\)). Notably, it does not depend on current, which would imply non-linear behavior.
To make the connection back to lumped-element transmission line model parameters (Sections 3.4 and 3.10), we simply divide by \(l\) to get the per-unit length parameter: \[\boxed{ L' = \frac{\mu}{2\pi} \ln\left(\frac{b}{a}\right) } \label{m0125_eLp} \] which has the expected units of H/m.
Example \(\PageIndex{1}\): Inductance of RG-59 coaxial cable
RG-59 coaxial cable consists of an inner conductor having radius \(0.292\) mm, an outer conductor having radius \(1.855\) mm, and polyethylene (a non-magnetic dielectric) spacing material. Estimate the inductance per length of RG-59.
Solution
From the problem statement, \(a=0.292\) mm, \(b=1.855\) mm, and \(\mu \cong \mu_0\) since the spacing material is non-magnetic. Using Equation \ref{m0125_eLp}, we find \(L'\cong 370\) nH/m.


1.8: Structuring the Plot
Plot is just another name for Story. Plot is the series of events that are structured to take the hero on a transformative journey throughout the story. Stories have been around for centuries and throughout that time the structure of storytelling has become solidified and with that there are essential components or rules you must follow. There are always attempts to break the form and some are done quite well, but if you just follow the structure and deliver a strong and well crafted story your audience will be receptive and happy. Each plot or story has essential components and below I will briefly outline them. You can choose to devise a cohesive story around one character’s journey or connect a series of scenes around a theme.
Opening Image
Every single story is about a transformation. Blake Snyder describes this beautifully in his book Save the Cat . He says “Every play is about the transformation of a caterpillar into a butterfly.” Every story is also a hero's journey.  Yet this hero has a goal and in order to achieve that goal they will be met with many obstacles. It is watching this character overcome the obstacles that transforms not only the hero but the audience. The opening image is where the story shows us the main character in the beginning or caterpillar stage of their journey. This is important because the opening image and final image are the things your audience will remember the most. It is the contrast of these two images that tell the story.
Setup/Prologue/Exposition
Every story has to have certain elements in the beginning in order for your audience to understand and follow the plot. In the set up you have to establish the setting or time and location, introduce the main characters, set up the rules of the world, and establish what a normal day is like in the main characters world.
Establish the Setting:
The audience needs to understand where and when the play is taking place. Remember that in a play the action and story is being conveyed through dialogue and will need to be clearly established up front. You even do this when telling a quick story to your friend: “Last I was walking into Wal-Mart and ____”
or in a joke: “2 guys walk into a bar.”
or in a Fairy Tale: “Once upon a time in a land far far away…”
In each instance, the Setting is established up front so the audience has a location in which to stage the rest of the story.
Introducing the Main Characters:
Within the first few pages of the story the audience needs to be introduced to the main characters they will be following for the journey of the play. The audience needs to know who to care about and identify with. If you want the audience to bond with your character, it helps to have the protagonist do a small act of kindness to allow us to see the characters potential as a hero. The audience does not need backstory, but they do need to be aware of the principal players in the world of the story.
If this is a love story, the audience needs to see a selfish and childish hero who avoids responsibility and only strives to have fun. We then need to see the over responsible counterpart that is overstressed, overworked, and avoids fun and risk at all cost. These characters behavior, flaws, and routines need to be established so that when they meet in the inciting incident, the audience understands the journey each character needs to undertake.
Setting Up the Rules of the World:
Every story takes place in an alternate reality and each reality has a set of rules that the characters must abide by. If your characters are in a world where ghosts or magic exists, you must establish this early so that when these elements are explored later the audience is with you. You have to clearly establish these rules early so that they can be used later in the story.
Even in situations where the story seems realistic there are rules. For example, every romance has the rule or convention that “love at first sight” exists as well as the rule/convention/concept of the existence of a “true love.” In a story the author needs the audience to root for the leads to get together and since the author is limited to a two hour narrative the easiest way to establish a strong connection is to create the existence of “Love at First Sight.” This rule is not realistic because the only human beings that truly believe in love at first sight are stalkers, but the convention or rule allows the author to establish a connection between the main characters and then use the rest of the time to focus on the plot elements that help tell the actual story.
The concept of “True Love” is also useful and equally counter to real life. The rule of true loves existence allows for the audience to understand that the main characters share a special bond and connection that is worth enduring the trials of the story. In real life, human beings have complicated relationships with everything.  Do you love your spouse or significant other? Do you love your children? Do you love food? Now do you love all of those things the same way? NO!! That is both disgusting and illegal, but you cannot say that your love for any of those things is less ‘true’ than the other. Nevertheless the concept of a true love allows the author to establish a worthy connection between the characters and convinces the audience to root for the characters to get together.
Establishing Normal:
For the purpose of telling your story, you need to establish what the rules of the imaginary world you are creating are and then show us what a normal and average day in this world, with these rules, looks like. It is very important to do this because when you introduce the conflict it will disrupt the normal flow of the world and both the characters and the audience will see it. It is also important to note that the world of the exposition is not maintainable and if life were to continue this way for the character the characters would either die or live wasted and meaningless lives. Even if their life is filled with happiness, it is unfulfilled in ways they are not yet aware of and that is why the story is worth telling.
Catalyst/Inciting Incident
The inciting incident or catalyst is the thing that happens that causes everything else to happen. This is the event that will introduce the conflict of the story. This can be a character being introduced, an event occuring, or something as small as a letter being dropped off. This is the plot device that disrupts the normal life you have established and sets the plot in motion. Ultimately, this is where the problem is introduced that you are going to spend the rest of the story solving. While there are many points in the rising action and prologue/exposition, there is only one inciting incident.
Debate
Once the Catalyst or Inciting Incident has occurred, the Hero contemplates taking the journey to confront the problem. Most human beings never feel ready to take on a difficult task, let alone a life defining journey. This plot point is really important to have because it highlights a truly human characteristic. Every great hero does not go willingly at first to their destiny, they doubt themselves. This contemplation is usually a discussion with a friend or mentor, but can also be a soliloquy.
Decision
The Hero does not always make the decision to go on a journey to confront the problem. This is the plot point where the decision is made. The Hero either decides to confront the problem willingly or the villian/circumstances make a decision that forces the hero into action.
Fun and Games/New World
The main character begins the journey to confront the problem and this forces the character to enter a new world. The new world is the antithesis of the previous world and all of the rules from the previous world are broken. The section is called ‘fun and games’ by some writers but this does not necessarily mean that the world the character is entering is fun. The fun comes from the exploration of the world. Your main character is clueless to the rules of this world, as is your audience, and both will discover the world together. If for example the world of the exposition is safe and predictable, the hero must enter a world that is dangerous and unpredictable. This section is rather long and allows for new characters to be introduced. This is also why your hero is clueless at the beginning of the story, because as you introduce the new world to them the audience is equally clueless and you can hide any further exposition in dialogue or actions by the hero asking questions or discovering things on their own.
