11.4: Art of Southeast Asia
Sculpture in Southeast Asia
The sculpture of Southeast Asia was heavily influenced by Indian religions and artistic styles.
Describe the Buddhist and Indian influences on Southeast Asian sculpture from 300–600 CE
Key Points
- The communities and cultures of Southeast Asia were in direct contact with India through trade routes. Several Indic kingdoms competed for dominance in the region between the 1st and 8th centuries CE, particularly the Cambodian Funan and the Burmese Mon .
- Most of the Southeast Asian sculpture of the period 300–600 CE was heavily influenced by the style of the Gupta Empire in India, which patronized Buddhist art in the Greco–Buddhist style.
- Southeast Asian Buddha statues of this period were characterized by a purity of statuary and a delicacy in portraying the folds of clothing. Somewhat less attention was paid to the realism of artistic details, and symbolic, shell-like curls were used to render the hair of the Buddha.
- The Indic civilization of Champa flourished along the coasts of what is now central and southern Vietnam from 500 CE onward and left an impressive artistic legacy of sandstone sculptures, both in the round and in relief .
- In later periods, Chinese influences predominated in Vietnam, Laos, and Cambodia.
Key Terms
- Indic : Pertaining to India or its people or culture; Indian.
- Mon : A people living primarily in Myanmar (also known as Burma) and Thailand; one of the earliest peoples to reside in Southeast Asia, they are responsible for the spread of Buddhism in Burma and Thailand.
Overview: Influences on Southeast Asia
The communities and cultures of Southeast Asia were in direct contact with India through trade routes and were heavily influenced by Indian religion and art. The Pali and Sanskrit languages, Indian script, and Hindu epic literature—such as the Ramayana and the Mahabharata—were transmitted to Southeast Asia during this period.
Hinduism and Buddhism were brought to the region and became the main religions practiced from about the 1st century BCE to the 13th century CE. These influences played a considerable role in shaping the art and sculpture of Southeast Asia.
Sculpture in Southeast Asia
Between the 1st and 8th centuries CE, several Indic kingdoms competed for dominance in Southeast Asia, particularly the Cambodian Funan and the Burmese Mon. Most of the Southeast Asian sculpture of the period 300–600 CE was heavily influenced by the style of the Gupta Empire in India, which patronized Buddhist art in the Greco–Buddhist style.
Buddhist art in Thailand was shaped both by direct contact with Indian traders and the expansion of the Mon kingdom. In later periods, Chinese influences predominated in Vietnam, Laos, and Cambodia, and more wooden sculpture survives from across the region.
Buddhist Sculptures
Southeast Asian Buddha statues of this period were characterized by a purity of statuary and a delicacy in portraying the folds of clothing. Symbolic, shell-like curls were used to render the hair of the Buddha. Somewhat less attention was paid to the realism of artistic details. A number of votive tablets and Sanskrit inscriptions are also found in the region.
Champa
The Indic civilization of Champa flourished along the coasts of what is now central and southern Vietnam from 500 CE onward. This civilization left an impressive artistic legacy of sandstone sculptures, both in the round and in relief. These sculptures expressed religious themes and synthesized elements of Hinduism, Buddhism, and indigenous cults. They depicted common themes, such as Hindu and Buddhist deities and icons , as well as scenes from daily life.
The Cham created freestanding sandstone sculptures in the round, as well as high and bas-relief carvings of sandstone. In general, they appear to have preferred sculpting in relief, and they excelled at sculpture in high relief .
Cham sculpture went through a marked succession of historical styles, the foremost of which produced some of the best works of Southeast Asian art. The subject-matter of Cham sculpture is drawn mostly from the legends and religion of Indian civilization. Many of the sculptures are representations of particular Hindu and Buddhist deities, most prominently Siva but also Lokesvara, Visnu, Brahma, Devi, and Shakti. Such sculptures may have served a religious purpose rather than being purely decorative.
Painting in Southeast Asia
Southeast Asian painting from 300–600 CE mostly took the form of frescoes and reflected Hindu and Buddhist themes.
Compare and contrast the Hindu and Buddhist art of Southeast Asia from 300–600 CE.
Key Points
- Very little Southeast Asian painting from 300–600 CE has survived to the present day, owing to the heat and humidity of tropical and subtropical weather.
