4.3: Language Barriers
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- Kristine Clancy, Tiffany Ruggeri-Dilello, & Kim Yee
- ASCCC Open Educational Resources Initiative (OERI)
Increasing Communication Competence
We encounter many language barriers. How we address or work with these barriers can increase our communication competence. Conversational skills are learned. Despite what we may have learned in the past, these skills are something we are consistently working on. We assume that because we have been communicating our whole lives that we have attained skills. However, communication does not mean that we simply get “better” with experience, without knowledge and practicing specific skills. In fact, when employers describe their ideal candidate, “communication skills” is on the top of the list. Therefore, it is important for us to learn about barriers we experience, skills that will improve our communication competence, and putting those skills to practice. One of the many barriers we experience is related to apprehension.
Communication Apprehension
Communication apprehension is the hesitancy or discomfort surrounding our communication events. This is common in oral communication (Loureiro, et al., 2020). One of our authors gives this example:
As a professor, I feel comfortable lecturing. However, when my family and I are ordering food at a restaurant, I am uncomfortable. In part, this is my lived experience. People can be rude when they are hungry and take it out on food servers.
Those who have worked in the food industry have been on the receiving end of this incivility. Our lived experience is one of many reasons for our communication apprehension. We may not fully understand the reason(s) that we have high communication apprehension in some realms but not others. However, James McCroskey designed an assessment tool, the PRCA-24, to help us determine where we experience different levels of communication apprehension in different spheres (public, interpersonal, group, and meetings). Knowing where we experience communication apprehension helps us appreciate the way we typically react and how to change these behaviors in an attempt to lower our communication apprehension.
Jung and McCroskey (2004) found that non-native English speakers experience communication apprehension at an increased rate. Moderating factors included the number of years that participants had lived in the US and the number of years of that they had spoken English. This means that when we communicate interculturally, communication apprehension is higher and that becomes a barrier. We can utilize intercultural communication competence tools to help us achieve a positive communication event.
Culture
There are many skills at our disposal that relate to culture. Culture is influenced by communication and our perceptions (Balakrishnan, et al., 2021). We can start by becoming more culturally empathetic. “Cultural empathy requires a perception of the needs of others, as well as the knowledge of their cultural specificities'' (Gonçalves, et al., 2020, p. 246). We should try to be attuned to the emotional needs of others and be able to practice perspective-taking—that is, attempt to understand the other person’s perspective.
Social intelligence and cultural sensitivity can help mitigate language barriers. Social intelligence in this context means that we seek understanding of someone’s language and culture in an attempt to reach communication competence. For example, we can learn about the rituals and customs of someone’s culture, which would help us to understand our communication partners and avoid language barriers. Cultural sensitivity relates to how we are open minded and overcome these barriers. When we seek out information about a culture without preconceived notions, we can be both culturally sensitive and more socially intelligent.
We can also avoid ethnocentric behaviors. Ethnocentrism is when we tend to see the world through our own cultural lens and judge others’ behaviors by the standards we hold. When we hold other people to our own standards, we are asserting our dominance and superiority, and we not taking into account cultural differences or barriers that exist. While it may sound straightforward to avoid ethnocentric behaviors, we must begin by becoming aware of what ethnocentric ideals we have and when we apply them. Only then can we start to eliminate our ethnocentric behaviors. This relates to the way we communicate and taking responsibility with our language use.
Consider This: Introducing English Idioms to Second-Language Speakers
One of our authors shares this example:
Since English is my second language, I translated very literally when I first began to learn English. I was confused by many of these strange American sayings—and I wasn’t the only one! A good friend of mine from graduate school was an international student from Japan who spoke very proficient English. She went to one of her advisor’s classes to talk about her thesis. Before leaving, her advisor asked her to come back and share the results of her thesis when she had finished. She agreed, and he responded, “It’s a date!” She was utterly baffled, but she did not want to disrespect her advisor. With a slow and reluctant nod, she answered, “OK… but I will have to ask my boyfriend first.” I later learned that some of these “strange American sayings” are referred to as idioms .
Idioms are figurative-language phrases that require understanding of cultural context to determine meaning—which is never the literal, dictionary definition of the strung-together phrases. Table \(\PageIndex{1}\) lists some common idioms that may cause confusion for English language learners.
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Idiom |
What a Non-native English Speaker Might Be Picturing |
Actual Meaning |
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Beat around the bush |
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To avoid getting to the point of an issue Example: “Stop beating around the bush, and tell me the bad news.” |
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Let the cat out of the bag |
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To disclose a secret Example: “I accidentally let the cat out of the bag, and now everyone knows you won the lottery!” |
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Break the ice |
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To get to know people; get a conversation going Example: “Let me break the ice and tell you a little something about myself and our company.” |
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Pull someone's leg |
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To exaggerate the truth, usually as a joke Example: “I was pulling his leg when I told him I swallowed a fly. I didn’t think he would be so grossed out.” |
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Beat a dead horse |
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To waste time on something that cannot be changed Example: “There is no point in beating a dead horse. It’s time to buy a new car.” |
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Barking up the wrong tree |
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To waste time by going after the wrong thing Example: “You’re barking up the wrong tree begging for money, because I’m broke!” |
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Bite the bullet |
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To force oneself to do something difficult or unpleasant Example: “I had to bite the bullet and attend traffic school.” |
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On the ball |
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Describes people who are efficient, effective, talented, and aware of what’s around them Example: “Danny was on top of the ball and graduated in three years instead of four.” |
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Straight from the horse’s mouth |
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Getting information directly from the source who has direct knowledge of something Example: “Unless I hear it straight from the horse’s mouth, I won’t believe she is retiring.” |
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Kill two birds with one stone |
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To do two things with one action Example: “We can kill two birds with one stone by using the soup base in two different recipes.” |
Table created by Kim Yee
CC BY-NC-SA
Clipart source: English idioms by Graphics RF from Vecteezy (Free License)
Discussion Questions/Journal Prompts:
- What other English idioms might confuse English language learners?
- If you speak another language, are there idioms used in your language you’d like to share?
- Why do we use idioms instead of saying what we mean?
Activities with Idioms:
- Pick an idiom and draw its literal and figurative meanings.
- Pick an idiom and act it out as a charade.
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Use idioms as discussion starters. For example:
- Tell us about a time you had to “burn the midnight oil.”
- Describe something that “gets under your skin.”
- Tell about a time when you were able to “kill two birds with one stone.”
- Describe a situation where you had to “bite the bullet.”
- What is the best way to “break the ice” when meeting new people?
- Describe a situation where “beating around the bush” is necessary.
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What are some idioms you can use to give advice? For example:
- Take what someone says with a grain of salt.
- You can catch more flies with honey than vinegar.
- You should bury the hatchet.
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What are some idioms we use during holidays?
- Valentine’s Day (to pop the question)
- Halloween (skeleton in the closet)
- Christmas (snowball effect)
Responsibility
When we are participating in a communication event, it is important to understand our role. In doing so, we must accept responsibility for the role that we play. We call this the language of responsibility . We know that using the language of responsibility in a corporate setting facilitates relationships (Joutsenvirta, 2009).
We can act responsibly by using it statements , and avoiding but statements and you statements . The use of pronouns confers blame; the language of responsibility aids us in avoiding blaming others while accepting responsibility for our own actions. When we remove pronouns completely and use the term “it,” we avoid pointing the finger at others and state the facts. For instance, instead of stating, “you forgot to take out the garbage,” I could state, “the garbage was not taken out,” or even “we forgot to take out the trash.” The latter infers that “we” are in this together and the issue is not about who did or did not take an action. This helps us to avoid “you statements.” Is there a way to reframe our language to avoid pointing fingers? By removing “you,” we can accept responsibility and not make someone else responsible for an action. With “but statements,” we essentially negate everything that came before the word but. For instance, a spouse states to another spouse, “I love you, but I really need you to take out the trash.” The way this is stated assumes that the idea of “love” is not the point, but rather that who takes out the trash is the pivotal part of the phrase. It can even infer that “love” is contingent on this action being taken. If we are on the opposite end of this statement, we may in fact, be concerned that the way this is exclaimed, points to the fact that the trash is more important than the relationship. We see this in many areas, specifically politics. Have you been on the opposite end of a non-apology? “I am sorry you feel that way.” I am not sorry for the action itself, but rather your reaction. By utilizing the language of responsibility, we can avoid hurting the relationship and promote relational and professional growth.
Defensive Communication
Defensiveness is the opposite of the language of responsibility, so it is important to start by recognizing when we use defensive language and then ensure we focus on the issue at hand instead of on the other person or their actions. John Gottman (2012), a leading researcher in marital relationships, found that defensive communication is one of the most destructive relationship behaviors and in fact a leading indicator of future divorce. This research has been extended to communication in a broad setting. Defensiveness is defined by the perception of threat or threatening behavior (Gibbs, 1965). Avoiding defensiveness and airing grievances can spur relational growth and relational satisfaction (Gottman & Silver, 1994). I focus on four main types of defensive communication in this section: dogmatism, superiority, indifference, and control messages.
Dogmatism
Dogmatism is when we use emotional justification rather than evidence-based justification (Harrison, 2021). We typically do not notice when we respond using a dogmatic perspective. We are inextricably connected to our emotions, and with dogmatism we are operating from our sense of self. This is defensive behavior: when we use emotions to justify our stance, we leave out empirical evidence. We lack connection to our relational communication partners. This dismissive behavior hurts the relationships involved. For example, in a professional environment I was explaining why I process I had developed was a better option than our previous process. Another person said to me, “Why does that make you correct?” I realized I was using my emotions rather than logic in my explanation. I switched my tone and focused on the evidence that I had. Focusing on evidence and the argument itself can be a strong skill to develop.
Superiority
According to Gottman (2012), any time we “act superior,” we are saying that we are correct and that the other(s) must be “damaged.” Therefore, superiority is a form of defensiveness. We can start by recognizing when we are using judging language and attempt to use the language of responsibility in its place. We can also try to focus on why we think our stance is superior and focus on those traits, using a discussion communication style rather than prescriptive communication style. If we create a habit of challenging why we believe we are correct, we can then focus on the issue at hand and avoid superiority.
Indifference
We display indifference when we show a lack of caring or connection to the other person(s) and/or the content of the message. At times we do not show our interest to our communication partners, which results in harm to the relationship(s). Indifference can take many forms. At times it is intentional and meant to be understood by our relational partners, and other times we simply do not see that we lack caring or connection. This is related to the different contexts of communication: culture, relational, and situational. These contexts play a role in how involved we are in the communication event. For example, if we are communicating with someone whom we do not care to continue in relationship with, we may intentionally use indifference. Whereas, perhaps our cultural lens is different than our communication partner and we do not see our communication behavior as indifferent, but culturally we are giving space to avoid being rude or engaging in cultural clashes. If we care about the relationship, we can show caring within these contexts. By engaging in a way that shows that we care, we promote the relationship—if we also keep in mind what is appropriate culturally, and what is appropriate for the situation.
Control Messages
Control messages are meant to coerce rather than persuade relational partners to an action. We have seen this in history. If you look at the Milgram experiment, participants were asked to shock people, by use of control and authoritarian messages. Participants did not know that there was no shock being administered. The experiment showed that humans are susceptible to control messages. When we are told what to do using manipulative or coercive messages, we tend to obey. This lack of critical thinking focuses on superiority and hurts all involved. The participants in the experiment disclosed being unable to resist the commands they received, and as a result became distraught and experienced trauma. This reflects what happens to us when we receive control messages. Therefore, we can avoid control messages and use ethical means of persuasion instead. For instance, when my son exhibits “bad behavior,” instead of focusing on consequences of his actions, I can focus on the benefits of positive behavior.
Bias
When discussing language barriers, we include our understanding of culture and our perceptions, which are ingrained in the language we use and its implications. When we discuss bias, we tend to focus on the negative connotations. Simply put, bias relates to our preferences. If I ask you for your favorite color, favorite musical artist, or favorite podcaster, you likely have a response.
When we discuss bias when it comes to culture, we are talking about when those preferences become negative. Perhaps you prefer to have your hair or romantic partner’s hair blond. This is a preference. When this becomes negative is when we think of and treat others differently as a result. When we put people into classifications, that is stereotyping . Is this negative? It depends on what transpires. If I am your professor, and I assume you are brilliant because you wear glasses and then hold you to a higher standard than other students, that is negative (and should never happen). This stereotyping could be positive, too: for example, perhaps you wear glasses and do identify as a brilliant person.
However, stereotyping is largely negative. We don’t like being put into boxes. These boxes can be understood as a wide array of categories from our race and ethnicity to our demographics and personality. It would be unfortunate if being put into a certain box prevented us from potential employment. For instance, if I choose a box that states that I have a disability, will that prevent potential employers from hiring me? Therefore, we must acknowledge our biases and try to avoid negative stereotypes.
Prejudice
Prejudice is when our stereotyping has turned from preferences into hate and hateful actions. Prejudice is defined as “a preconceived judgment or opinion, usually based on limited information” (Tatum, 2017, p. 85). Many prejudices are influenced by misinformation, We have prejudiced action when we use hostile statements based on group membership (Cary & Chasteen, 2021). Prejudice is experienced in face-to-face interactions as well as online. As we are communicating more through online means, we see prejudice in many platforms and contexts. Recent research has shown that prejudice is found in gaming (Cary & Chasteen, 2022), television news (Takano, et al., 2021), social media (Capobianco, 2020), and even academia (Hurd, et al., 2022; Ruf, 2020). On gaming platforms, gamers communicate. As this is an open forum, prejudicial comments are made and are in fact commonplace. Cary, Axt, and Chasteen (2020) found that early interventions reduced these actions. Let’s say you are gaming and someone says something prejudicial (i.e., a comment about you that relates to your group membership). If the comments are confronted, there is a better chance that prejudicial actions will be reduced. If the comments are not confronted, then the other person may feel like this is banter and playful instead of hurtful, thereby continuing prejudicial actions. Prejudice often turns into hate speech.
Hate Speech
Marginalized people are people that are excluded and discriminated against due to their group membership, typically related to race, sexual orientation, gender and/or gender identity, socioeconomic status, age, disability, and others. Microaggressions are hostile or negative messages towards people within marginalized groups. Microaggressions are defined as brief and commonplace daily verbal, behavioral, or environmental indignities, whether intentional or unintentional, that communicate hostile, derogatory, or negative slights and insults (Wing, 2007). These microaggressions can grow into hate speech. Hate speech consists of abhorrent messages used to hurt, incite violence, and inflict psychological harm (ŞtefÃniÞÃ & Buf, 2021). It is used in multiple contexts and against many types of targets, but most commly against marginalized people. The receipt of hate speech results in fear and anxiety, which ultimately increases stress (Crichton, 2019). Targeted marginalized groups are often subject to violent acts, known as hate crimes . While hate speech in many forms is protected under the First Amendment, hate crimes are not.
Racism and Sexism
Stereotyping can lead to racism and sexism. Racism is a systemic form of oppression using power dominance to control people based on race. Cultural racism is defined as “cultural images and messages that affirm the assumed superiority of Whites and the assumed inferiority of people of color” (Tatum, 2017, p. 86). Racism is rooted in oppression and oppressive systems put in place by dominant groups to control marginalized groups. Oppression occurs when the dominant group reinforces their power while withholding or suppressing marginalized groups from power. This may start with ethnocentrism , when we unconsciously see the world through our personal lens and fail to perspective take or see things from another’s perspective. These unconscious biases and assumptions are reinforced through structural processes. This occurs through institutional practices and policies based on cultural images and ideals that affirms systemic racism. When we pair oppression and racism with hierarchical structures, we reinforce systemic racism.
Sexism is defined as discrimination or prejudice based on gender. When we discuss sexism we are often talking about women being treated with less respect than would a male counterpart. This can start with a generalization that turns into discrimination or prejudice. We have heard the joke that women are “bad” drivers. This may have started as a joke but become prejudicial quickly.
Sexism is pervasive and widespread, especially in the workplace. There is a disparity in wages between males and females. We are seeing more discriminatory sexist acts in the workplaces that take other forms, such as who gets parental leave when a child is born, how much time, the argument of fairness to childless persons, or the assumption that in a two-person relationship if a child is sick, a female in the relationship is responsible and thereby is a riskier hire than a male. Sexism affects those in the LGBTQIA+ community as well.
When we discuss gender, a lot of the time we discuss it in terms of biological sex, but it includes gender expression and identity. Take for instance a person who is transgender and how they may be treated in terms of biological sex norms. These "traditional" and antiquated views lead to the mistreatment of individuals based on their gender. Sexism is prevalent and we can begin to stem the tide by reviewing the language we use and exclude language rooted in negative and potentially untrue stereotypes and sexist language.
Muted group theory studies the differences in dominant and subdominant groups and explains how subdominant groups have less power and access than the dominant group. In the development of this theory, researchers started out by examining gender differences and the power dynamics in males versus females. However, muted group theory was not intended to primarily be about gender, but to lay the foundation to understand differences in the power of dominant and subdominant groups (Smith Barkman, 2018). Essentially, we are looking at why some people have power, and when they do, how they use it to keep that power and “mute” or prevent subdominant groups from attaining the same power. Marginalized or muted groups often do not see themselves as muted or do not feel they have power to change their position. According to Linda Smith Barkman (2018), there are four primary ways we can avoid people being muted:
- We have different forms of expression, which also relates to power and how it is perceived. When we do not understand others’ forms of expression, divides are created in our communication. We can learn about different forms of expression to avoid these pitfalls.
- We can start to recognize and understand others’ cultures and group membership and how that relates to their power or lack thereof.
- Speaking up for marginalized others in an appropriate and nuanced way can benefit both dominant and muted groups.
- It is sometimes difficult to see that a group is marginalized. It is therefore imperative to stay open-minded and ask ourselves what is happening related to group membership and power.