Midpoint
The midpoint of the play is where the main character has adequately explored the new world and has unlocked a piece of hidden potential. This event usually manifests itself into a ‘false victory’ or ‘false defeat.’ This means that the character has a minor confrontation with another character and through the confrontation learns a new skill. This confrontation can either be a victory or defeat, but it is labeled as ‘false’ because there is much more to learn and the final confrontation is yet to come. Immediately following the midpoint is the intermission.
Bad Guys Close In/Tensions Rise
Following the victory or defeat at the midpoint, the Antagonist becomes aware of the hero and adds pressure. If the hero is part of a team, the team begins to struggle and fall apart. Relationships that were strong in Act 1 now begin to fracture. In addition the hero begins to doubt their abilities, the process, and begins losing faith in themselves and their team.
All is Lost
This is a major defeat at the hands of the antagonist causing the team to completely fracture and in the process destroys the heroes confidence. This can be the perceived loss of the goal, a character dying, something being lost, etc. In life, each person is going to be hit with events that crush any semblance of what we thought we were. This plot point illustrates that in story form and has characters struggle with an equal loss. Due to this loss, all progress made feels lost and the final problem appears insurmountable.
Dark Night of the Soul
This is the part of the story where the Hero contemplates giving up and in some cases even ending their life. As the Hero decides whether or not to continue, he or she is offered guidance by a sidekick or mentor and through this conversation the hero finally understands the deeper meaning of the theme. At the end of the scene, the hero decides to meet the antagonist in a final confrontation. The parallel in life is that only once we hit rock bottom can we begin anew and if you build on the lessons you have learned along your journey you can become the unstoppable force you were looking for all along.
Finale
The finale is the strongest point of conflict where the hero and antagonist collide. The more the hero has struggled throughout the journey, the more the audience has anticipated this confrontation, so make sure to capitalize on this by increasing the heroes struggles leading up to this point. At this point, the protagonist and antagonist are equally matched and now the hero is armed with experience and an unwavering understanding of the theme of the story. The protagonist and antagonist collide and the finale should be designed to test both characters to their breaking points.
Climax
The climax is the highest point of tension. Tension is created when we are anticipating an event but it has yet to occur. This is the moment where you extend the anticipation of victory or defeat, because once the event occurs it becomes part of the resolution and the audience will begin the process of disengaging from the performance and reconnecting with their lives.
Denouement / Falling Action / Resolution
Denouement means to unravel or unknot. The plot has been building tension and tying knots in the audiences stomachs and denouement is where the story receives closure and the knots are untied. In the Denouement, the conflict is resolved,the hero is either victorious or fails, life returns to as normal as possible for the characters following the events of the plot, and the audience begins to internalize the themes and knowledge gained from the story. This should not take too long, and do get ‘precious’ or sentimental with the subject matter.
Closing Image
This is our final moment or image of the character as the transformed butterfly. This image should be powerful and will be the image your audience will carry with them out of the theater.
The Process
As a group, select an inciting incident then select a climax. Once you know what starts your conflict and what you are building to, the rest of your process is creating events that logically build tension and take you from point A to point B in a clear progression. Have each member of the group present the ideas and then go through the voting process to narrow choices.
You can also break up the play into a series of scenes that all relate to a common theme and then structure them in a way that they follow a narrative arc.


22.10A: Anatomy of the Small Intestine
The small intestine is the part of the gastrointestinal tract where much of the digestion and absorption of food takes place.
- Diagram the anatomy of the small intestine
Key Points
- The small intestine is the part of the gastrointestinal tract that follows the stomach, which is in turn followed by the large intestine.
- The average length of the small intestine in an adult human male is 6.9 m (22 feet, 6 inches), and in the adult female 7.1 m (23 feet, 4 inches).
- The small intestine is divided into the duodenum, jejunum, and ileum.
- Much of the small intestine is covered in projections called villi that increase the surface area of the tissue available to absorb nutrients from the gut contents.
Key Terms
- duodenum : The first part of the small intestine that starts at the lower end of the stomach and extending to the jejunum.
- ileum : The last, and usually the longest, division of the small intestine; the part between the jejunum and large intestine.
- small intestine : The upper part of the intestine, between the stomach and the large intestine, that is divided into the duodenum, the jejunum, and the ileum.
- jejunum : The central of the three divisions of the small intestine that lies between the duodenum and the ileum.
The Small Intestine
The small intestine is the part of the gastrointestinal tract that follows the stomach, which is in turn followed by the large intestine. The small intestine is the site where almost all of the digestion and absorption of nutrients and minerals from food takes place.
Small intestine : An illustration of the small intestine with the duodenum, jejunum, and ileum labeled.
The average length of the small intestine in an adult human male is 6.9 m (22 feet, 6 inches), and in the adult female 7.1 m (23 feet, 4 inches). It can vary greatly, from as short as 4.6 m (15 feet) to as long as 9.8 m (32 feet). The small intestine is approximately 2.5–3 cm in diameter, and is divided into three sections:
- The duodenum is the first section of the small intestine and is the shortest part of the small intestine. It is where most chemical digestion using enzymes takes place.
- The jejunum is the middle section of the small intestine. It has a lining which is designed to absorb carbohydrates and proteins. The inner surface of the jejunum, its mucous membrane, is covered in projections called villi, which increase the surface area of tissue available to absorb nutrients from the gut contents. The epithelial cells which line these villi possess even larger numbers of microvilli. The transport of nutrients across epithelial cells through the jejunum includes the passive transport of some carbohydrates and the active transport of amino acids, small peptides, vitamins, and most glucose. The villi in the jejunum are much longer than in the duodenum or ileum.
- The ileum is the final section of the small intestine. The function of the ileum is mainly to absorb vitamin B12, bile salts, and any products of digestion that were not absorbed by the jejunum. The wall itself is made up of folds, each of which has many tiny finger-like projections known as villi on its surface. The ileum has an extremely large surface area both for the adsorption of enzyme molecules and for the absorption of products of digestion.
The Villi
The villi contain large numbers of capillaries that take the amino acids and glucose produced by digestion to the hepatic portal vein and the liver. Lacteals are the small lymph vessels that are present in villi. They absorb fatty acids and glycerol, the products of fat digestion, into direct circulation.
Layers of circular and longitudinal smooth muscle enable the digested food to be pushed along the ileum by waves of muscle contractions called peristalsis. The undigested food (waste and water) are sent to the colon.