- Artists worked in many perishable mediums, painting mostly on wood, cloth, and palm leaf . Frescoes , usually executed on cave temple or monastery walls, are the most common form of Southeast Asian painting to have survived.
- Hinduism and Buddhism were introduced to Southeast Asia during the 1st century BCE. There were also several powerful Indic kingdoms in the region, which practiced Hinduism and Buddhism and produced art that reflected their religious beliefs.
- Hindu art commonly depicted figures from the Hindu pantheon that includes Shiva, Vishnu, and Shakti. Buddhist art depicted images of the Buddha, the Bodhisattvas or enlightened beings, apsaras or celestial dancers, and tales and parables from Buddhist lore, including the Jataka tales .
- The most famous surviving examples of Southeast Asian- style frescoes are found in the rock fortress and palace ruins of Sigiriya in Sri Lanka. These date from about the 5th–6th centuries CE and depict graceful female figures bearing flowers.
Key Terms
- pantheon : All the gods of a particular people or religion, considered as a group.
Southeast Asian Painting from 300–500 CE
Very little Southeast Asian painting from 300–600 CE has survived to the present day, owing to the heat and humidity of tropical and subtropical weather. One can only hypothesize the styles and techniques that painters would have used based on evidence gleaned from sculptures (which are far more durable and have survived), contemporary painting styles in India (which played a large role in influencing Southeast Asian art), and literary texts that talk about painting. The few examples of painting that do survive are frescoes on cave or temple walls.
Approaches
Artists worked in perishable mediums, painting mostly on wood, cloth, and palm leaf, none of which have withstood the rigors of the Southeast Asian climate. The most durable forms of Southeast Asian art are sculpture and architecture in stone.
It is likely that stone sculptures, both in relief and in the round , were originally painted in bright colors, but these have worn away over the course of time, leaving the underlying stone exposed. Frescoes, usually executed on cave temple or monastery walls, were the most common form of Southeast Asian painting.
Themes
The themes most commonly depicted would probably mimic those found in contemporary Southeast Asian sculpture—primarily religious themes from Hindu and Buddhist mythology. Hinduism and Buddhism, both of which originated in the Indian subcontinent, were introduced to Southeast Asia from the 1st century BCE to the 1st century CE.
Southeast Asia subsequently came under the influence of several powerful Indic dynasties , which established kingdoms, practiced and spread Hinduism and Buddhism through the region, and patronized art that reflected their religious beliefs.
Hindu art commonly depicts figures from the Hindu pantheon, including the gods Shiva and Vishnu and the divine female creative principle or Shakti. Buddhist art depicted images of the Buddha, the Bodhisattvas or enlightened beings, apsaras or celestial dancers, and tales and parables from Buddhist lore, including the Jataka tales—stories about the previous incarnations of the Buddha, both in human and in animal form.
It is also possible that Southeast Asian painting would have depicted court and battle scenes, animals both real and mythical, and scenes from daily life. In Thai art, the most frequent narrative subjects for paintings included the Jataka stories, episodes from the life of the Buddha, the Buddhist heavens and hells, and scenes of daily life. Some of the scenes are influenced by Thai folklore instead of following strict Buddhist iconography .
Frescoes in Sigiriya
The most famous surviving examples of Southeast Asian-style frescoes are to be found in the rock fortress and palace ruin of Sigiriya in Sri Lanka. These date from about the 5th–6th centuries CE and depict graceful female figures bearing flowers. These figures are hypothesized to be apsaras , or women of the king’s court.
The paint is applied in sweeping strokes, using more pressure on one side than the other and results in deep colors toward the edge. These frescoes are reminiscent of the contemporary frescoes in the Ajanta caves in India, which are masterpieces of Buddhist religious art and depict figures from the Buddhist pantheon and scenes from the Jataka tales.
Northern-Style Temples in Southeast Asia
North Indian temples are often simpler and less orthodox than those in South, and all people are permitted to worship within them.
Distinguish the simplicity of northern-style Nagara temple from the complexity of the southern Dravida temple
Key Points
- India’s temple architecture is developed from the creativity of Sthapathis and Shilpis, both of whom belong to the larger community of craftsmen and artisans called Vishwakarma .
- A small Hindu temple consists of an inner sanctum; the garbha graha or womb-chamber in which the idol or deity is housed; a congregation hall; and sometimes an antechamber and porch.