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3: Static Equilibrium of Rigid Bodies
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Video introduction to the topics covered in Chapter 3, delivered by Dr. Jacob Moore. YouTube source:
https://youtu.be/x_L6S6ohu-k
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3.1: Moment of a Force (Scalar)
Further explanation of the moment of a force, and the equation for calculating a moment. How to calculate the moment using only scalar quantities, for two- and three-dimensional systems. Includes several worked examples.
3.2: Varignon's Theorem
Using Varignon's Theorem as an alternative to finding perpendicular distances, in scalar moment calculations. Includes several worked examples.
3.3: Couples
Definition of a force couple, and calculation of the moment produced by a couple.
3.4: Equilibrium Analysis for a Rigid Body
Using the definition of static equilibrium to set up equations that allow for the analysis of rigid body systems. Includes several worked examples.
3.5: Chapter 3 Homework Problems


12: Microsoft Access Basic Skills Last updated Save as PDF Page ID 46388 Learning Objectives Create database tables Work with fields Set table primary key Create indexes 12.1: Why It Matters- Microsoft Access Basic Skills 12.2: Introduction to Creating Database Tables 12.3: Creating a Database 12.4: Naming Databases 12.5: Creating an Access Table 12.6: Introduction to Fields in Access 12.7: Field Properties 12.8: Field Names and Data Types 12.9: Introduction to Table Primary Key 12.10: Primary Key 12.11: Set a Primary Key 12.12: Introduction to Indexes 12.13: What is an Index? 12.14: To Index or Not to Index? 12.15: Putting It Together- Microsoft Access Basic Skills 12.16: Assignment- Create New Access Database 12.17: Discussion- Selecting Primary Key Tips


1.2: Recap- Position and Momentum States
Before proceeding, let us review the properties of quantum particles in free space. In a \(d\) -dimensional space, a coordinate vector \(\mathbf{r}\) is a real vector of \(d\) components. A quantum particle can be described by the position basis—a set of quantum states \(\{|\mathbf{r}\rangle\}\) , one for each possible \(\mathbf{r}\) . If we are studying a particle trapped in a finite region (e.g., a particle in a box), \(\mathbf{r}\) is restricted to that region; otherwise, \(\mathbf{r}\) is any real \(d\) -dimensional vector. In either case, the \(\mathbf{r}\) ’s are continuous, so the position eigenstates form an uncountably infinite set.
The position eigenstates are assumed to span the state space, so the identity operator can be resolved as
\[\hat{I} = \int d^dr \, |\mathbf{r}\rangle \,\langle\mathbf{r}|,\]
where the integral is taken over all allowed \(\mathbf{r}\) . It follows that
\[\langle \mathbf{r} | \mathbf{r}' \rangle = \delta^d(\mathbf{r}-\mathbf{r}').\]
The position eigenstates are thus said to be “delta-function normalized”, rather than being normalized to unity. In the above equation, \(\delta^d(\cdots)\) denotes the \(d\) -dimensional delta function; for example, in 2D,
\[\langle x,y \,|\, x',y' \rangle = \delta(x-x') \, \delta(y-y').\]
The position operator \(\hat{\mathbf{r}}\) is defined by taking \(|\mathbf{r}\rangle\) and \(\mathbf{r}\) as its eigenstates and eigenvalues:
\[\hat{\mathbf{r}} |\mathbf{r}\rangle \,=\, \mathbf{r}\, |\mathbf{r}\rangle.\]
Momentum eigenstates are constructed from position eigenstates via Fourier transforms. First, suppose the allowed region of space is a box of length \(L\) on each side, with periodic boundary conditions in every direction. Define the set of wave-vectors \(\mathbf{k}\) corresponding to plane waves satisfying the periodic boundary conditions at the box boundaries:
\[\nonumber\Big\{\mathbf{k} \; \Big| \;\; k_j = 2\pi m/L\;\;\mathrm{for} \;m\in\mathbb{Z}, \; j = 1, \dots,d\, \Big\}.\]
So long as \(L\) is finite, the \(\mathbf{k}\) vectors are discrete. Now define
\[|\mathbf{k}\rangle = \frac{1}{L^{d/2}} \, \int d^dr \; e^{i\mathbf{k}\cdot\mathbf{r}} |\mathbf{r}\rangle,\]
where the integral is taken over the box. These can be shown to satisfy
\[\langle\mathbf{k}|\mathbf{k}'\rangle = \delta_{\mathbf{k},\mathbf{k}'}, \quad \langle\mathbf{r}|\mathbf{k}'\rangle = \frac{1}{L^{d/2}} e^{i\mathbf{k}\cdot\mathbf{r}}, \quad I = \sum_{\mathbf{k}} |\mathbf{k}\rangle\,\langle\mathbf{k}|.\]
The momentum operator is defined so that its eigenstates are \(\{|\mathbf{k}\rangle\}\) , with \(\hbar\mathbf{k}\) as the corresponding eigenvalues:
\[\hat{\mathbf{p}} |\mathbf{k}\rangle \,=\, \hbar \mathbf{k}\, |\mathbf{k}\rangle.\]
Thus, for finite \(L\) , the momentum eigenstates are discrete and normalizable to unity. The momentum component in each direction is quantized to a multiple of \(\Delta p = 2\pi\hbar/L\) .
We then take the limit of an infinite box, \(L \rightarrow \infty\) . In this limit, \(\Delta p \rightarrow 0\) , so the momentum eigenvalues coalesce into a continuum. It is convenient to re-normalize the momentum eigenstates by taking
\[|\mathbf{k}\rangle \rightarrow \left(\frac{L}{2\pi}\right)^{d/2} |\mathbf{k}\rangle.\]
In the \(L\rightarrow\infty\) limit, the re-normalized momentum eigenstates satisfy
Definition: Re-normalized momentum eigenstates
\[\begin{align} |\mathbf{k}\rangle &= \frac{1}{(2\pi)^{d/2}} \, \int d^dr \; e^{i\mathbf{k}\cdot\mathbf{r}} |\mathbf{r}\rangle, \\ |\mathbf{r}\rangle &= \frac{1}{(2\pi)^{d/2}} \, \int d^dk \; e^{-i\mathbf{k}\cdot\mathbf{r}} |\mathbf{k}\rangle, \\ \langle\mathbf{k}|\mathbf{k}'\rangle = \delta^d(\mathbf{k}-\mathbf{k}'),& \quad \langle\mathbf{r}|\mathbf{k}\rangle = \frac{1}{(2\pi)^{d/2}} e^{i\mathbf{k}\cdot\mathbf{r}}, \quad I = \int d^dk \;|\mathbf{k}\rangle\,\langle\mathbf{k}|. \end{align}\]
The above integrals are taken over infinite space, and the position and momentum eigenstates are now on a similar footing: both are delta-function normalized. In deriving the above equations, it is helpful to use the formula
\[\int_{-\infty}^\infty dx\; \exp(ikx) \;=\; 2\pi\, \delta(k).\]
For an arbitrary quantum state \(|\psi\rangle\) , a wavefunction is defined as the projection onto the position basis: \(\psi(\mathbf{r}) = \langle \mathbf{r}|\psi\rangle\) . Using the momentum eigenstates, we can show that
\[\begin{align} \begin{aligned}\langle \mathbf{r}|\hat{\mathbf{p}}|\psi\rangle &= \int d^dk \; \langle\mathbf{r}|\mathbf{k}\rangle \; \hbar\mathbf{k} \; \langle\mathbf{k}|\psi\rangle \\ &= \int \frac{d^dk}{(2\pi)^{d/2}}\; \hbar\mathbf{k} \;e^{i\mathbf{k}\cdot\mathbf{r}} \langle\mathbf{k}|\psi\rangle \\ &= -i\hbar\nabla \int \frac{d^dk}{(2\pi)^{d/2}}\; \;e^{i\mathbf{k}\cdot\mathbf{r}} \langle\mathbf{k}|\psi\rangle \\ &= -i\hbar \nabla\psi(\mathbf{r}).\end{aligned}\end{align}\]
This result can also be used to prove Heisenberg’s commutation relation \([\hat{r}_i, \hat{p}_j] = i\hbar\delta_{ij}\) .