8.1: Police Subculture
At the root of all that is good and bad in law enforcement, there is a strong subculture that permeates most agencies. While a common theme in academic discourse is that police culture is negative, entrenched in cynicism, masochism, loyalty above all else, and an “us versus them” mentality, it has positive aspects that are often overlooked. Members of the law enforcement subculture share values that enable officers to survive what at times is a difficult and emotionally taxing job. Values such as supportiveness, teamwork, perseverance, empathy, and caring enable officers to cope with post-traumatic stress; they are part of team of colleagues who care for their coworkers. The support received from other officers is the result of shared values within the culture. Officers who are faced with dangerous situations are able to rely on their comrades because of other values they believe these members also possess. Values such as bravery, camaraderie, and sacrifice will embolden members to place themselves in harm’s way.
The following table outlines both positive and negative attributes within the police culture.
| Positive attributes | Negative attributes |
|---|---|
| Safety | Cynicism |
| Camaraderie | Close-mindedness |
| Empathy | Biases |
| Support | Prejudice |
| Caring | Non-scientific tactics |
| Teamwork | Overly conservative |
| Loyalty | Loyalty |
| Sacrifice | Alienated |
| Suspicion | |
| Authoritarianism |
In spite of the positive aspects of police subculture, what society may define as ethical or good conduct may not be viewed within the subculture as relevant to the task, which is, among other things, to continue the mission of “safe-guarding social order” (Reiner, 2010, p.120). The tactics that are relevant to the police subculture may include using trickery and lies to elicit confessions and receiving minor gratuities to foster community relations (Reiner, 2010). Examining ethics and its relation to the police subculture is important to help delineate not only the grey area of ethics but also the grey area within which the police operate.
Once selected and hired by municipal police agencies, police recruits are exposed to police subculture during their training partially due to the instruction they receive from police officers who are recently retired or seconded to the police academy. However, the choice to become a police officer is not made in a vacuum. When recruits start their training, they often think like police officers on a visceral level, because generally certain individuals are drawn to the occupation (Conti, 2010). In an ethnographic study observing police recruits at an American police academy, Conti (2010) observed that the evolution of recruits into members who reflect the police mindset likely started at an early age when they formed the belief that they would become police officers. As potential officers enter the selection process, they become involved in an extensive application process, which is their first introduction into the police subculture. Rokeach, Miller, and Snyder (1971) concluded that a police personality distinct from others does exist, and proposed the idea that individuals come into an occupation with predetermined attributes that are identified with their new occupation. However, Rokeach et al. (1971) also found that this distinct police personality is attributed to predispositions of personality that are present before the recruits’ induction into the police subculture. These distinct predispositions are conducive to a career in policing and allow the individuals to comfortably choose and fit into the subculture (Conti, 2010; Rokeach et al., 1971). While the police subculture is distinct, at times it does attempt to catch up to the norms of the mainstream culture and can shift from negative attributes to positive attributes (Skolnick, 2008).
A historical look at the police subculture offers a view into the changing nature of how police officers see the world. In analyzing the police subculture in the 1940s, Myrdal (1964) observed in an ethnographic study of police officers in America that officers behaved in an overtly bigoted fashion toward African Americans. Myrdal (1964) observed that these were the norms of the day and that the police subculture reflected the attitude of mainstream society toward African Americans. While not supported empirically, it would be a logical conclusion that police recruits or rookie police officers would have shared the same cultural bigotry as mainstream society and their fellow police officers. More recently, when we see and question incidents involving police use of force on racial minorities, it is important to look broadly at society as well. The shooting of Michael Brown in Ferguson, Missouri, is an example where prominent civic leaders pointed out that the incident was merely a manifestation of a broader issue of racism that is widespread throughout the United States.
As society has evolved so too have law enforcement agencies. Ethical conduct and diversity play a large role in recruiting and are considered important attributes of potential officers. Crank and Caldero (2010) have concluded that due to society’s emphasis on ethics and the stringent hiring process, recruits are typically very ethical. The subculture, they argue, is not only present but also highly influential; the recruits’ ethical orientations are formed earlier, well before their application process commences (Crank and Caldero, 2010). Conversely, Conti (2010) and Banish and Ruiz (2003) argue that the police subculture is present when the officers start at the police academy and that its influence on recruits’ ethics is negative and destructive.
Conti views a recruit’s induction into the police academy as a transformation of the recruit into the “organisational ideal” (Conti, 2010). It is in this way, Conti (2010) argues, that the police subculture, ever-present at the police academy, assists in the conversion of the recruit from civilian to police officer. These cultural nuances are passed on through a variety of means such as:
- Parades and drills (Campbell, 2007)
- Marching (Davis, 1996)
- Storytelling (Banish and Ruiz, 2003; Ford, 2003; Newburn and Reiner, 2007).
Storytelling by instructors in the police academy can be a valuable and effective teaching tool, as demonstrated by Conti’s (2010) study of an American police academy. Stories told by trainers must reflect ethical conduct and be relatable to the lesson plan goals and outcomes. Conversely, stories by instructors may inflate the recruits’ perception of danger (Banish and Ruiz, 2003) or cynicism (Ford, 2003), but instructors’ stories can also serve to relay positive outcomes, such as surviving life and death situations confronted by police (Conti, 2010). Ultimately, storytelling perpetuates the police subculture by passing on both truisms as well as not-so-true legends (Newburn and Reiner, 2007). Banish and Ruiz (2003) further contend that storytelling affects the police culture negatively by instilling negative traits of cynicism, suspicion, conservatism, and authoritarianism.
These negative traits are often associated with a police subculture that affects senior police officers, and it is specifically these traits that define an individual as a police officer. Skolnick (2008) considers the police vocation as being similar to that of a priest or the clergy: the culture wholly defines what it means to be a police officer by the traits that police officers share. These traits, according to Skolnick, include “skepticism, cynicism, mistrust of outsiders—all are traits observers of police apply to them and that they apply to themselves” (2008, p.36). Twersky-Glasner (2005) concurs, noting that the police are members of a unique occupation in which they are the insiders and the rest of society are the outsiders. The insiders are those who are trustworthy while outsiders are viewed with suspicion (Skolnick, 2008). This is reflected not only in the culture in which recruits find themselves, but also in the training they receive and the way in which they as civilians are accepted into the academy to begin training.
In a qualitative study of police officers, Loftus (2010) followed officers on the street and determined that two characteristics are ever-present in the police culture: cynicism and moral conservatism. While older officers exhibit these traits, Loftus (2010) did observe that newer officers are hired from a more diverse background that includes different sexual orientations, cultures, and races. This may enable the police subculture to adapt and overcome its more negative characteristics.