- In contrast to the elaborate appearance of South Indian temples, most temples in North India are very simple in design. This is thought to be due, at least in part, to the attacks that Hindus have historically suffered from Muslims in this region.
- Darśana is a Sanskrit term meaning sight, vision, apparition, or glimpse; North Indian temples have a specific hall where devotees can receive this blessing.
- Śikhara, a Sanskrit word that means mountain peak, refers to the tallest tower in Northern Hindu temple architecture.
- The Somnath Mandir, a temple in Gujarat, is considered to be one of the twelve jyotirlinga shrines of Lord Shiva and has a history dating as far back as the beginning of the common area.
Key Terms
- circumambulation : The act of walking around something in a circle, especially for a ritual purpose.
- Vishwakarma : A group that is sometimes referred to as a caste and which comprises five sub-groups (carpenters, blacksmiths, bell metalworkers, goldsmiths, and stonemasons) who describe themselves as descendants of the Hindu god by the same name.
- sikhara : A Sanskrit word that means mountain peak, refers to the rising tower in the Hindu temple architecture of North India.
Introduction
India’s temple architecture was developed from the creativity of Sthapathis and Shilpis, both of whom belong to the larger community of craftsmen and artisans called Vishwakarma. A small Hindu temple consists of an inner sanctum; the garbha graha or womb-chamber in which the idol or deity is housed; a congregation hall; and sometimes an antechamber and porch.
The garbhagriha is crowned by a tower-like shikara. At the turn of the first millennium CE, two major types of temples existed:
- The northern or Nagara style .
- The southern or Dravida type of temple.
The Northern Style
In contrast to the elaborate appearance of South Indian temples, most temples in North India are very simple in design. This is thought to be due, at least in part, to the constant attacks Hindus have suffered from Muslims historically in this region. North Indian temples also tend to be less orthodox than those in the south. In many cases, all castes and categories of people are permitted to enter the innermost sanctum of these temples and worship the deity personally. However, in such cases, the deities are not adorned with valuable jewelry.
The innermost heart of this type of temple is a sanctum where a deity (usually cast in fixed stone) is present, followed by a large hall where lay worshipers can stand and obtain darśana, or divine audience.
Darśana is a Sanskrit term meaning sight, vision, apparition, or glimpse, and is most commonly used for visions of the divine in Hindu worship. These visions are either of a deity, a very holy person, or an artifact . An individual can receive darśana, or the blessing of a particular deity, within the temple, or from a saintly person, such as a great guru.
In this type of temple, there may or may not be a number of additional corridors and halls, in addition to these aforementioned worship rooms. However, in all cases, there will be space for devotees to circulate the temple in a clockwise fashion. In Hindu culture , this kind of circumambulation is a mark of respect. Many of these temples were destroyed during the Islamic rule of India during the Mughal Empire .
Śikhara, a Sanskrit word that means mountain peak, refers to the tallest tower in Northern Hindu temple architecture. Because the sikhara tower is always situated over the sanctum sanctorum, where the presiding deity is enshrined, it is the most prominent and visible part of this temple’s architecture. Originally, the sikharas were homogeneous in design but, over time, secondary sikharas, which are smaller and narrower, have been plated on to the sides of many of these main sikharas.
Notable Northern Temples
The Somnath Mandir, a temple in Gujarat, is considered to be one of the twelve jyotirlinga shrines of Lord Shiva and has a history dating as far back as the beginning of the common area. Delhi’s Chhatarpur Temple, notable for its size, is one of the largest Hindu temple complexes in India.
Southern-Style Temples in Southeast Asia
South India gave rise to the Dravida style of architecture and is where most of the largest Hindu temples are found.
Identify the primary temple styles of the various regions of India and southeast Asia
Key Points
- The main architectural styles of Indian temples are the Dravida style of South India and the Nagara style of North India.
- The Dravida style of architecture consists primarily of temples with pyramid-shaped towers constructed of sandstone, soapstone, or granite.
- Most of the largest Hindu temples are found in South India, and specifically in Tamil Nadu. Many large bannabs (grand stone temples) still stand in South India.
- Temple ritual tends to be orthodox and elaborate, especially in the large vedic brahmincal temples, which follow the pan-Indian Sanskrit agama scriptural traditions.