9.3: Adding and Editing Equations
Our adding equations video tutorial on YouTube:
youtu.be/74b5lf-4Y4A
Equations in LibreTexts are handled by the MathJax add-on (based on LaTeX). This page contains guidelines and references on how to use this system. It is fairly straight-forward for anyone who has ever worked with code (markup).
Inline vs. Displaymode Equations
To add an equation, authors have two choices: either (1) inline or (2) display . The difference is demonstrated here:
An inline equation looks like this, \(E=mc^2,\) and is meant to be read with the paragraph. An equation that is presented in display mode looks like this
\[E=mc^2\]
with the equation taking up the entire line. Note that the displaystyle equation is automatically numbered.
To invoke an inline equation add \( \backslash( \) and \(\backslash)\) around the LaTeX code (discussed below). To invoke the display mode add \(\backslash[\) and \(\backslash]\) around the LaTeX code. The code can be somewhat complicated and must be formulated properly or the MathJax add-on will not render the equation (or will do so improperly). Examples of MathJax code, including some useful templates, can be found at the bottom of this page.
MathJax is a little overwhelming at first but thankfully there is a tool that makes it super easy called MathPix Snip.
- Download the tool from https://mathpix.com/ and install it on your computer
- When you need to transfer an equation press ctrl+alt+m and select the equation that you want to copy.
- Use the first line that doesn't have any environment coding and add the \( or \[ for inline or display mode environment manually.
- If the snipper got a few details wrong you can edit the code and check that it renders before copying it over
- If it is a complicated equation zoom in on it using your browser before using the tool. If it still doesn't work consider snipping the equation in several parts.
Intrapage Numbering and Referencing of Equations
The MathJax system is ideal for referencing equations on a page. Equations in displaymode (i.e., used with \[ \] delimiters) are automatically numbered, inline equations (using \( \) delimiters) are not. If you want to reference a displayed equation you give it a label by writing \label{name} after the equation but before \]. For example, the following code
\[ E =mc^2 \label{Einstein} \] See Equation \ref{Einstein}
renders as
\[ E =mc^2 \label{Einstein} \] See Equation \ref{Einstein}
If you do not want a display equation to be numbered you can use \nonumber after the equation (and drop the label text since that is unneeded.
\[ E =mc^2 \nonumber\]
renders as
\[ E =mc^2 \nonumber\]
Interpage Numbering and Referencing of Equations
Interpage referencing is a bit more complicated. The best option currently involves HTML rather than MathJax.
- Source page: This is the page you want to link to.  Then select Elements on the menu and choose the Anchor option. You can use it like you would \label in LaTeX/MathJax. It adds a little flag that is visible when in Edit mode.
- Reference page: This is the page on which you wish to place a link to an anchor on another page. Go to this page and open the edit view. Highlight text you want to be a link. Go to Elements menu again and click Link (there is also a button on the blue WYSIWYG editor bar for this). Select the source page and at the bottom entry titled "Jump to anchor or section", you will see the text you used for the label on the source page. Save this link and save the page.
If the equation shows up just under the floating blue bar, you can move the anchor to the paragraph before to make it display as desired (small issues until we get it fixed).
When editing math pages
- Be sure to use the escape character \(\backslash\) in front of all trigonometric functions (\( \sin x, \cos x, \tan x,\)) natural logarithms (\(\ln x\)), etc. Then leave a space between the function and its argument unless there is a parentheses there. For example, \( \backslash\text{sin}\,\text{x}\) or \( \backslash\text{sin}(\text{x})\) (rendering as \( \sin x \) and \( \sin (x) \), respectively). If you don't do this, the results may look like this: \( sinx, cosx, \sinx, lnx, \lnx\).
- Use the escape character \(\backslash\) in front of "lim" for limits.
- If limits, integrals or sigma notation (sums) are to be used as part of inline mathematics, be sure to add \(\backslash\text{displaystyle}\) as the first commands of these inline math expressions. Here is what a limit should look like when rendered correctly: \(\displaystyle \lim_{x \to 0} \frac{\sin x}{x}\), and incorrectly: \(lim_{x \to 0} \frac{\sin x}{x}.\)
- Display mode mathematics will never need \(\backslash\text{displaystyle}\).
- Do not use \(\backslash\text{displaystyle}\) unless it is helpful for making one of the above mathematical notations render nicely.
- To obtain larger fractions in an inline setting like \(\dfrac{2x}{x^2 + 1}\) instead of small ones like \(\frac{2x}{x^2 + 1}\), use \(\backslash\text{dfrac}\) instead of using \(\backslash\text{displaystyle}\) with \(\backslash\text{frac}\) which would have the same effect, but is overkill unless the \(\backslash\text{displaystyle}\) is needed for another reason like those listed above.
- When using display mode mathematics between \(\backslash[\) and \(\backslash]\), consider whether the equation is an important result that may be referred to later in the text or not. By default display mode mathematics lines will be aut\nonumbered. If the equation is not a main result, it should typically not be numbered, so you will need to insert a \(\backslash\text{n\nonumber}\) at the end of it but within the expression.
- Do not use any extra commands, like \(\backslash\text{mathrm}\), in front of every instance of math in a page to try to get text to look a certain way. There should be a better way to get the desired effect.
- When editing integrals, please use a spacing element before the differential. For example, \(\backslash\text{displaystyle }\backslash\text{int x^2 }\backslash,\text{dx}\).
- en.wikibooks.org/wiki/LaTeX/Mathematics
- This link www.codecogs.com/latex/eqneditor.php(external link) is an online WYSIWYG LaTeX editor that creates math expressions using templates. The LaTeX is automatically written and a preview is displayed. Just copy the code and paste it. Be sure to enclose the LaTeX in dollar signs ($).
- en.wikibooks.org/wiki/LaTeX/Mathematics
- https://www.cs.odu.edu/~zeil/cs390/l...math/demo.html
- http://www.artofproblemsolving.com/W...LaTeX:Commands
- http://www.onemathematicalcat.org/Ma.../TeXSyntax.htm
- https://en.wikibooks.org/wiki/LaTeX/Advanced_Mathematics
DO NOT USE EXTERNAL SITES LIKE www.codecogs.com/latex TO RENDER THE EQUATION FOR YOU VIA PROVIDING AN IMAGE FILE. You can use them to build LaTex equations and then copy-paste the LaTex into your page.
| Feature | Syntax | How it looks rendered |
|---|---|---|
| Superscript |
a^2
|
\(a^2\) |
| Subscript |
a_2
|
\(a_2\) |
| Grouping |
a^{2+2}
|
\(a^{2+2}\) |
a_{i,j}
|
\(a_{i,j}\) | |
| Combining sub & super |
x_2^3
|
\(x_2^3\) |
| Preceding and/or Additional sub & super |
\sideset{_1^2}{_3^4}\prod_a^b
|
\(\sideset{_1^2}{_3^4}\prod_a^b\) |
{}_1^2\!\Omega_3^4
|
\({}_1^2\!\Omega_3^4\) | |
| Stacking |
\overset{\alpha}{\omega}
|
\(\overset{\alpha}{\omega}\) |
\underset{\alpha}{\omega}
|
\(\underset{\alpha}{\omega}\) | |
\overset{\alpha}{\underset{\gamma}{\omega}}
|
\(\overset{\alpha}{\underset{\gamma}{\omega}}\) | |
\stackrel{\alpha}{\omega}
|
\(\stackrel{\alpha}{\omega}\) | |
| Derivative (forced PNG) |
x', y
, f', f
\!
|
|
| Derivative (f in italics may overlap primes in HTML) |
x', y
, f', f
|
\(x', y'', f', f''\) |
| Derivative (wrong in HTML) |
x^\prime, y^{\prime\prime}
|
\(x^\prime, y^{\prime\prime}\) |
| Derivative (wrong in PNG) |
x\prime, y\prime\prime
|
\(x\prime, y\prime\prime\) |
| Derivative dots |
\dot{x}, \ddot{x}
|
\(\dot{x}, \ddot{x}\) |
| Underlines, overlines, vectors |
\hat a \ \bar b \ \vec c
|
\(\hat a \ \bar b \ \vec c\) |
\overrightarrow{a b} \ \overleftarrow{c d} \ \widehat{d e f}
|
\(\overrightarrow{a b} \ \overleftarrow{c d} \ \widehat{d e f}\) | |
\overline{g h i} \ \underline{j k l}
|
\(\overline{g h i} \ \underline{j k l}\) | |
| Arrows |
A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C
|
\( A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C\) |
| Overbraces |
\overbrace{ 1+2+\cdots+100 }^{5050}
|
\(\overbrace{ 1+2+\cdots+100 }^{5050}\) |
| Underbraces |
\underbrace{ a+b+\cdots+z }_{26}
|
\(\underbrace{ a+b+\cdots+z }_{26}\) |
| Sum |
\sum_{k=1}^N k^2
|
\(\sum_{k=1}^N k^2\) |
Sum (force
\displaystyle
for inline use)
|
\displaystyle \sum_{k=1}^N k^2
|
\(\displaystyle \sum_{k=1}^N k^2\) |
| Product |
\prod_{i=1}^N x_i
|
\(\prod_{i=1}^N x_i\) |
Product (force
\displaystyle
for inline use)
|
\displaystyle \prod_{i=1}^N x_i
|
\(\displaystyle \prod_{i=1}^N x_i\) |
| Coproduct |
\coprod_{i=1}^N x_i
|
\(\coprod_{i=1}^N x_i\) |
Coproduct (force
\displaystyle
for inline use)
|
\displaystyle \coprod_{i=1}^N x_i
|
\(\displaystyle \coprod_{i=1}^N x_i\) |
| Limit |
\lim_{n \to \infty}x_n
|
\(\lim_{n \to \infty}x_n\) |
Limit (force
\displaystyle
for inline use)
|
\displaystyle \lim_{n \to \infty}x_n
|
\(\displaystyle \lim_{n \to \infty}x_n\) |
| Integral |
\int\limits_{-N}^{N} e^x\, dx
|
\(\int\limits_{-N}^{N} e^x\, dx\) |
Integral (force
\displaystyle
for inline use)
|
\displaytstyle \int\limits_{-N}^{N} e^x\, dx
|
\(\displaystyle \int\limits_{-N}^{N} e^x\, dx\) |
| Double integral |
\iint\limits_{D} \, dx\,dy
|
\(\iint\limits_{D} \, dx\,dy\) |
| Triple integral |
\iiint\limits_{E} \, dx\,dy\,dz
|
\(\iiint\limits_{E} \, dx\,dy\,dz\) |
| Quadruple integral |
\iiiint\limits_{F} \, dx\,dy\,dz\,dt
|
\(\iiiint\limits_{F} \, dx\,dy\,dz\,dt\) |
| Path integral |
\oint\limits_{C} x^3\, dx + 4y^2\, dy
|
\(\oint\limits_{C} x^3\, dx + 4y^2\, dy\) |
| Intersections |
\bigcap_1^{n} p
|
\(\bigcap_1^{n} p\) |
| Unions |
\bigcup_1^{k} p
|
\(\bigcup_1^{k} p\) |
| Cancelling factors |
\dfrac{\cancel{(x-3)}(x+5)}{\cancel{(x-3)}
|
\(\dfrac{\cancel{(x-3)}(x+5)}{\cancel{(x-3)}}\) |
| Feature | Syntax | How it looks rendered |
|---|---|---|
| Fractions |
\frac{2}{4}=0.5
|
\(\frac{2}{4}=0.5\) |
| Small Fractions |
\tfrac{2}{4} = 0.5
|
\(\tfrac{2}{4} = 0.5\) |
| Large (normal) Fractions |
\dfrac{2}{4} = 0.5
|
\(\dfrac{2}{4} = 0.5\) |
| Large (nested) Fractions |
\cfrac{2}{c + \cfrac{2}{d + \cfrac{2}{4}}} = a
|
\(\cfrac{2}{c + \cfrac{2}{d + \cfrac{2}{4}}} = a\) |
| Binomial coefficients |
\binom{n}{k}
|
\(\binom{n}{k}\) |
| Small Binomial coefficients |
\tbinom{n}{k}
|
\(\tbinom{n}{k}\) |
| Large (normal) Binomial coefficients |
\dbinom{n}{k}
|
\(\dbinom{n}{k}\) |
| Matrices |
\begin{matrix}
x & y \\
z & v
\end{matrix}
|
\(\begin{matrix} x & y \\ z & v \end{matrix}\) |
\begin{vmatrix}
x & y \\
z & v
\end{vmatrix}
|
\(\begin{vmatrix} x & y \\ z & v \end{vmatrix}\) | |
\begin{Vmatrix}
x & y \\
z & v
\end{Vmatrix}
|
\(\begin{Vmatrix} x & y \\ z & v \end{Vmatrix}\) | |
\begin{bmatrix}
0 & \cdots & 0 \\
\vdots & \ddots & \vdots \\
0 & \cdots & 0
\end{bmatrix}
|
\(\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0\end{bmatrix} \) | |
\begin{Bmatrix}
x & y \\
z & v
\end{Bmatrix}
|
\(\begin{Bmatrix} x & y \\ z & v \end{Bmatrix}\) | |
\begin{pmatrix}
x & y \\
z & v
\end{pmatrix}
|
\(\begin{pmatrix} x & y \\ z & v \end{pmatrix}\) | |
\bigl( \begin{smallmatrix}
a&b\\ c&d
\end{smallmatrix} \bigr)
|
\( \bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr) \) | |
| Case distinctions |
f(n) =
\begin{cases}
n/2, & \mbox{if }n\mbox{ is even} \\
3n+1, & \mbox{if }n\mbox{ is odd}
\end{cases}
|
\(f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases} \) |
|
Multiline equations (use * for n\nonumber)
[4pt] or [5pt], etc., are used to space between lines |
\begin{align*}
f(x) & = (a+b)^2 \\[4pt]
& = a^2+2ab+b^2
\end{align*}
|
\( \begin{align*} f(x) & = (a+b)^2 \\[4pt]
&= a^2+2ab+b^2 \end{align*} \) |
| Multiline equations with explanations |
\begin{align*}
f(x) &= (a+b)^2 & &
\text{Original binomial squared}
\\[5pt]
&= a^2+2ab+b^2 & &\text{Expand.}
\end{align*}
|
\( \begin{align*} f(x) & = (a+b)^2 & & \text{Original binomial squared}\\[5pt]
&= a^2+2ab+b^2 & & \text{Expand.} \end{align*} \) |
| Multiline equations (must define number of colums used ({lcr}) (should not be used unless needed) |
\begin{array}{lcl}
z & = & a \\
f(x,y,z) & = & x + y + z
\end{array}
|
\(\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}\) |
| Multiline equations (more) |
\begin{array}{lcr}
z & = & a \\
f(x,y,z) & = & x + y + z
\end{array}
|
\(\begin{array}{lcr} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}\) |
| Breaking up a long expression so that it wraps when necessary |
<math>f(x) \,\!</math>
<math>= \sum_{n=0}^\infty a_n x^n </math>
<math>= a_0+a_1x+a_2x^2+\cdots</math>
|
\(f(x) \,\!\)\(= \sum_{n=0}^\infty a_n x^n \)\(= a_0 +a_1x+a_2x^2+\cdots\) |
| Simultaneous equations |
\begin{cases}
3x + 5y + z \\
7x - 2y + 4z \\
-6x + 3y + 2z
\end{cases}
|
\(\begin{cases} 3x + 5y + z \\ 7x - 2y + 4z \\ -6x + 3y + 2z \end{cases}\) |
| Long Division |
\require{enclose} \begin{array}{r}x+\phantom{0}4\hspace{.5em}\\[-3pt] x+5\enclose{longdiv}{x^2+\phantom{00}9x+20\phantom{0}}\\[-3pt] \underline{{\color{red}-}x^2+({\color{red}-}5x)}\hspace{2.1em}\\[-3pt] 4x+20\hspace{.5em}\\[-3pt] \underline{{\color{red}-}4x+({\color{red}-}20)}\\[-3pt] 0\hspace{.33em}\end{array} |
\(\require{enclose} \begin{array}{r}x+\phantom{0}4\hspace{.5em}\\[-3pt] x+5\enclose{longdiv}{x^2+\phantom{00}9x+20\phantom{0}}\\[-3pt] \underline{{\color{red}-}x^2+({\color{red}-}5x)}\hspace{2.1em}\\[-3pt] 4x+20\hspace{.5em}\\[-3pt] \underline{{\color{red}-}4x+({\color{red}-}20)}\\[-3pt] 0\hspace{.33em}\end{array}\) |
| Synthetic Division |
{\begin{array}{c}-2\\ \\ \\ \end{array}} {\begin{align*}&\\[0pt] &{\begin{array}{r|}\\[0pt] \\[0pt] \end{array}}\\[1pt] & \\[2pt]& \end{align*}}\!\! {\begin{array}{rrrr} 4 & 10 & -6 & -20\\ & -8 & -4 & 20 \\ \hline 4 & 2& -10 & 0 \end{array}} |
\({\begin{array}{c}-2\\ \\ \\ \end{array}}{\begin{align*}&\\[0pt]&{\begin{array}{r|}\\[0pt] \\[0pt] \end{array}}\\[1pt]& \\[2pt]& \end{align*}}\!\!
{\begin{array}{rrrr} 4 & 10 & -6 & -20\\ & -8 & -4 & 20 \\ \hline 4 & 2& -10 & 0 \end{array}}\) |
Chemistry Package
The Libretexts libraries have also installed the mhchem, a tool for writing beautiful chemical equations easily. The syntax for this powerful add-on can be found here: https://mhchem.github.io/MathJax-mhchem/