5.4: Exercises
Write the following permutations in cycle notation.
-
\[ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 4 & 1 & 5 & 3 \end{pmatrix} \nonumber \]
-
\[ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 5 & 1 & 3 \end{pmatrix} \nonumber \]
-
\[ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 5 & 1 & 4 & 2 \end{pmatrix} \nonumber \]
-
\[ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 3 & 2 & 5 \end{pmatrix} \nonumber \]
Compute each of the following.
- \(\displaystyle (1345)(234)\)
- \(\displaystyle (12)(1253)\)
- \(\displaystyle (143)(23)(24)\)
- \(\displaystyle (1423)(34)(56)(1324)\)
- \(\displaystyle (1254)(13)(25)\)
- \(\displaystyle (1254) (13)(25)^2\)
- \(\displaystyle (1254)^{-1} (123)(45) (1254)\)
- \(\displaystyle (1254)^2 (123)(45)\)
- \(\displaystyle (123)(45) (1254)^{-2}\)
- \(\displaystyle (1254)^{100}\)
- \(\displaystyle |(1254)|\)
- \(\displaystyle |(1254)^2|\)
- \(\displaystyle (12)^{-1}\)
- \(\displaystyle (12537)^{-1}\)
- \(\displaystyle [(12)(34)(12)(47)]^{-1}\)
- \(\displaystyle [(1235)(467)]^{-1}\)
Express the following permutations as products of transpositions and identify them as even or odd.
- \(\displaystyle (14356)\)
- \(\displaystyle (156)(234)\)
- \(\displaystyle (1426)(142)\)
- \(\displaystyle (17254)(1423)(154632)\)
- \(\displaystyle (142637)\)
Find \((a_1, a_2, \ldots, a_n)^{-1}\text{.}\)
List all of the subgroups of \(S_4\text{.}\) Find each of the following sets:
- \(\displaystyle \{ \sigma \in S_4 : \sigma(1) = 3 \}\)
- \(\displaystyle \{ \sigma \in S_4 : \sigma(2) = 2 \}\)
- \(\{ \sigma \in S_4 : \sigma(1) = 3\) and \(\sigma(2) = 2 \}\text{.}\)
Are any of these sets subgroups of \(S_4\text{?}\)
Find all of the subgroups in \(A_4\text{.}\) What is the order of each subgroup?
Find all possible orders of elements in \(S_7\) and \(A_7\text{.}\)
Show that \(A_{10}\) contains an element of order \(15\text{.}\)
Does \(A_8\) contain an element of order \(26\text{?}\)
Find an element of largest order in \(S_n\) for \(n = 3, \ldots, 10\text{.}\)
What are the possible cycle structures of elements of \(A_5\text{?}\) What about \(A_6\text{?}\)
Let \(\sigma \in S_n\) have order \(n\text{.}\) Show that for all integers \(i\) and \(j\text{,}\) \(\sigma^i = \sigma^j\) if and only if \(i \equiv j \pmod{n}\text{.}\)
Let \(\sigma = \sigma_1 \cdots \sigma_m \in S_n\) be the product of disjoint cycles. Prove that the order of \(\sigma\) is the least common multiple of the lengths of the cycles \(\sigma_1, \ldots, \sigma_m\text{.}\)
Using cycle notation, list the elements in \(D_5\text{.}\) What are \(r\) and \(s\text{?}\) Write every element as a product of \(r\) and \(s\text{.}\)
If the diagonals of a cube are labeled as Figure \(5.28\), to which motion of the cube does the permutation \((12)(34)\) correspond? What about the other permutations of the diagonals?
Find the group of rigid motions of a tetrahedron. Show that this is the same group as \(A_4\text{.}\)
Prove that \(S_n\) is nonabelian for \(n \geq 3\text{.}\)
Show that \(A_n\) is nonabelian for \(n \geq 4\text{.}\)
Prove that \(D_n\) is nonabelian for \(n \geq 3\text{.}\)
Let \(\sigma \in S_n\) be a cycle. Prove that \(\sigma\) can be written as the product of at most \(n-1\) transpositions.
Let \(\sigma \in S_n\text{.}\) If \(\sigma\) is not a cycle, prove that \(\sigma\) can be written as the product of at most \(n - 2\) transpositions.
If \(\sigma\) can be expressed as an odd number of transpositions, show that any other product of transpositions equaling \(\sigma\) must also be odd.
If \(\sigma\) is a cycle of odd length, prove that \(\sigma^2\) is also a cycle.
Show that a \(3\)-cycle is an even permutation.
Prove that in \(A_n\) with \(n \geq 3\text{,}\) any permutation is a product of cycles of length \(3\text{.}\)
Prove that any element in \(S_n\) can be written as a finite product of the following permutations.
- \(\displaystyle (1 2), (13), \ldots, (1n)\)
- \(\displaystyle (1 2), (23), \ldots, (n- 1,n)\)
- \(\displaystyle (12), (1 2 \ldots n )\)
Let \(G\) be a group and define a map \(\lambda_g : G \rightarrow G\) by \(\lambda_g(a) = g a\text{.}\) Prove that \(\lambda_g\) is a permutation of \(G\text{.}\)
Prove that there exist \(n!\) permutations of a set containing \(n\) elements.