- Apart from the main fixed stone deities , processional deities made of panchaloha (an alloy of gold, silver, copper, zinc, and tin) are bathed, dressed, and decorated with valuables and then taken out in processions for festivals throughout the year.
- The richer the temple, the more elaborate the festivals. However, many ancient temples in small villages with great architectural and historical heritage value languish for lack of funds for maintenance.
Key Terms
- murtis : Images that express a divine spirit.
- dravida : The main architectural style in South India.
Introduction
A Hindu temple is a place of worship for followers of Hinduism . A characteristic of most temples is the presence of murtis (statues) of the Hindu deity to whom the temple is dedicated. They are usually dedicated to one primary deity, the presiding deity, and other deities associated with the main deity. However, some temples are dedicated to several deities, and others are dedicated to murtis in an iconic form .
Many temples are in key geographical points, such as on a hilltop or near waterfalls, caves, and rivers. The main architectural styles in India are the Nagara style of North India and the Dravida style of South India.
The Dravidian Style
Dravidian architecture was an architectural idiom that emerged in the Southern part of the Indian subcontinent, or South India. This style of architecture consists primarily of temples with pyramid-shaped towers constructed of sandstone, soapstone, or granite.
Dravidian-style temples consist almost invariably of the three following parts, arranged in differing manners according to the age in which they were executed.
- The porches, or mantapas, which always cover and precede the door leading to the cell.
- Gate-pyramids, or gopuras, which are the principal features in the quadrangular enclosures that surround the more notable temples.
- Pillared halls (chaultris or chawadis), which are used for many purposes and are the invariable accompaniments of these temples.
In addition to these, a South Indian temple usually has a tank, called the kalyani or pushkarni, to be used for sacred purposes or for the convenience of the priests. Dwellings for the priesthood are typically attached to this kalyani.
Rituals within these temples tend to be orthodox and elaborate, especially in the large vedic brahminical temples that follow the pan-Indian Sanskrit agama scriptural traditions. Apart from the main fixed stone deities, processional deities made of panchaloha (an alloy of gold, silver, copper, zinc, and tin) are bathed, dressed, decorated with valuables, and taken out in processions for festivals throughout the year.
The richer the temple, the more elaborate the festivals. However, many ancient temples in small villages with great architectural and historical heritage value languish for lack of funds for maintenance.
Notable Temples in South India
Most of the largest Hindu Temples are found in South India, and specifically in Tamil Nadu. Many large bannabs (grand stone temples) still stand in South India. Famous South Indian temples include the Tirumala Venkateswara Temple in Andhra Pradesh, the Guruvayur temple in Kerala, the Chettikulangara Devi temple in Kerala, among others.
Monumental Reliefs in Southeast Asia
Sculpture and architecture were intimately connected in Southeast Asia, and monumental reliefs were used to decorate the walls of buildings.
Discuss the techniques, themes, and common subjects of the monumental reliefs of Borobudur and the Khmer temples
Key Points
- Relief is a sculptural technique that gives the impression that the sculpted material has been raised above the background plane .
- Relief sculptures were used to decorate the walls of buildings, particularly Hindu and Buddhist temples, and were very large.
- In Southeast Asia, this style of sculpture was greatly influenced by Indian techniques and generally portrayed religious themes with high iconographic precision. Other themes included court and battle scenes, scenes depicting daily life and the customs of the people, and animals both real and mythical.
- Most ancient Southeast Asian relief sculpture was done in bas-relief , where the projecting images have shallow overall depth; however, the kingdom of Champa in southern and central Vietnam excelled in haut-relief sculpture, which was marked by much greater depth and undercut areas.
- The most famous examples of monumental relief sculptures in Southeast Asia are to be found in the 9th-century Buddhist temple of Borobudur in Java, Indonesia.
- The Khmer of Cambodia were also renowned for their monumental bas-reliefs, which usually took narrative form to depict stories from history and mythology. The most famous example of Khmer sculpture is the 12th-century Hindu temple of Angkor Wat .
Key Terms
- iconographic : Related to the branch of art history that studies the identification, description, and interpretation of the content of images, including the subjects depicted, the particular compositions and details used to do so, and other elements that are distinct from artistic style.
- bas-relief : A low or mostly flat sculpture that is carved into a wall or is in the form of a tile mounted flat to a wall, rather than a fully three-dimensional, free-standing figure.