17.3.1: Glycolysis
If a person is in a catabolic state or in need of energy, such as during fasting, glucose will be used for glycolysis.
Glycolysis is the metabolic process of breaking down of one glucose molecule (6 carbons) into two pyruvate molecules (3 carbons). During this process, a net of two ATPs and two NADHs are also produced.
Figure <IP_ADDRESS> Glycolysis 1
3 steps of Glycolysis
1. Energy investment step - 2 ATP are added to the 6 carbon molecule.
Figure <IP_ADDRESS> Glycolysis step 1, energy investment 1
2. Glucose Split - The 6 carbon molecule is split into two 3 carbon molecules.
Figure <IP_ADDRESS> Glycolysis step 2, glucose split 1
3. Energy harvesting step - 1 NADH and 2 ATPs are produced from each 3 carbon molecule (there are two 3 carbon molecules formed from each glucose).
Figure <IP_ADDRESS> Glycolysis step 3, energy harvesting 1
Thus, from a molecule of glucose, the harvesting step produces a total of four ATPs and two NADHs. Subtracting the harvesting from the investment step, the net output from one molecule of glucose is two ATPs and two NADHs. 3
The figure below shows the stages of glycolysis, as well as the transition reaction, citric acid cycle, and electron transport chain that are utilized by cells to produce energy. They are also the focus of the next 3 sections.
Figure <IP_ADDRESS> Glycolysis, transition reaction, citric acid cycle, and the electron transport chain 2
Video Animation
If the above was 'too techy' for you, please watch this short video:
References & Links
- en.Wikipedia.org/wiki/File:Glycolysis.svg
- en.Wikipedia.org/wiki/File:CellRespiration.svg
- Berg JM, Tymoczko JL, Stryer L. Biochemistry. 5th edition. New York: W H Freeman; 2002. Section 16.1, Glycolysis Is an Energy-Conversion Pathway in Many Organisms. Available from: https://www.ncbi.nlm.nih.gov/books/NBK22593/


Skip to main content
24.10: Thinking about the material
-
-
Last updated
-
-
Save as PDF
-
Reflect and research
-
How did Michelson and Morley demonstrate that the ether does not exist?
-
Why is 1905 the “year of physics”?
-
Give an example of a device that you use that is affected by relativistic effects.
-
How do you resolve the twin paradox? Can you show it on a space-time diagram?
-
What did Lorentz do and when?
-
Apart from the space-time interval,
\(s\)
, what else is Lorentz invariant?
-
What is Cherenkov radiation?
To try at home
-
Build a particle accelerator.
-
Look up a video illustrating the barn paradox, and other relativistic effects.
To try in the lab
-
Propose an experiment to measure the speed of light.
-
Propose an experiment to test relativistic effects with electro magnetism.