Recall that the center of a group \(G\) is
\[ Z(G) = \{ g \in G : gx = xg \text{ for all } x \in G \}\text{.} \nonumber \]
Find the center of \(D_8\text{.}\) What about the center of \(D_{10}\text{?}\) What is the center of \(D_n\text{?}\)
Let \(\tau = (a_1, a_2, \ldots, a_k)\) be a cycle of length \(k\text{.}\)
-
Prove that if \(\sigma\) is any permutation, then
\[ \sigma \tau \sigma^{-1 } = ( \sigma(a_1), \sigma(a_2), \ldots, \sigma(a_k)) \nonumber \]
is a cycle of length \(k\text{.}\)
- Let \(\mu\) be a cycle of length \(k\text{.}\) Prove that there is a permutation \(\sigma\) such that \(\sigma \tau \sigma^{-1 } = \mu\text{.}\)
For \(\alpha\) and \(\beta\) in \(S_n\text{,}\) define \(\alpha \sim \beta\) if there exists an \(\sigma \in S_n\) such that \(\sigma \alpha \sigma^{-1} = \beta\text{.}\) Show that \(\sim\) is an equivalence relation on \(S_n\text{.}\)
Let \(\sigma \in S_X\text{.}\) If \(\sigma^n(x) = y\) for some \(n \in \mathbb Z\text{,}\) we will say that \(x \sim y\text{.}\)
- Show that \(\sim\) is an equivalence relation on \(X\text{.}\)
-
Define the
orbit
of \(x \in X\) under \(\sigma \in S_X\) to be the set
\[ {\mathcal O}_{x, \sigma} = \{ y : x \sim y \}\text{.} \nonumber \]
Compute the orbits of each element in \(\{1, 2, 3, 4, 5\}\) under each of the following elements in \(S_5\text{:}\)
\begin{align*} \alpha & = (1254)\\ \beta & = (123)(45)\\ \gamma & = (13)(25)\text{.} \end{align*}
- If \({\mathcal O}_{x, \sigma} \cap {\mathcal O}_{y, \sigma} \neq \emptyset\text{,}\) prove that \({\mathcal O}_{x, \sigma} = {\mathcal O}_{y, \sigma}\text{.}\) The orbits under a permutation \(\sigma\) are the equivalence classes corresponding to the equivalence relation \(\sim\text{.}\)
- A subgroup \(H\) of \(S_X\) is transitive if for every \(x, y \in X\text{,}\) there exists a \(\sigma \in H\) such that \(\sigma(x) = y\text{.}\) Prove that \(\langle \sigma \rangle\) is transitive if and only if \({\mathcal O}_{x, \sigma} = X\) for some \(x \in X\text{.}\)
Let \(\alpha \in S_n\) for \(n \geq 3\text{.}\) If \(\alpha \beta = \beta \alpha\) for all \(\beta \in S_n\text{,}\) prove that \(\alpha\) must be the identity permutation; hence, the center of \(S_n\) is the trivial subgroup.
If \(\alpha\) is even, prove that \(\alpha^{-1}\) is also even. Does a corresponding result hold if \(\alpha\) is odd?
If \(\sigma \in A_n\) and \(\tau \in S_n\text{,}\) show that \(\tau^{-1} \sigma \tau \in A_n\text{.}\)
Show that \(\alpha^{-1} \beta^{-1} \alpha \beta\) is even for \(\alpha, \beta \in S_n\text{.}\)
Let \(r\) and \(s\) be the elements in \(D_n\) described in Theorem \(5.21\)
- Show that \(srs = r^{-1}\text{.}\)
- Show that \(r^k s = s r^{-k}\) in \(D_n\text{.}\)
- Prove that the order of \(r^k \in D_n\) is \(n / \gcd(k,n)\text{.}\)


4.4: Columns
Learning Outcomes
- Create columns.
Columns can help you fit more text on a page or provide better readability. Some documents, such as a newsletter or newspaper, are customarily formatted with columns. Columns are typically used for shorter pieces of text (like dictionary definitions or short articles) rather than longer readings.
To add columns, either select the text you want to arrange into columns or place your cursor where you want columns to start. Then, go to Layout>Columns and select the number of columns. (For more than three columns, select More Columns… at the bottom of the menu.)
You can also end a column with a column break. Place the cursor where you want to end a column, then select Layout>Breaks>Column Break.
Adding a column break will end the column and send text to the next column. Note that a column break is tied to the location within the text, not the location on the page. Putting a column break before the Mobile Commerce headline will put a column break before the Mobile Commerce headline, regardless of how much text comes before or after the Mobile Commerce headline.
Contributors and Attributions
- Columns. Authored by : Lumen Learning. License : CC BY: Attribution
- Letter R dictionary. Authored by : Gabrielle_RRI. Located at : pixabay.com/en/letter-r-dictionary-bezel-2351442/. License : CC0: No Rights Reserved


4.E: Line and Surface Integrals (Exercises)
4.1: Line Integrals
A
For Exercises 1-4, calculate \(\int_C f (x, y)\,ds\) for the given function \(f (x, y)\) and curve \(C\).
4.1.1. \(f (x, y) = x y;\quad C : x = \cos t, y = \sin t, 0 ≤ t ≤ π/2\)
4.1.2. \(f (x, y) = \frac{x}{ x^ 2} +1 ;\quad C : x = t, y = 0, 0 ≤ t ≤ 1\)
4.1.3. \(f (x, y) = 2x+ y;\quad C:\text{ polygonal path from }(0,0) \text{ to }(3,0)\text{ to }(3,2)\)
4.1.4. \(f (x, y) = x + y^2 ;\quad C: \text{ path from }(2,0)\text{ counterclockwise along the circle }x^ 2 + y^ 2 = 4 \text{ to the point }(−2,0)\text{ and then back to }(2,0)\text{ along the }x\text{-axis}\)
4.1.5. Use a line integral to find the lateral surface area of the part of the cylinder \(x^ 2 + y^ 2 = 4\) below the plane \(x+2y+ z = 6\) and above the \(x y\)-plane.
For Exercises 6-11, calculate \(\int_C \textbf{f}· d\textbf{r}\) for the given vector field \(\textbf{f}(x, y)\) and curve \(C\).
4.1.6. \(\textbf{f}(x, y) = \textbf{i}−\textbf{j};\quad C : x = 3t, y = 2t, 0 ≤ t ≤ 1\)
4.1.7. \(\textbf{f}(x, y) = y\textbf{i}− x\textbf{j};\quad C : x = \cos t, y = \sin t, 0 ≤ t ≤ 2π\)
4.1.8. \(\textbf{f}(x, y) = x\textbf{i}+ y\textbf{j};\quad C : x = \cos t, y = \sin t, 0 ≤ t ≤ 2π\)
4.1.9. \(\textbf{f}(x, y) = (x^ 2 − y)\textbf{i}+(x− y^ 2 )\textbf{j};\quad C : x = \cos t, y = \sin t, 0 ≤ t ≤ 2π\)
4.1.10. \(\textbf{f}(x, y) = x y^2 \textbf{i}+ x y^3 \textbf{j};\quad C :\text{ the polygonal path from }(0,0)\text{ to }(1,0)\text{ to }(0,1)\text{ to }(0,0)\)
4.1.11. \(\textbf{f}(x, y) = (x^ 2 + y^ 2 )\textbf{i};\quad C : x = 2+\cos t, y = \sin t, 0 ≤ t ≤ 2π\)
B
4.1.12. Verify that the value of the line integral in Example 4.1 is unchanged when using the parametrization of the circle \(C\) given in formulas Equation (4.8).
4.1.13. Show that if \(\textbf{f} ⊥ \textbf{r} ′ (t)\) at each point \(\textbf{r}(t)\) along a smooth curve \(C\), then \(\int_C \textbf{f}· d\textbf{r} = 0\).