- sunken relief : A type of sculpture largely restricted to the art of Ancient Egypt, in which the image is made by cutting the sculpture itself into a flat surface.
Overview: Relief Sculpture
Relief is a sculptural technique that gives the impression that the sculpted material has been raised above the background plane. This is accomplished by cutting into a flat surface of stone or wood, thereby lowering the field and leaving the unsculpted parts seemingly raised.
Reliefs depicting figures that are at least life-size or bigger or are attached to monuments of some sort are termed monumental reliefs by art historians, thus distinguishing them from small metal or ivory reliefs, portable sculptures, and diptychs.
Monumental Reliefs in Southeast Asia
Monumental reliefs represent an important facet of ancient Southeast Asian art, where sculpture and architecture were intimately connected with one another. As a result, relief sculpture was generally used to decorate the walls of buildings—particularly Hindu and Buddhist temples—and was accomplished on a very large scale.
It was greatly influenced by Indian styles and techniques, and it generally portrayed religious themes with high iconographic precision. Court and battle scenes, scenes depicting daily life and the customs of the people, and animals (both real and mythical) were other common subjects.
Most of ancient Southeast Asian relief sculpture was done in bas-relief, where the projecting images have shallow overall depth, although the kingdom of Champa in southern and central Vietnam excelled in haut-relief sculpture, which was marked by much greater depth and undercut areas.
Notable examples of monumental reliefs include Borobodur in Java, Angkor Wat in Cambodia, the Sanchi case in India, and many South Indian temples, including the Unakoti group of sculptures at Kailashahar, Tripura, India.
Borobudur
The most famous examples of monumental relief sculptures in Southeast Asia are to be found in the 9th-century Buddhist temple of Borobudur in Java, Indonesia. Built during the rein of the Indic Sailendra Dynasty , the temple is constructed to reveal many different levels of terraces, many of which are heavily ornamented with intricate bas-reliefs.
In total, there are 2,672 individual bas-reliefs, 1,460 of which depict narratives from Buddhist lore, including the birth and life of the Buddha. The remaining 1,212 are solely decorative.
Khmer Temples
The Khmer of Cambodia were also renowned for their monumental bas-reliefs, which usually took narrative form to depict stories from history and mythology. They decorated the tympana (semi-circular arched spaces above a doorway), walls, and ceilings with complex scenes.
The earliest surviving example of Khmer narrative bas-relief sculpture comes from the 10th-century Hindu temple of Banteay Srei, which has carved tympana and towers that depict scenes from the great Hindu epics—the Ramayana and the Mahabharata.
The most famous example of Khmer bas-relief sculpture is undoubtedly at the 12th-century Hindu temple of Angkor Wat, which has 13,000 square meters of narrative bas-reliefs on the walls of its outer gallery.
Themes
The reliefs have a diverse range of themes. They depict mythical, spiritual beings from the Buddhist pantheon such as bodhisattvas , apsaras (celestial dancers or musicians), gandharvas (male nature spirits), and asuras (demons). They also depict images of people, such as the king and queen, princes, priests, courtiers, soldiers, servants, and commoners.
Many of these reliefs provide glimpses of scenes of daily life; for example, the relief sculptures from Borobudur depict scenes from 8th-century Java, including courtly palace life, a hermit in the forest, commoners in the village, temple and marketplace scenes, native vernacular architecture , and flora and fauna. These bas-relief sculptures have served as a reference for historians in the study of ancient Javanese architecture, weaponry, fashion, and transportation.
In addition, a group of 160 panels of monumental relief sculpture provides a complete illustration of the law of karma or the principles of cause and effect. There are depictions of both praiseworthy activities (including charity and pilgrimage) and blameworthy activities (ranging from gossip to murder), with their corresponding rewards and punishments. These panels provide particularly complex scenes of daily life, depicting the full panorama of samsara (the endless cycle of birth and death).