15.21A: Rabies
- Examine the causes and symptoms associated with infection by the rabies virus
Rabies is a viral disease that causes acute encephalitis (inflammation of the brain) in warm-blooded animals. Rabies literally means “madness” in Latin. The disease is zoonotic and can be transmitted from one species to another, commonly by a bite from an infected animal. In humans, rabies is almost invariably fatal if postexposure prophylaxis is not administered prior to the onset of severe symptoms.
The rabies virus infects the central nervous system, travelling from the peripheral nerves to the brain. In humans, the incubation period between infection and the first sign of symptoms is typically two to 12 weeks, although periods as short as four days and longer than six years have been documented. Incubation period depends on the quantity of virus introduced and the distance it must travel to reach the central nervous system. Once there, symptoms begin to show and the infection is virtually untreatable. Early-stage symptoms include malaise, headache and fever, violent movements, uncontrolled excitement, depression, confusion, agitation, anxiety, and hydrophobia. Late stage symptoms extend to paranoia, terror, mania, and hallucinations progressing into delirium. Once symptoms have presented, survival is rare. Death almost invariably occurs within two to 10 days. Treatment with human rabies immunoglobulin (HRIG) and rabies vaccine is highly successful if administered before the onset of symptoms.
Rabies causes about 55,000 human deaths annually worldwide, with 95% of human deaths occurring in Asia and Africa. Roughly 97% of human rabies cases result from dog bites. In the U.S., animal control and vaccination programs have effectively eliminated domestic dogs as reservoirs of rabies. In several countries, including Australia and Japan, rabies carried by terrestrial animals has been eliminated entirely. While rabies was once eradicated in the United Kingdom, infected bats have recently been found in Scotland. In the U.S., the widespread vaccination of domestic dogs and cats and the development of effective human vaccines and immunoglobulin treatments has dropped the number of recorded human deaths from 100 or more annually in the early 20 th century, to one to two per year (mostly caused by bat bites). Modern cell-based vaccines are similar to flu shots in terms of pain and side effects. The old nerve-tissue-based vaccinations that require multiple painful injections into the abdomen with a large needle are cheap, but are being phased out and replaced by affordable World Health Organization intradermal vaccination regimens.
Rabies may be diagnosed by PCR or viral culture of brain samples after death, or from skin samples taken before. Diagnosis can be made from saliva, urine, and cerebrospinal fluid samples with less accuracy. Cheaper rabies diagnosis will become possible for low-income settings using basic light microscopy techniques.
Key Points
- The rabies virus infects the central nervous system, travelling from the peripheral nerves to the brain.
- Symptoms include hydrophobia, paranoia, terror, mania, hallucinations, and delirium.
- Once symptoms have presented, survival is rare, but treatment administered before the onset of symptoms is highly successful.
Key Terms
- hydrophobia : An aversion to water, as a symptom of rabies.
- zoonotic : of or relating to zoonosis, the transmission of an infectious disease between species.


1.1: Introduction
Maternal-newborn and women’s health nursing encompasses the lifespan of those assigned female at birth (AFAB). This care can focus on contraception and preconception care as well as preventive health care. The nurse can provide education on health promotion and disease prevention during an annual exam. For those persons who become pregnant, perinatal nurses provide education, support, referrals, and care in the office, the hospital, and the community. After the newborn arrives, the perinatal nurse cares for the parent-newborn couplet, providing breast-feeding education and support, parenting education, postpartum depression screening, and anticipatory guidance of the postpartum period.
Nurses learn the fundamentals of maternal-newborn and women’s health care to provide quality, safe care. Many nurses will care for a parent-baby couplet and ensure the antepartum, intrapartum, and postpartum person and newborn are transitioning well. Other nurses care for persons assigned female at birth (AFAB) throughout the lifespan in a gynecologic office.
Maternal-newborn and women’s health nurses are advocates for their patients. These nurses screen for intimate partner violence, human trafficking, substance use disorders, depression, and discrimination. The nurse advocates for persons of color, LGBTQIA+ (lesbian, gay, bisexual, transgender, queer, intersex, and asexual) persons, marginalized families, and people experiencing homelessness. Through patient circumstances involving ethical dilemmas, the nurse provides supportive, nonjudgmental care. Nurses also serve on hospital committees and work with organizations to influence policies and laws regarding health issues affecting persons AFAB.
Special populations require individualized care. The nurse tailors care to the individual needs of the person. The care plan is created with the person’s physical, mental, social, and emotional needs along with cultural and religious considerations. The care of persons AFAB is enhanced by the care provided by maternal-newborn and women’s health nurses.


5.4: Post-Lab 5 Questions
(3 point)
Last Name: _______________________ First Name: _______________________
-
List the structures in each layer of the eye. (0.5 points)
- Fibrous tunic:
- Vascular tunic:
- Neural tunic:
- What is the olfactory pathway, starting from odorant to the primary olfactory area? How does this pathway differ from other sensory pathways? (0.5 points)
-
Match the following structures with their corresponding descriptions. (1 point)
Name of Structure Descriptions No. of Structure 1. Optic disc an area where odorants bind to receptors to produce a sensation that will be perceived as smell 2. Round window contains approximately 100 taste buds 3. Fungiform papillae location of no visual activity, known as the “blind spot” 4. Vitreous humor contains the organs that sense dynamic equilibrium 5. Olfactory epithelium contains the organs that sense static equilibrium 6. Filiform papillae jelly-like mass that provides stability and structure to the eye 7. Semicircular canals provide friction, contains no taste buds 8. Retina contains approximately 5 taste buds 9. Auditory ossicles (malleus, incus, and stapes) the smallest bones in the body; transmits vibrations that are key to hearing 10. Circumvallate papillae possess the following layers to allow for the transmission of stimuli to the optic nerve; pigmented layer, photoreceptor layer, outer synaptic layer, bipolar cell layer, inner synaptic layer, ganglion layer 11. Vestibule membrane between the inner and middle ear to allow for pressure changes to equilibrate - Describe the path of sound traveling through the ear to CN (VIII). (list structures)(0.5 points)
-
Describe the function of the following muscles. Do they assist in intorsion, extortion, abduction, adduction, elevation and/or depression of the eye? (0.5 points)
- Superior Rectus-
- Inferior Rectus-
- Medial Rectus-
- Lateral Rectus-
- Superior Oblique-
- Inferior Oblique-


13.8: Summary of Phrases in Combination
13.8 Summary of Phrases in Combination
| Number of Phrases | Form | Melodic Scheme | Cadence Info |
| 2 phrases | Parallel Period | a a' |
Less conclusive then
more conclusive |
| Contrasting Period | a b |
Less conclusive then
more conclusive |
|
| Phrase Group | a a' | Ends with HC | |
| Phrase Chain | a b | Ends with HC | |
| Repeated Phrase | a a | Same cadence twice | |
| 3 phrases | Asymmetrical period |
a a b
or
a b b |
Ends with most
conclusive |
| Phrase group |
a a b
or
a b b |
Ends with HC | |
| Phrase Chain | a b c | Ends with HC | |
| 4 phrases | Double Period |
a b a b'
or
a b a c or a a' a a" or a a' a b |
HC–HC–HC–AC or
IAC–IAC–IAC–PAC or IAC–HC–IAC–PAC |
| Repeated period |
a b a b
or
a b a c or a a' a a' |
Less conclusive then
more conclusive formula repeated |


6.1.1: Relationships between Quantities
Lesson
Let's try to solve some new kinds of problems.
Exercise \(\PageIndex{1}\): Pricing Theater Popcorn
A movie theater sells popcorn in bags of different sizes. The table shows the volume of popcorn and the price of the bag.
Complete one column of the table with prices where popcorn is priced at a constant rate. That is, the amount of popcorn is proportional to the price of the bag. Then complete the other column with realistic example prices where the amount of popcorn and price of the bag are not in proportion.
| volume of popcorn (ounces) | price of bag, proportional ($) | price of bag, not proportional ($) |
|---|---|---|
| \(10\) | \(6\) | \(6\) |
| \(20\) | ||
| \(35\) | ||
| \(48\) |
Exercise \(\PageIndex{2}\): Entrance Fees
A state park charges an entrance fee based on the number of people in a vehicle.
- How much do you think a bus containing 30 people would be charged?
- If a bus is charged $122, how many people do you think it contains?
- What rule do you think the state park uses to decide the entrance fee for a vehicle?
Exercise \(\PageIndex{3}\): Making Toast
A toaster has 4 slots for bread. Once the toaster is warmed up, it takes 35 seconds to make 4 slices of toast, 70 seconds to make 8 slices, and 105 seconds to make 12 slices.
- How long do you think it will take to make 20 slices?
- If someone makes as many slices of toast as possible in 4 minutes and 40 seconds, how many slices do think they can make?
Are you ready for more?
What is the smallest number that has a remainder of 1, 2, and 3 when divided by 2, 3, and 4, respectively? Are there more numbers that have this property?
Summary
In much of our previous work that involved relationships between two quantities, we were often able to describe amounts as being so much more than another, or so many times as much as another. We wrote equations like \(x+3=8\) and \(4x=20\) and solved for unknown amounts.
In this unit, we will see situations where relationships between amounts involve more operations. For example, a pizza store might charge the amounts shown in the table for delivering pies.
| number of pies | total cost in dollars |
|---|---|
| \(1\) | \(13\) |
| \(2\) | \(23\) |
| \(3\) | \(33\) |
| \(5\) | \(53\) |
We can see that each additional pie adds $10 to the total cost, and that each total includes a $3 additional cost, maybe representing a delivery fee. In this situation, 8 pies will cost \(8\cdot 10+3\) and a total cost of $63 means 6 pies were ordered.
In this unit, we will see many situations like this one, and will learn how to use diagrams and equations to answer questions about unknown amounts.
Practice
Exercise \(\PageIndex{4}\)
Lin and Tyler are drawing circles. Tyler's circle has twice the diameter of Lin’s circle. Tyler thinks that his circle will have twice the area of Lin’s circle as well. Do you agree with Tyler?
(From Unit 3.2.2)
Exercise \(\PageIndex{5}\)
Jada and Priya are trying to solve the equation \(\frac{2}{3}+x=4\).
- Jada says, “I think we should multiply each side by \(\frac{3}{2}\) because that is the reciprocal of \(\frac{2}{3}\).”
- Priya says, “I think we should add \(-\frac{2}{3}\) to each side because that is the opposite of \(\frac{2}{3}\).”
- Which person’s strategy should they use? Why?
- Write an equation that can be solved using the other person’s strategy.
(From Unit 5.5.1)
Exercise \(\PageIndex{6}\)
What are the missing operations?
- \(48 ? (-8) = (-6)\)
- \((-40)?8=(-5)\)
- \(12?(-2)=14\)
- \(18?(-12)=6\)
- \(18?(-20)=-2\)
- \(22?(-0.5)=-11\)
(From Unit 5.4.1)
Exercise \(\PageIndex{7}\)
In football, the team that has the ball has four chances to gain at least ten yards. If they don't gain at least ten yards, the other team gets the ball. Positive numbers represent a gain and negative numbers represent a loss. Select all of the sequences of four plays that result in the team getting to keep the ball.
- \(8, -3, 4, 21\)
- \(30, -7, -8, -12\)
- \(2, 16, -5, -3\)
- \(5, -2, 20, -1\)
- \(20, -3, -13, 2\)
(From Unit 5.4.2)
Exercise \(\PageIndex{8}\)
A sandwich store charges $20 to have 3 turkey subs delivered and $26 to have 4 delivered.
- Is the relationship between number of turkey subs delivered and amount charged proportional? Explain how you know.
- How much does the store charge for 1 additional turkey sub?
- Describe a rule for determining how much the store charges based on the number of turkey subs delivered.
Exercise \(\PageIndex{9}\)
Which question cannot be answered by the solution to the equation \(3x=27\)?
- Elena read three times as many pages as Noah. She read 27 pages. How many pages did Noah read?
- Lin has 27 stickers. She gives 3 stickers to each of her friends. With how many friends did Lin share her stickers?
- Diego paid $27 to have 3 pizzas delivered and $35 to have 4 pizzas delivered. What is the price of one pizza?
- The coach splits a team of 27 students into 3 groups to practice skills. How many students are in each group?