4.1.14. Show that if \(\textbf{f}\) points in the same direction as \(\textbf{r} ′ (t)\) at each point \(\textbf{r}(t)\) along a smooth curve \(C\), then \(\int_C \textbf{f}· d\textbf{r} = \int_C \norm{ \textbf{f}}\,ds\).
c
4.1.15. Prove that \(\int_C f (x, y)\,ds = \int_{−C} f (x, y)\,ds\). ( Hint: Use formulas Equation (4.9) .)
4.1.16 . Let \(C\) be a smooth curve with arc length \(L\), and suppose that \(\textbf{f}(x, y) = P(x, y)\textbf{i}+Q(x, y)\textbf{j}\) is a vector field such that \(\norm{\textbf{f}(x, y)} ≤ M\) for all \((x, y) \text{ on }C\). Show that \(\left | \int_C \textbf{f}· d\textbf{r} \right | ≤ ML\). ( Hint: Recall that \(\left | \int_a^b g(x)\,dx \right | ≤ \int_a^b | g(x) |\,dx\) for Riemann integrals. )
4.1.17. Prove that the Riemann integral \(\int_a^b f (x)\,dx\) is a special case of a line integral.
4.2: Properties of Line Integrals
A
4.2.1. Evaluate \(\oint_C (x^ 2 + y^ 2 )\,dx+2x y\, d y\) for \(C : x = \cos t, y = \sin t, 0 ≤ t ≤ 2π.\)
4.2.2. Evaluate \(\int_C (x^2 + y^ 2 )\,dx+2x y\, d y\) for \(C : x = \cos t, y = \sin t, 0 ≤ t ≤ π\)
4.2.3. Is there a potential \(F(x, y) \text{ for }\textbf{f}(x, y) = y\textbf{i}− x\textbf{j}\)? If so, find one.
4.2.4. Is there a potential \(F(x, y) \text{ for }\textbf{f}(x, y) = x\textbf{i}− y\textbf{j}\)? If so, find one.
4.2.5. Is there a potential \(F(x, y) \text{ for }\textbf{f}(x, y) = x y^2 \textbf{i}+ x^ 3 y\textbf{j}\)? If so, find one.
B
4.2.6. Let \(\textbf{f}(x, y) \text{ and }\textbf{g}(x, y)\) be vector fields, let \(a \text{ and }b\) be constants, and let \(C\) be a curve in \(\mathbb{R}^ 2\). Show that
\[\int_C (a\textbf{f}± b\textbf{g})· d\textbf{r} = a \int_C \textbf{f}· d\textbf{r} ± b \int_C \textbf{g}· d\textbf{r} .\]
4.2.7.  Show that \(\int_C 1\,ds = L\).
4.2.8. Let \(f (x, y) \text{ and }g(x, y)\) be continuously differentiable real-valued functions in a region \(R\). Show that
\[\oint_C f ∇g · d\textbf{r} = −\oint_C g∇f · d\textbf{r}\]
for any closed curve \(C\) in \(R\). ( Hint: Use Exercise 21 in Section 2.4. )
4.2.9. Let \(\textbf{f}(x, y) = \frac{−y}{ x^ 2+y^ 2} \textbf{i}+ \frac{x}{ x^ 2+y^ 2} \textbf{j}\) for all \((x, y) \neq (0,0)\), and \(C : x = \cos t, y = \sin t, 0 ≤ t ≤ 2π.\)
(a) Show that \(\textbf{f} = ∇F,\text{ for }F(x, y) = \tan^{−1} (y/x)\).
(b) Show that \(\oint_C \textbf{f}· d\textbf{r} = 2π\). Does this contradict Corollary 4.6? Explain.
C
4.2.10. Let \(g(x) \text{ and }h(y)\) be differentiable functions, and let \(\textbf{f}(x, y) = h(y)\textbf{i}+ g(x)\textbf{j}\). Can f have a potential \(F(x, y)\)? If so, find it. You may assume that \(F\) would be smooth. ( Hint: Consider the mixed partial derivatives of \(F\).)
4.3: Green’s Theorem
A
For Exercises 1-4, use Green’s Theorem to evaluate the given line integral around the curve \(C\), traversed counterclockwise.
4.3.1. \(\oint_C (x^ 2 − y^ 2 )\,dx+2x y \,d y; C\text{ is the boundary of }R = {(x, y) : 0 ≤ x ≤ 1, 2x^ 2 ≤ y ≤ 2x}\)
4.3.2. \(\oint_C x^ 2 y \,dx+2x y \,d y; C \text{ is the boundary of }R = {(x, y) : 0 ≤ x ≤ 1, x^ 2 ≤ y ≤ x}\)
4.3.3. \(\oint_C 2y\, dx−3x\, d y; C \text{ is the circle }x^ 2 + y^ 2 = 1\)
4.3.4. \(\oint_C (e^{ x^ 2} + y^ 2 )\,dx + (e^{ y^ 2} + x^ 2 )\,d y; C \text{ is the boundary of the triangle with vertices }(0,0), (4,0) \text{ and }(0,4)\)
4.3.5. Is there a potential \(F(x, y) \text{ for }\textbf{f}(x, y) = (y^ 2 +3x^ 2 )\textbf{i}+2x y\textbf{j}\)? If so, find one.
4.3.6. Is there a potential \(F(x, y) \text{ for }\textbf{f}(x, y) = (x^ 3 \cos (x y) + 2x\sin (x y))\textbf{i} + x^ 2 y\cos (x y)\textbf{j}\)? If so, find one.
4.3.7. Is there a potential \(F(x, y) \text{ for }\textbf{f}(x, y) = (8x y+3)\textbf{i}+4(x^ 2 + y)\textbf{j}\)? If so, find one.
4.3.8. Show that for any constants \(a, b \text{ and any closed simple curve }C, \oint_C a\, dx+ b \,d y = 0\).
B
4.3.9. For the vector field f as in Example 4.8, show directly that\(\oint_C \textbf{f}· d\textbf{r} = 0\), where \(C\) is the boundary of the annulus \(R = {(x, y) : 1/4 ≤ x^ 2 + y^ 2 ≤ 1}\) traversed so that \(R\) is always on the left.
4.3.10. Evaluate \(\oint_C e^ x \sin y \,dx+(y^ 3 + e^ x \cos y)\,d y\), where \(C\) is the boundary of the rectangle with vertices \((1,−1), (1,1), (−1,1) \text{ and }(−1,−1)\), traversed counterclockwise.
C
4.3.11. For a region \(R\) bounded by a simple closed curve \(C\), show that the area \(A \text{ of }R\) is
\[A = −\oint_C y\, dx = \oint_C x \,d y = \frac{1}{ 2}\oint_C x \,d y− y \,dx ,\]
where \(C\) is traversed so that \(R\) is always on the left. ( Hint: Use Green’s Theorem and the fact that \(A = \iint\limits_R 1\,d A.\))
4.4: Surface Integrals and the Divergence Theorem
A
For Exercises 1-4, use the Divergence Theorem to evaluate the surface integral \(\iint_Σ \textbf{f}· d\textbf{σ}\) of the given vector field \(\textbf{f}(x, y, z)\) over the surface \(Σ\).
4.4.1. \(\textbf{f}(x, y, z) = x\textbf{i}+2y\textbf{j}+3z\textbf{k}, Σ : x^ 2 + y^ 2 + z^ 2 = 9\)
4.4.2. \(\textbf{f}(x, y, z) = x\textbf{i}+ y\textbf{j}+ z\textbf{k}, Σ :\text{ boundary of the solid cube }S = {(x, y, z) : 0 ≤ x, y, z ≤ 1}\)
4.4.3. \(\textbf{f}(x, y, z) = x^ 3 \textbf{i}+ y^ 3 \textbf{j}+ z^ 3\textbf{k}, Σ : x^ 2 + y^ 2 + z^ 2 = 1\)
4.4.4. \(\textbf{f}(x, y, z) = 2\textbf{i}+3\textbf{j}+5\textbf{k}, Σ : x^ 2 + y^ 2 + z^ 2 = 1\)
B
4.4.5. Show that the flux of any constant vector field through any closed surface is zero.
4.4.6. Evaluate the surface integral from Exercise 2 without using the Divergence Theorem, i.e. using only Definition 4.3, as in Example 4.10. Note that there will be a different outward unit normal vector to each of the six faces of the cube.
4.4.7. Evaluate the surface integral \(\iint_Σ \textbf{f}· d\textbf{σ}\), where \(\textbf{f}(x, y, z) = x^ 2 \textbf{i}+ x y\textbf{j}+ z\textbf{k}\text{ and }Σ\) is the part of the plane \(6x + 3y + 2z = 6 \text{ with }x ≥ 0, y ≥ 0, \text{ and }z ≥ 0\), with the outward unit normal n pointing in the positive \(z\) direction.
4.4.8. Use a surface integral to show that the surface area of a sphere of radius \(r\) is \(4πr^2\). ( Hint: Use spherical coordinates to parametrize the sphere. )
4.4.9. Use a surface integral to show that the surface area of a right circular cone of radius \(R\) and height \(h\) is \(πR \sqrt{ h^ 2 + R^2}\). ( Hint: Use the parametrization \(x = r \cos θ, y = r \sin θ, z = \frac{h}{ R} r, \text{ for }0 ≤ r ≤ R \text{ and }0 ≤ θ ≤ 2π.\))
4.4.10. The ellipsoid \(\frac{x^ 2}{ a^ 2} + \frac{y^ 2}{ b^ 2} + \frac{z^ 2}{ c^ 2} = 1\) can be parametrized using ellipsoidal coordinates
\[x = a\sin φ \cos θ ,\, y = b\sin φ \sin θ , \,z = c \cos φ ,\text{ for } 0 ≤ θ ≤ 2π \text{ and }0 ≤ φ ≤ π.\]
Show that the surface area \(S\) of the ellipsoid is
\[S = \int_0^π \int_0^2π \sin φ \sqrt{ a^ 2 b^ 2 \cos^2 φ+ c^ 2(a^ 2 \sin^2 θ + b^ 2 \cos^2 θ)\sin^2 φ}\, dθ\, dφ .\]
(Note: The above double integral can not be evaluated by elementary means. For specific values of \(a, b \text{ and }c\) it can be evaluated using numerical methods. An alternative is to express the surface area in terms of elliptic integrals . )
C
4.4.11. Use Definition 4.3 to prove that the surface area \(S\) over a region \(R\) in \(\mathbb{R}^ 2\) of a surface \(z = f (x, y)\) is given by the formula
\[S = \iint\limits_R \sqrt{ 1+ \left ( \frac{∂f}{ ∂x} \right )^2 + \left ( \frac{ ∂f}{ ∂y} \right )^2} \,d A .\]
( Hint: Think of the parametrization of the surface. )
4.5: Stokes’ Theorem
A
For Exercises 1-3, calculate \(\int_C f (x, y, z)\,ds\) for the given function \(f (x, y, z)\) and curve \(C\).
4.5.1. \(f (x, y, z) = z; \quad C : x = \cos t, y = \sin t, z = t, 0 ≤ t ≤ 2π\)
4.5.2. \(f (x, y, z) = \frac{x}{ y} + y+2yz;\quad C : x = t^ 2 , y = t, z = 1, 1 ≤ t ≤ 2\)
4.5.3. \(f (x, y, z) = z^ 2 ;\quad C : x = t\sin t, y = t \cos t, z = \frac{2 \sqrt{ 2}}{ 3} t^{ 3/2}, 0 ≤ t ≤ 1\)
For Exercises 4-9, calculate \(\int_C \textbf{f}· d\textbf{r}\) for the given vector field \(\textbf{f}(x, y, z) \text{ and curve }C\).
4.5.4. \(\textbf{f}(x, y, z) = \textbf{i}−\textbf{j}+\textbf{k};\quad C : x = 3t, y = 2t, z = t, 0 ≤ t ≤ 1\)
4.5.5. \(\textbf{f}(x, y, z) = y\textbf{i}− x\textbf{j}+ z\textbf{k};\quad C : x = \cos t, y = \sin t, z = t, 0 ≤ t ≤ 2π\)
4.5.6. \(\textbf{f}(x, y, z) = x\textbf{i}+ y\textbf{j}+ z\textbf{k};\quad C : x = \cos t, y = \sin t, z = 2, 0 ≤ t ≤ 2π\)
4.5.7. \(\textbf{f}(x, y, z) = (y−2z)\textbf{i}+ x y\textbf{j}+(2xz + y)\textbf{k};\quad C : x = t, y = 2t, z = t^ 2 −1, 0 ≤ t ≤ 1\)
4.5.8. \(\textbf{f}(x, y, z) = yz \textbf{i}+ xz \textbf{j}+ x y\textbf{k};\quad C : \text{ the polygonal path from }(0,0,0) \text{ to }(1,0,0)\text{ to }(1,2,0)\)
4.5.9. \(\textbf{f}(x, y, z) = x y\textbf{i}+(z − x)\textbf{j}+2yz\textbf{k};\quad C :\text{ the polygonal path from }(0,0,0) \text{ to }(1,0,0) \text{ to }(1,2,0) \text{ to }(1,2,−2)\)
For Exercises 10-13, state whether or not the vector field \(\textbf{f}(x, y, z)\) has a potential in \(\mathbb{R}^ 3\) (you do not need to find the potential itself).