20.24: Dietary Supplements
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Learning Objectives
- Dietary supplements and health risk behaviors
Research conducted by
Wen-Bin Chiou, Chao-Chin Yang, and Chin-Sheng Wan
Case study prepared by
Robert F. Houser and Georgette Baghdady
Overview
Although the dietary-supplement market in the U.S. is enormous, there is no apparent association between the use of dietary supplements and improved public health. The researchers of this study explored this paradox under the hypothesis that taking dietary supplements triggers a phenomenon called the “licensing effect,” namely, the tendency for positive choices to license subsequent self-indulgent, risky or unhealthful choices. The researchers hypothesized that supplement use confers “perceived health credentials,” leading people to feel invulnerable to health hazards and thus more likely to engage in risky, health-related behaviors.
The study involved two experiments. In the first experiment, \(82\) participants were randomly assigned to either a vitamin-pill (multivitamin) group or control (placebo) group and were told the kind of pill they would be taking. However, only the control group was given correct information. In actuality, both groups received the placebo pill. After taking the pills, the participants completed a survey on leisure-time activities, rating the desirability of nine hedonic (pleasurable) activities, such as excessive drinking and wild parties, and nine exercise activities, such as yoga and running, on \(7\)-point scales. The survey also included a general invulnerability scale to assess a participant’s perceived invulnerability to harm and disease. After completing the survey, the participants were offered a free lunch, choosing freely between a buffet and a healthful, organic meal.
The second experiment involved different participants. The vitamin-pill (multivitamin) group again unknowingly took placebo pills. After completing a questionnaire that included the general invulnerability scale and reading a medical report on the health benefits of walking, the distance participants walked in one hour was measured with a pedometer.
Questions to Answer
Does taking dietary supplements disinhibit unhealthy behaviors, such as eating unhealthful meals? Is the study sufficiently powered to detect significant differences between males and females?
Design Issues
The research was conducted in Taiwan, where cultural attitudes and behaviors related to dietary supplements may differ from those in the U.S. It is possible that the results might not generalize to other countries, so more research is needed. Participants in \(\text{Experiment 1}\) had a wide range in age, from \(18\) to \(46\) years, with a mean (\(SD\)) of \(30.9\) (\(7.8\)) years. It would be helpful to consider age in the analysis, especially if age is associated with invulnerability scores. Leisure-time activities and invulnerability were assessed only post-intervention; future studies should also measure these variables before the intervention to see if the two groups had similar scores at the start of the study. The general invulnerability scale used to assess perceived invulnerability to harm and disease has been validated only for adolescents.
Descriptions of Variables
| VARIABLE | DESCRIPTION |
| Experimental condition | Vitamin-pill (multivitamin) condition or Control (placebo) condition |
| Meal choice | Either a buffet meal or a healthful, organic meal |
| Gender | The sex of participants |
Links
The licensing effect
No Significant Difference … Says Who?
References
- Chiou, WB, Yang, CC, Wan, CS. (2011). Ironic effects of dietary supplementation: Illusory invulnerability created by taking dietary supplements licenses health-risk behaviors. Psychological Science, 22, 1081-1086.
- Trout, A. T., Kaufmann, T. J., Kallmes, D. F. (2007). No significant difference … Says who? Editorial. American Journal of Neuroradiology, 28, 195-197.


15.15: Gap 4- Service Delivery vs. External Communications
- Outline how a retailer can close the service delivery and communications gap
Gap between service delivery and external communication: Consumer expectations are highly influenced by statements made by company representatives and advertisements. The gap arises when these assumed expectations are not fulfilled at the time of delivery of the service. For example – a hospital printed on its brochure may have clean and furnished rooms but in reality, it may be poorly maintained – in this case, the patient’s expectations are not met.
The fourth gap in the model is the communication gap, reflecting the difference between the level of service promised and what is received. Consumer expectations are shaped by a range of communications including promotional advertising, website copy and photographs and statements made by company representatives. This gap is commonly due to over-promising or, at the extreme, false advertising (i.e., bait and switch). The classic “under promise and over deliver” advice is a way to avoid this gap. For perspective on this point, read marketing guru Tom Peter’s Under Promise, Over Deliver rant. Key takeaway: Quality, responsiveness and price are all important elements. What’s most important, however, is keeping your word. That will keep customers satisfied, and ensure that you keep your customers.
Specific actions to close this gap:
- Review communications across all touch-points, including digital (email, website, social media and partner sites), print, and on-premise materials, to ensure accuracy and consistency
- Implement a “reality check” review and approval process for new campaigns and updates.
- Conduct mock customer interactions and listen in and provide coaching on point-of-contact conversations to ensure services are being represented realistically
- Incorporate employee input and customer reviews to make advertising real. Similarly, incorporating actual employees and customers in advertising puts a real face on the business.
Contributors and Attributions
- Gap 4: Service Delivery vs. External Communications. Authored by : Nina Burokas. Provided by : Lumen Learning. License : CC BY: Attribution
- Introduction to Gap 4. Provided by : Boundless. Located at : https://courses.lumenlearning.com/boundless-marketing/chapter/service-quality/ . License : CC BY-SA: Attribution-ShareAlike


4.7: Discussion- The Strategy Behind Your Favorite Product
Post 1: Initial Post
Instructions
This posting should be a minimum of one short paragraph and a maximum of two paragraphs. Word totals for this post should be in the 100–200-word range. Whether you agree or disagree, explain why with supporting evidence and concepts from the readings or a related experience. Include a reference, link, or citation when appropriate.
Preparation
Think about a product or service you really love–something that you’ve recommended to your friends or family, or that you’d get excited talking about to a stranger at the bus stop. Maybe it’s LaCroix soft drinks, your most comfortable pair of shoes, the double-chocolate muffin at your favorite bakery, or the convenient rides you get through your Lyft app. With that product or service in mind, think about the business strategy behind it. Conduct research on the business, gathering as much information about its strategy as possible: visit its website, search for news stories talking about the product or service, and search social media for mentions of the business. While you won’t be able to learn everything about a business’s strategy that an insider would know, you should be able to get a general idea of how the business operates and what the industry environment is like.
For Discussion
Choose ONE of the following questions to respond to in your initial post.
- Identify the business’s main competitors. Explain how the business differentiates its product or service from those competitors’ offerings. Identify its competitive advantage. Do you think the advantage is sustainable? Why or why not?
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Conduct a customer analysis on the business’s product or service. Your analysis should include ALL of the following:
- Identify the target customer. Is this customer base growing or is it decreasing? What are the customer demographics (age, income, location, gender, politics, etc.)? What is the revenue of these customers? How much discretionary income do they have?
- Understand the specific customer needs. Why do they buy certain brands? How do make their purchasing decisions? Do they purchase in person or online?
- Show if and how the product or service you chose meets those needs.
- Evaluate the environment the business is operating in based on the four factors of change discussed in your text: stability, complexity, resource scarcity, and uncertainty. Based on your analysis of these factors, advise one or two areas where you believe the business should focus its strategy in the next two years.
Posts 2 and 3: Respond to Classmates’ Posts
Instructions
After you have created your initial post, look over the discussion posts of your classmates and give at least two thoughtful responses to two different classmates (one per classmate).
A response posting should be a minimum of one short paragraph. Word totals for these posts should be in the 75–100-word range. Whether you agree or disagree, explain why with supporting evidence and concepts from the readings or a related experience. Include a reference, link, or citation when appropriate. The goal of your response posts is to extend discussions already taking place or pose new possibilities or opinions not previously voiced. Your goal should be to motivate the group discussion and present a creative approach to the topic.
Contributors and Attributions
- Discussion: The Strategy Behind Your Favorite Product. Authored by : Lumen Learning. License : CC BY: Attribution


16.5: The Culturally Responsive Lens
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You can also use the Culturally Responsive Strength-Based (CRSB) Framework, which is used by Head Start, to address diversity. This framework presents the big picture and identifies the program pieces that support the growth and development of all children.
Figure 16.3: The elements of the CRSB Framework. [265]
Coupled with a culturally responsive approach, the CRSB Framework is a strength-based approach. The focus is on what children know and can do as opposed to what they cannot do or what they do not know. Cultural, family, and individual strengths are emphasized, not just the negative and proposed interventions to “fix the problem” that resides with the children, their families, and/or their communities.
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The strengths approach has a contagious quality and it intuitively makes sense to those who reflect a “cup half full” attitude in life. — Hamilton & Zimmerman, 2012 If we ask people to look for deficits, they will usually find them, and their view of the situation will be colored by this. If we ask people to look for successes, they will usually find it, and their view of the situation will be colored by this. — Kral, 1989 (as cited in Hamilton & Zimmerman, 2012) [266] |
The CRSB Framework should be used with the understanding that children are influenced by many environments, as represented in a bioecological systems model. A bioecological systems model captures the variety of environments that impact individual development over the course of a lifetime. Young children do not live in a vacuum, but co-exist in many environments that affect their development, starting with the family, extending into the community, and reaching out into the economic and political spheres.
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Think About It… There is also privilege surrounding socioeconomic status. How does the following excerpt change your perspective on future and current goals of children with different socioeconomic statuses? How will you as a teacher think about this? A study done by Dumais (2005) found that:
56% of parents from the lowest SES expected their children will get a bachelor’s degree, while 95% of those from the highest SES expected the same [267] |


20.5: Funding and Sustainability
By the end of this section, you should be able to:
- Define sustainability.
- Consider the influence of efficiency, value, and cost on program continuity and sustainability.
- Examine how external and internal funding streams and levels can influence program continuity.
- Explain the relevance of creating a sustainability plan early in the planning and implementation stages.
The goal of health promotion and disease prevention is to deliver community health programs that effectively and efficiently meet community needs. Funding and resources are required to deliver community health programs. The program team must address funding needs and program sustainability at the beginning of the program planning process and before the end of the initial funding cycle. Sustainability is the continuation of community health programs because the program is valued, cost and resource efficient, effective, and supported by the community (Georgia Health Policy Center, 2011; U.S. Department of Housing and Urban Development [HUD], 2012). The program team enhances sustainability by decreasing dependence on one funding source and shifting to one or more new funding streams. Sustainability goals may differ depending upon the developmental stage of the program. For example, newer programs focus on the continuation of activities once the initial funding stream ends, and experienced programs may focus on expanding program reach , building new partnership s, and promoting policy initiatives (HUD, 2012). Additionally, sustainability does not refer to the continuation of a program using the original strategies and activities. Most often sustainable programs evolve over time to adjust to changing funding, support, and community needs (Georgia Health Policy Center, 2011). Sustainability is influenced by (HUD, 2012):
- Efficiency and effectiveness of program activities
- Community support
- Partnerships that maximize use of resources
- Funding diversification
- Ability to draw new funds and partnerships
- Understanding of the overall cost savings associated with program implementation and improvement of community health status
External and Internal Funding Streams
The program team must identify funding to support new and continuing community health programs. The nurse must be aware of possible external and internal funding streams and have experience with soliciting funds for community health programs (see Table 20.9). Frequently, community health programs begin with funding support from one stream that does not continue throughout the program’s lifetime, so it is necessary to identify diverse funding streams to enhance program sustainability.
| Funding Stream | Description | Nurse’s Role in Acquiring Funds |
|---|---|---|
| Grants | Funds provided by a government source or private foundation for a specific purpose and for a specified time frame | The nurse may locate potential grants, write grant applications, communicate with funders, and carry out activities, or deliverables, specified within the grant. |
| Indirect funding | Resources, such as volunteers, equipment, materials, or funding, provided by an external source | The nurse may solicit or recruit volunteers, materials, etc. |
| Contributions or sponsorships | Businesses, social and civic clubs, churches, or individuals make financial or resource donations. |
The nurse may provide presentations or communicate with potential sponsors to solicit funds.
Sponsorships and contributions are often communicated as funders in program messaging and materials. |
| Government budgets | Local, state, and federal monies are factored into the annual budget and are set aside for community health program funding. The funds are usually acquired via taxes and fees to community residents. | The nurse may advocate for additional annual monies to be provided for community health programs and health promotion activities. |
| Fundraising events | Sporting events, such as golf outings or marathons, or lunch/dinner events are held to raise funds to support community health programs. | The nurse may plan and implement events. |
| Earned income | Funds earned through internal streams, such as fee for service, consultation, reimbursement from second party payers, or product sales are used to support community health programs. | The nurse may provide services that contribute to the internal streams, such as vaccination clinics. |
The nurse must understand how to procure funds for public health programs. This video introduces funding for public health projects and describes how funding agencies and funding cycles are structured.
Watch the video, and then respond to the following questions.
- Why should the nurse understand the funding agency’s objective before applying for a grant?
- What is the general process of grant application and funding?
Sustainability Planning
The program team begins intentionally planning for sustainability during program planning and development. The sustainability plan is supported by program evaluation and achievement of program objectives. The program team uses evaluation data to decide which program activities will continue following a grant period or when initial funds have been depleted (Georgia Health Policy Center, 2011). Sustainability efforts are strengthened if a program demonstrates effectiveness, positive results, strong leadership, and community engagement (HUD, 2012). Program growth and sustainability are promoted when the program team focuses on evaluation and continuous quality improvement , organizational capacity, expansion of partnerships, systems or policy change , identification of new funding streams, and diversification of sources of funding (RHIhub, 2022). Studies have shown that sustainable programs are adaptable, have leadership that develops a clear, strategic purpose for grants and commit to providing resources, include cross-sector partnerships, consider community needs and demand, use data to show program impact, and align with the current social and political environment (Georgia Health Policy Center, 2020).
The nurse and program team can use the Georgia Health Policy Center (2011) step-by-step workbook to guide sustainability planning. Generally, the program team answers the following questions when planning for sustainability (Georgia Health Policy, 2011; HUD, 2012):
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What criteria will be used to determine continuation? For example, criteria could include that the program
- meets community needs,
- is valued by the community,
- has positively impacted participants and the community,
- achieves objectives and goals,
- uses cost-effective program delivery,
- has a return on investment,
- has sufficient partner and community support, and
- can access needed resources.
- Has the program met sustainability criteria?
- What will continue to be offered—all or part of a program?
- Who is needed to continue the program (leadership, staffing, and partnerships)?
- Can support for the program and partnerships be expanded? How?
- What is the cost of the program (budgeted line items and overall cost)?
- What are current and potential funding strategies?
- Are the funding strategies diversified?
RHIhub (2022) offers several resources for sustainability planning , including toolkits for specific types of community health programs. For example, the maternal health toolkit lists potential federal, state, and other funding sources.
To successfully implement and sustain community health programs, the program team must consider facilitators and barriers to program implementation to plan interventions that will enhance facilitators and reduce barriers. Participant recruitment and retention is an important consideration. Without program participants, a program would not effectively meet its goals and objectives. Recruitment and retention are facilitated when sociocultural and linguistic needs are met. Systematic evaluation of the community health program provides evidence of efficacy, effectiveness, and efficiency in order to make decisions about the current and future programs. Program evaluation strategies depend upon the needs of the program team and community partners and on the developmental level of the program. Outcome evaluation should occur regardless of the developmental level. Most often process evaluation occurs along with outcome evaluation so that the program team can make inferences about outcome results. Effective communication, diverse funding streams, and sustainability planning are key in promoting and maintaining community health programs.


1.8: Vector-Valued Functions
Now that we are familiar with vectors and their operations, we can begin discussing functions whose values are vectors.
Definition 1.10
A vector-valued function of a real variable is a rule that associates a vector \(\textbf{f}(t)\) with a real number \(t\), where \(t\) is in some subset \(D\) of \(\mathbb{R}^1\) (called the domain of \(f\)). We write f : \(D → \)\(\mathbb{R}^ 3\) to denote that f is a mapping of \(D\) into \(\mathbb{R}^ 3\).
For example, \(\textbf{f}(t) = t\textbf{i}+ t^ 2 \textbf{j}+ t^ 3\textbf{k}\) is a vector-valued function in \(\mathbb{R}^ 3\), defined for all real numbers \(t\). We would write f : \(\mathbb{R} → \mathbb{R}^ 3\). At \(t = 1\) the value of the function is the vector i + j + k , which in Cartesian coordinates has the terminal point \((1,1,1)\).
A vector-valued function of a real variable can be written in component form as
\[\nonumber \textbf{f}(t) = f_1(t)\textbf{i}+ f_2(t)\textbf{j}+ f_3(t)\textbf{k}\]
or in the form
\[\nonumber \textbf{f}(t) = (f_1(t), f_2(t), f_3(t))\]
for some real-valued functions \(f_1(t), f_2(t), f_3(t)\), called the component functions of f . The first form is often used when emphasizing that \(\textbf{f}(t)\) is a vector, and the second form is useful when considering just the terminal points of the vectors. By identifying vectors with their terminal points, a curve in space can be written as a vector-valued function.
Example 1.35
Define f : \(\mathbb{R} → \mathbb{R}^ 3\) by \(\textbf{f}(t) = (\cos t,\sin t,t)\). This is the equation of a helix (see Figure 1.8.1). As the value of \(t\) increases, the terminal points of \(\textbf{f}(t)\) trace out a curve spiraling upward. For each \(t\), the \(x- \text{ and }y-\)coordinates of \(\textbf{f}(t) \text{ are }x = \cos t \text{ and }y = \sin t\), so
\[\nonumber x^2 + y^ 2 = \cos^2 t+\sin^2 t = 1\]
Thus, the curve lies on the surface of the right circular cylinder \(x^ 2 + y^ 2 = 1\).
It may help to think of vector-valued functions of a real variable in \(\mathbb{R}^ 3\) as a generalization of the parametric functions in \(\mathbb{R}^ 2\) which you learned about in single-variable calculus. Much of the theory of real-valued functions of a single real variable can be applied to vector-valued functions of a real variable. Since each of the three component functions are real-valued, it will sometimes be the case that results from single-variable calculus can simply be applied to each of the component functions to yield a similar result for the vector-valued function. However, there are times when such generalizations do not hold (see Exercise 13). The concept of a limit, though, can be extended naturally to vector-valued functions, as in the following definition.
Definition 1.11
Let \(\textbf{f}(t)\) be a vector-valued function, let \(a\) be a real number and let c be a vector. Then we say that the limit of \(\textbf{f}(t)\) as \(t\) approaches \(a\) equals c , written as \(\lim\limits_{t \to a} \textbf{f}(t) = \textbf{c}\), if \(\lim\limits_{t \to a} \norm{\textbf{f}(t)−\textbf{c}} = 0\). If \(\textbf{f}(t) = (f_1(t), f_2(t), f_3(t))\), then
\[\nonumber \lim\limits_{t \to a}\textbf{f}(t) = \left ( \lim\limits_{t \to a}f_1(t),\,\lim\limits_{t \to a}f_2(t),\,\lim\limits_{t \to a}f_3(t)\right ) \]
provided that all three limits on the right side exist.
The above definition shows that continuity and the derivative of vector-valued functions can also be defined in terms of its component functions.
Definition 1.12
Let \(\textbf{f}(t) = (f_1(t), f_2(t), f_3(t))\) be a vector-valued function, and let \(a\) be a real number in its domain. Then \(\textbf{f}(t)\) is continuous at \(a\) if \(\lim\limits_{t \to a} \textbf{f}(t) = \textbf{f}(a)\). Equivalently, \(\textbf{f}(t)\) is continuous at \(a\) if and only if \(f_1(t), f_2(t), \text{ and }f_3(t)\) are continuous at \(a\).
The derivative of \(\textbf{f}(t)\) at \(a\), denoted by \(\textbf{f}′(a)\) or \(\dfrac{d\textbf{f}}{ dt} (a)\), is the limit
\[\nonumber \textbf{f} ′ (a) = \lim\limits_{h \to 0} \dfrac{\textbf{f}(a+ h)−\textbf{f}(a)}{ h}\]
if that limit exists. Equivalently, \(\textbf{f} ′ (a) = (f_1 ′ (a), f_2 ′ (a), f_3 ′ (a))\), if the component derivatives exist. We say that \(\textbf{f}(t)\) is differentiable at \(a\) if \(\textbf{f} ′ (a)\) exists.
Recall that the derivative of a real-valued function of a single variable is a real number, representing the slope of the tangent line to the graph of the function at a point. Similarly, the derivative of a vector-valued function is a tangent vector to the curve in space which the function represents, and it lies on the tangent line to the curve (see Figure 1.8.2).
Example 1.36
Let \(\textbf{f}(t) = (\cos t,\sin t,t)\). Then \(\textbf{f} ′ (t) = (−\sin t,\cos t,1) \text{ for all }t\). The tangent line \(L\) to the curve at \(\textbf{f}(2π) = (1,0,2π) \text{ is }L = \textbf{f}(2π)+ s \textbf{f} ′ (2π) = (1,0,2π)+ s(0,1,1)\), or in parametric form: \(x = 1, y = s, z = 2π+ s \text{ for }−∞ < s < ∞\).
A scalar function is a real-valued function. Note that if \(u(t)\) is a scalar function and \(\textbf{f}(t)\) is a vector-valued function, then their product, defined by (\(u\textbf{f})(t) = u(t)\textbf{f}(t)\) for all \(t\), is a vector-valued function (since the product of a scalar with a vector is a vector).
The basic properties of derivatives of vector-valued functions are summarized in the following theorem.
Theorem 1.20
Let \(\textbf{f}(t) \text{ and }\textbf{g}(t)\) be differentiable vector-valued functions, let \(u(t)\) be a differentiable scalar function, let \(k\) be a scalar, and let c be a constant vector. Then
- \(\dfrac{d}{ dt} (\textbf{c}) = \textbf{0}\)
- \(\dfrac{d}{ dt} (k\textbf{f}) = k \dfrac{d\textbf{f}}{ dt}\)
- \(\dfrac{d}{ dt} (\textbf{f}+\textbf{g}) = \dfrac{d\textbf{f}}{ dt} + \dfrac{d\textbf{g}}{ dt}\)
- \(\dfrac{d}{ dt} (\textbf{f−g}) = \dfrac{d\textbf{f}}{ dt} − \dfrac{d\textbf{g}}{ dt}\)
- \(\dfrac{d}{ dt} (u\textbf{f}) = \dfrac{du}{ dt} \textbf{f} + u \dfrac{d\textbf{f}}{ dt}\)
- \(\dfrac{d}{ dt} (\textbf{f}· \textbf{g}) = \dfrac{d\textbf{f}}{ dt} · \textbf{g} + \textbf{f}· \dfrac{d\textbf{g}}{ dt}\)
- \(\dfrac{d}{ dt} (\textbf{f} × \textbf{g}) = \dfrac{d\textbf{f}}{ dt} × \textbf{g} + \textbf{f} × \dfrac{d\textbf{g}}{ dt}\)
Proof
The proofs of parts (a)-(e) follow easily by differentiating the component functions and using the rules for derivatives from single-variable calculus. We will prove part (f), and leave the proof of part (g) as an exercise for the reader.
(f) Write \(\textbf{f}(t) = (f_1(t), f_2(t), f_3(t)) \text{ and }\textbf{g}(t) = (g_1(t), g_2(t), g_3(t))\), where the component functions \(f_1(t), f_2(t), f_3(t), g_1(t), g_2(t), g_3(t)\) are all differentiable real-valued functions. Then
\[\nonumber \begin{align}\dfrac{d}{dt} (\textbf{f}(t)· \textbf{g}(t))&=\dfrac{d}{ dt} (f_1(t) g_1(t)+ f_2(t) g_2(t)+ f_3(t) g_3(t)) \\[4pt] \nonumber &= \dfrac{d}{ dt} (f_1(t) g_1(t))+ \dfrac{d}{ dt} (f_2(t) g_2(t))+ \dfrac{d}{ dt} (f_3(t) g_3(t)) \\[4pt] \nonumber &=\dfrac{d f_1}{ dt} (t) g_1(t)+ f_1(t)\dfrac{ d g_1}{ dt} (t)+ \dfrac{d f_2}{ dt} (t) g_2(t)+ f_2(t) \dfrac{d g_2}{ dt} (t)+ \dfrac{d f_3}{ dt} (t) g_3(t)+ f_3(t) \dfrac{d g_3}{ dt} (t) \\[4pt] \nonumber &=\left ( \dfrac{d f_1}{ dt} (t),\dfrac{d f_2}{ dt} (t), \dfrac{d f_3}{ dt} (t) \right ) · (g_1(t), g_2(t), g_3(t)) \\[4pt] \nonumber &\quad +(f_1(t), f_2(t), f_3(t))· \left ( \dfrac{d g_1}{ dt} (t), \dfrac{d g_2}{ dt} (t), \dfrac{d g_3}{ dt} (t)\right ) \\[4pt] \nonumber &=\dfrac{d\textbf{f}}{ dt} (t)· \textbf{g}(t) + \textbf{f}(t)· \dfrac{d\textbf{g}}{ dt} (t) \text{ for all }t.\tag{\(\textbf{QED}\)} \\[4pt] \end{align}\]
\(\square\)
Example 1.37
Suppose \(\textbf{f}(t)\) is differentiable. Find the derivative of \(\norm{\textbf{f}(t)}\).
Since \(\norm{\textbf{f}(t)}\) is a real-valued function of \(t\), then by the Chain Rule for real-valued functions, we know that \(\dfrac{d}{ dt} \norm{\textbf{f}(t)}^2 = 2\norm{\textbf{f}(t)} \dfrac{d}{ dt} \norm{\textbf{f}(t)}\).
But \(\norm{\textbf{f}(t)}^ 2 = \textbf{f}(t)·\textbf{f}(t)\), so \(\dfrac{d}{ dt} \norm{\textbf{f}(t)}^2 = \dfrac{d}{ dt} (\textbf{f}(t)·\textbf{f}(t))\). Hence, we have