4.5.10. \(\textbf{f}(x, y, z) = y\textbf{i}− x\textbf{j}+ z\textbf{k}\)
4.5.11. \(\textbf{f}(x, y, z) = a\textbf{i}+ b\textbf{j}+ c\textbf{k} (a, b, c \text{ constant})\)
4.5.12. \(\textbf{f}(x, y, z) = (x+ y)\textbf{i}+ x\textbf{j}+ z^ 2\textbf{k}\)
4.5.13. \(\textbf{f}(x, y, z) = x y\textbf{i}−(x− yz^2 )\textbf{j}+ y^ 2 z\textbf{k}\)
B
For Exercises 14-15, verify Stokes’ Theorem for the given vector field \(\textbf{f}(x, y, z)\) and surface \(Σ\).
4.5.14. \(\textbf{f}(x, y, z) = 2y\textbf{i}− x\textbf{j}+ z\textbf{k};\quad Σ : x ^2 + y^ 2 + z^ 2 = 1, z ≥ 0\)
4.5.15. \(\textbf{f}(x, y, z) = x y\textbf{i}+ xz \textbf{j}+ yz\textbf{k};\quad Σ : z = x^ 2 + y^ 2 , z ≤ 1\)
4.5.16. Construct a Möbius strip from a piece of paper, then draw a line down its center (like the dotted line in Figure 4.5.3(b)). Cut the Möbius strip along that center line completely around the strip. How many surfaces does this result in? How would you describe them? Are they orientable?
4.5.17. Use Gnuplot (see Appendix C) to plot the Möbius strip parametrized as:
\[\textbf{r}(u,v) = \cos u(1+ v\cos \frac{u}{ 2} )\textbf{i}+\sin u(1+ v\cos \frac{u}{ 2} )\textbf{j}+ v\sin \frac{u}{ 2} \textbf{k} ,\, 0 ≤ u ≤ 2π , -\frac{1}{ 2} ≤ v ≤ \frac{1}{ 2}\]
C
4.5.18. Let \(Σ\) be a closed surface and \(\textbf{f}(x, y, z)\) a smooth vector field. Show that \(\iint\limits_Σ (\text{curl } \textbf{f})· \textbf{n}dσ = 0\). ( Hint: Split \(Σ\) in half .)
4.5.19. Show that Green’s Theorem is a special case of Stokes’ Theorem.
4.6: Gradient, Divergence, Curl, and Laplacian
A
For Exercises 1-6, find the Laplacian of the function \(f (x, y, z)\) in Cartesian coordinates.
4.6.1. \(f (x, y, z) = x+ y+ z \)
4.6.2. \(f (x, y, z) = x^ 5\)
4.6.3. \(f (x, y, z) = (x^ 2 + y^ 2 + z^ 2 )^{ 3/2}\)
4.6.4. \(f (x, y, z) = e^{ x+y+z}\)
4.6.5. \(f (x, y, z) = x^ 3 + y^ 3 + z^ 3\)
4.6.6. \(f (x, y, z) = e^{ −x^ 2−y^ 2−z^ 2}\)
4.6.7. Find the Laplacian of the function in Exercise 3 in spherical coordinates.
4.6.8. Find the Laplacian of the function in Exercise 6 in spherical coordinates.
4.6.9. Let \(f (x, y, z) = \frac{z}{ x^ 2 + y^ 2}\) in Cartesian coordinates. Find \(∇f\) in cylindrical coordinates.
4.6.10. For \(\textbf{f}(r,θ, z) = r\textbf{e}_r + z \sin θ \textbf{e}_θ + rz\textbf{e}_z\) in cylindrical coordinates, find div f and curl f .
4.6.11. For \(\textbf{f}(ρ,θ,φ) = \textbf{e}_ρ +ρ \cos θ \textbf{e}_θ +ρ \textbf{e}_φ\) in spherical coordinates, find div f and curl f .
B
For Exercises 12-23, prove the given formula (\(r = \norm{\textbf{r}}\) is the length of the position vector field \(\textbf{r}(x, y, z) = x\textbf{i}+ y\textbf{j}+ z\textbf{k}\)).
4.6.12 . \(∇(1/r) = −\textbf{r}/r^ 3\)
4.6.13. \(∆(1/r) = 0\)
4.6.14. \(∇· (\textbf{r}/r^ 3 ) = 0\)
4.6.15. \(∇(\ln r) = \textbf{r}/r^ 2\)
4.6.16. div \((\textbf{F}+\textbf{G}) = \text{div }\textbf{F} + \text{div }\textbf{G}\)
4.6.17. curl \((\textbf{F}+\textbf{G}) = \text{curl }\textbf{F} + \text{curl }\textbf{G}\)
4.6.18. div \((f \textbf{F}) = f \text{div }\textbf{F} + \textbf{F}· ∇f\)
4.6.19. div \((\textbf{F}× \textbf{G}) = \textbf{G}· \text{curl }\textbf{F} − \textbf{F}· \text{curl }\textbf{G}\)
4.6.20. div \((∇f × ∇g) = 0\)
4.6.21. curl \((f \textbf{F}) = f \text{curl }\textbf{F} + (∇f )× \textbf{F}\)
4.6.22. curl (curl \(\textbf{F}\)) = \(∇(\text{div }\textbf{F}) − ∆\textbf{F}\)
4.6.23. \(∆(f g) = f ∆ g + g∆ f + 2(∇f · ∇g)\)
C
4.6.24. Prove Theorem 4.17.
4.6.25. Derive the gradient formula in cylindrical coordinates: \(∇F = \frac{∂F}{ ∂r} \textbf{e}_r + \frac{1}{ r} \frac{∂F}{ ∂θ} \textbf{e}_θ + \frac{∂F}{ ∂z} \textbf{e}_z\)
4.6.26. Use \(\textbf{f} = u∇v\) in the Divergence Theorem to prove:
(a) Green’s first identity: \(\iiint\limits_S (u∆v + (∇u)· (∇v))\,dV = \iint\limits_Σ (u∇v)· dσ\)
(b) Green’s second identity: \(\iiint\limits_S (u∆v − v∆u)\,dV = \iint\limits_Σ (u∇v − v∇u)· dσ\)
4.6.27. Suppose that \(∆u = 0\) (i.e. \(u\) is harmonic ) over \(\mathbb{R}^ 3\). Define the normal derivative \(\frac{∂u}{ ∂n}\) of \(u\) over a closed surface \(Σ\) with outward unit normal vector n by \(\frac{∂u}{ ∂n} = D_n u = \textbf{n}· ∇u\). Show that \(\iint\limits_Σ \frac{∂u}{ ∂n}\, dσ = 0\). ( Hint: Use Green’s second identity .)