\[\nonumber \begin{align} 2\norm{\textbf{f}(t)} \dfrac{d}{ dt} \norm{\textbf{f}(t)} &= \dfrac{d}{ dt} (\textbf{f}(t)·\textbf{f}(t)) = \textbf{f} ′ (t)·\textbf{f}(t) + \textbf{f}(t)·\textbf{f} ′ (t)\text{ by Theorem 1.20(f), so} \\[4pt] \nonumber &= 2\textbf{f} ′ (t)·\textbf{f}(t) , \text{ so if }\norm{\textbf{f}(t)} \neq 0 \text{ then} \\[4pt] \nonumber \dfrac{d}{ dt} \norm{\textbf{f}(t)} &= \dfrac{\textbf{f} ′ (t)·\textbf{f}(t)}{ \norm{\textbf{f}(t)}} \\[4pt] \end{align}\]
We know that \(\norm{\textbf{f}(t)}\) is constant if and only if \(\dfrac{d}{ dt} \norm{\textbf{f}(t)} = 0\) for all \(t\). Also, \(\textbf{f}(t) ⊥ \textbf{f} ′ (t) \text{ if and only if }\textbf{f} ′ (t)·\textbf{f}(t) = 0\). Thus, the above example shows this important fact:
If \(\norm{\textbf{f}(t)} \neq 0, \text{ then }\norm{\textbf{f}(t)} \text{ is constant if and only if }\textbf{f}(t) ⊥ \textbf{f} ′ (t)\text{ for all }t\).
This means that if a curve lies completely on a sphere (or circle) centered at the origin, then the tangent vector \(\textbf{f} ′ (t)\) is always perpendicular to the position vector \(\textbf{f}(t)\).
Example 1.38
The spherical spiral \(\textbf{f}(t) = \left ( \dfrac{\cos t}{ \sqrt{ 1+ a^ 2t^ 2}} , \dfrac{\sin t}{ \sqrt{ 1+ a^ 2t^ 2}} , \dfrac{−at}{ \sqrt{ 1+ a^ 2t^ 2}} \right ) \), for \(a \neq 0\).
Figure 1.8.3 shows the graph of the curve when \(a = 0.2\). In the exercises, the reader will be asked to show that this curve lies on the sphere \(x^ 2 + y^ 2 + z^ 2 = 1\) and to verify directly that \(\textbf{f} ′ (t)·\textbf{f}(t) = 0 \text{ for all }t\).
Just as in single-variable calculus, higher-order derivatives of vector-valued functions are obtained by repeatedly differentiating the (first) derivative of the function:
\[\nonumber \textbf{f} ′′(t) = \dfrac{d}{ dt} \textbf{f} ′ (t) ,\qquad \textbf{f} ′′′(t) = \dfrac{d}{ dt} \textbf{f} ′′(t) , ... , \dfrac{d^ n \textbf{f}}{ dt^n} = \dfrac{d}{ dt} \left ( \dfrac{d^{n−1}\textbf{f}}{ dt^{n−1}} \right ) \text{ (for \(n = 2,3,4,...\))}\]
We can use vector-valued functions to represent physical quantities, such as velocity, acceleration, force, momentum, etc. For example, let the real variable \(t\) represent time elapsed from some initial time (\(t = 0\)), and suppose that an object of constant mass \(m\) is subjected to some force so that it moves in space, with its position (\(x, y, z\)) at time \(t\) a function of \(t\). That is, \(x = x(t), y = y(t), z = z(t)\) for some real-valued functions \(x(t), y(t), z(t)\). Call \(\textbf{r}(t) = (x(t), y(t), z(t))\) the position vector of the object. We can define various physical quantities associated with the object as follows:
\[\nonumber \begin{align}\textit{position}:\, \textbf{r}(t) &= (x(t), y(t), z(t)) \\[4pt] \nonumber\textit{velocity}:\, \textbf{v}(t) &= \dot{\textbf{r}}(t) = \textbf{r} ′ (t) = \dfrac{d\textbf{r}}{ dt} \\[4pt] \nonumber &= (x ′ (t), y ′ (t), z ′ (t)) \\[4pt] \nonumber \textit{acceleration}:\, \textbf{a}(t) &= \dot{\textbf{v}}(t) = \textbf{v} ′ (t) = \dfrac{d\textbf{v}}{ dt} \\[4pt] \nonumber &=\ddot{\textbf{r}}(t) = \textbf{r} ′′(t) = \dfrac{d^ 2 \textbf{r}}{ dt^2} \\[4pt] \nonumber &=(x ′′(t), y ′′(t), z ′′(t)) \\[4pt] \nonumber \textit{momentum}:\, \textbf{p}(t) &= m\textbf{v}(t) \\[4pt] \nonumber \textit{force}:\, \textbf{F}(t) &= \dot{\textbf{p}} (t) = \textbf{p} ′ (t) = \dfrac{d\textbf{p}}{ dt} \text{ (Newton’s Second Law of Motion)} \\[4pt] \end{align}\]
The magnitude \(\norm{\textbf{v}(t)}\) of the velocity vector is called the speed of the object. Note that since the mass \(m\) is a constant, the force equation becomes the familiar \(\textbf{F}(t) = m\textbf{a}(t)\).
Example 1.39
Let \(\textbf{r}(t) = (5\cos t,3\sin t,4\sin t)\) be the position vector of an object at time \(t ≥ 0\). Find its (a) velocity and (b) acceleration vectors.
Solution
- \(\textbf{v}(t) = \dot{\textbf{r}}(t) = (−5\sin t,3\cos t,4\cos t)\)
- \(\textbf{a}(t) = \dot{\textbf{v}}(t) = (−5\cos t,−3\sin t,−4\sin t)\)
Note that \(\norm{\textbf{r}(t)} = \sqrt{ 25\cos^2 t+25\sin^2 t} = 5\) for all \(t\), so by Example 1.37 we know that \(\textbf{r}(t)· \dot{\textbf{r}}(t) = 0 \text{ for all }t\) (which we can verify from part (a)). In fact, \(\norm{\textbf{v}(t)} = 5\) for all \(t\) also. And not only does \(\textbf{r}(t)\) lie on the sphere of radius 5 centered at the origin, but perhaps not so obvious is that it lies completely within a circle of radius 5 centered at the origin. Also, note that \(\textbf{a}(t) = −\textbf{r}(t)\). It turns out (see Exercise 16) that whenever an object moves in a circle with constant speed, the acceleration vector will point in the opposite direction of the position vector (i.e. towards the center of the circle).
Recall from Section 1.5 that if \(\textbf{r}_1 , \textbf{r}_2\) are position vectors to distinct points then \(\textbf{r}_1+t(\textbf{r}_2−\textbf{r}_1)\) represents a line through those two points as \(t\) varies over all real numbers. That vector sum can be written as \((1 − t)\textbf{r}_1 + t\textbf{r}_2\). So the function \(\textbf{l}(t) = (1 − t)\textbf{r}_1 + t\textbf{r}_2\) is a line through the terminal points of \(\textbf{r}_1 \text{ and }\textbf{r}_2\), and when \(t\) is restricted to the interval \([0,1]\) it is the line segment between the points, with \(\textbf{l}(0) = \textbf{r}_1 \text{ and }\textbf{l}(1) = \textbf{r}_2\).
In general, a function of the form \(\textbf{f}(t) = (a_1 t+b_1 ,a_2 t+b_2 ,a_3 t+b_3)\) represents a line in \(\mathbb{R}^ 3\). A function of the form \(\textbf{f}(t) = (a_1 t^ 2 + b_1 t + c_1 ,a_2 t^ 2 + b_2 t + c_2 ,a_3 t^ 2 + b_3 t + c_3)\) represents a (possibly degenerate) parabola in \(\mathbb{R}^ 3\).
Example 1.40: Bézier curves
Bézier curves are used in Computer Aided Design (CAD) to approximate the shape of a polygonal path in space (called the Bézier polygon or control polygon ). For instance, given three points (or position vectors) \(\textbf{b}_0 , \textbf{b}_1 , \textbf{b}_2 \text{ in }\)\(\mathbb{R}^ 3\), define
\[\nonumber \begin{align} \textbf{b}_0^1 (t) &= (1− t)\textbf{b}_0 + t\textbf{b}_1 \\[4pt] \nonumber \textbf{b}_1^1 (t) &= (1− t)\textbf{b}_1 + t\textbf{b}_2 \\[4pt] \nonumber \textbf{b}_0^2 (t) &= (1− t)\textbf{b}_0^1 (t)+ t\textbf{b}_1^1 (t) \\[4pt] \nonumber &=(1− t)^2 \textbf{b}_0 +2t(1− t)\textbf{b}_1 + t^2\textbf{b}_2 \\[4pt] \end{align}\]
for all real \(t\). For \(t\) in the interval \([0,1]\), we see that \(\textbf{b}_0^1 (t)\) is the line segment between \(\textbf{b}_0 \text{ and }\textbf{b}_1\), and \(\textbf{b}_1^1 (t)\) is the line segment between \(\textbf{b}_1 \text{ and }\textbf{b}_2\). The function \(\textbf{b}_0^2 (t)\) is the Bézier curve for the points \(\textbf{b}_0 , \textbf{b}_1 , \textbf{b}_2\). Note from the last formula that the curve is a parabola that goes through \(\textbf{b}_0\) (when \(t = 0\)) and \(\textbf{b}_2\) (when \(t = 1\)).
As an example, let \(\textbf{b}_0 = (0,0,0),\, \textbf{b}_1 = (1,2,3), \text{ and }\textbf{b}_2 = (4,5,2)\). Then the explicit formula for the Bézier curve is \(\textbf{b}_0^2 (t) = (2t +2t^ 2 ,4t + t^ 2 ,6t −4t^ 2 )\), as shown in Figure 1.8.4, where the line segments are \(\textbf{b}_0^1 (t)\text{ and }\textbf{b}_1^1 (t)\), and the curve is \(\textbf{b}_0^2 (t)\).
In general, the polygonal path determined by \(n ≥ 3\) noncollinear points in \(\mathbb{R}^ 3\) can be used to define the Bézier curve recursively by a process called repeated linear interpolation . This curve will be a vector-valued function whose components are polynomials of degree \(n − 1\), and its formula is given by de Casteljau’s algorithm . In the exercises, the reader will be given the algorithm for the case of \(n = 4\) points and asked to write the explicit formula for the Bézier curve for the four points shown in Figure 1.8.5.


2.10: Overview- Reading and Comprehension
Now that you have been briefly introduced to scholarly writing, it is important to reflect on your own reading and comprehension skills—these skills are important to develop in order to advance your scholarly writing capacity.
Comprehension is an important part of reading so that you understand and grasp the key ideas. Many strategies are available to help you read so that you optimize your comprehension. Some strategies involve taking certain steps before you start reading a text. Others involve taking notes and learning to read critically. In this chapter, you will learn about these strategies and how to critique a text and reflect and summarize as part of your journey to grasping the ideas.
Let’s begin!