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2.3: Tangent Plane to a Surface
In the previous section we mentioned that the partial derivatives \(\dfrac{∂f}{∂x}\) and \(\dfrac{∂f}{∂y}\) can be thought of as the rate of change of a function \(z = f (x, y)\) in the positive \(x\) and \(y\) directions, respectively. Recall that the derivative \(\dfrac{dy}{dx}\) of a function \(y = f (x)\) has a geometric meaning, namely as the slope of the tangent line to the graph of \(f\) at the point \((x, f (x))\) in \(\mathbb{R}^2\). There is a similar geometric meaning to the partial derivatives \(\dfrac{∂f}{∂x}\) and \(\dfrac{∂f}{∂y}\) of a function \(z = f (x, y)\): given a point \((a,b)\) in the domain \(D\) of \(f (x, y)\), the trace of the surface described by \(z = f (x, y)\) in the plane \(y = b\) is a curve in \(\mathbb{R}^ 3\) through the point \((a,b, f (a,b))\), and the slope of the tangent line \(L_x\) to that curve at that point is \(\dfrac{∂f}{∂x} (a,b)\). Similarly, \(\dfrac{∂f}{∂y} (a,b)\) is the slope of the tangent line \(L_y\) to the trace of the surface \(z = f (x, y)\) in the plane \(x = a\) (see Figure 2.3.1).
Since the derivative \(\dfrac{dy}{dx}\) of a function \(y = f (x)\) is used to find the tangent line to the graph of \(f\) (which is a curve in \(\mathbb{R}^2\)), you might expect that partial derivatives can be used to define a tangent plane to the graph of a surface \(z = f (x, y)\). This indeed turns out to be the case. First, we need a definition of a tangent plane. The intuitive idea is that a tangent plane “just touches” a surface at a point. The formal definition mimics the intuitive notion of a tangent line to a curve.
 Let \(T\) be a plane which contains the point \(P\), and let \(Q = (x, y, z)\) represent a generic point on the surface \(S\). If the (acute) angle between the vector \(\vec{PQ}\) and the plane \(T\) approaches zero as the point \(Q\) approaches \(P\) along the surface \(S\), then we call \(T\) the tangent plane to \(S\) at \(P\).
Note that since two lines in \(\mathbb{R}^ 3\) determine a plane, then the two tangent lines to the surface \(z = f (x, y)\) in the \(x\) and \(y\) directions described in Figure 2.3.1 are contained in the tangent plane at that point, if the tangent plane exists at that point . The existence of those two tangent lines does not by itself guarantee the existence of the tangent plane. It is possible that if we take the trace of the surface in the plane \(x − y = 0\) (which makes a 45◦ angle with the positive \(x\)-axis), the resulting curve in that plane may have a tangent line which is not in the plane determined by the other two tangent lines, or it may not have a tangent line at all at that point. Luckily, it turns out that if \(\dfrac{∂f}{∂x}\) and \(\dfrac{∂f}{∂y}\) exist in a region around a point \((a,b)\) and are continuous at \((a,b)\) then the tangent plane to the surface \(z = f (x, y)\) will exist at the point \((a,b, f (a,b))\). In this text, those conditions will always hold.
Suppose that we want an equation of the tangent plane \(T\) to the surface \(z = f (x, y)\) at a point \((a,b, f (a,b))\). Let \(L_x\) and \(L_y\) be the tangent lines to the traces of the surface in the planes \(y = b\) and \(x = a\), respectively (as in Figure 2.3.2), and suppose that the conditions for \(T\) to exist do hold. Then the equation for \(T\) is
\[A(x− a)+B(y− b)+C(z − f (a,b)) = 0 \label{Eq2.4}\]
where \(n = (A,B,C)\) is a normal vector to the plane \(T\). Since \(T\) contains the lines \(L_x\) and \(L_y\), then all we need are vectors \(v_x\) and \(v_y\) that are parallel to \(L_x \text{and } L_y\), respectively, and then let \(n = v_x × v_y\).
Since the slope of \(L_x\) is \(\dfrac{∂f}{∂x} (a,b)\), then the vector \(v_x = (1,0, \dfrac{∂f}{∂x} (a,b))\) is parallel to \(L_x\) (since \(v_x\) lies in the \(xz\)-plane and lies in a line with slope \(\dfrac{\dfrac{∂f}{∂x}(a,b)}{1} = \dfrac{∂f}{∂x} (a,b)\). See Figure 2.3.3). Similarly, the vector \(v_y = (0,1, \dfrac{∂f}{∂y} (a,b))\) is parallel to \(L_y\). Hence, the vector
\[\textbf{n} = \mathbf{v_x} × \mathbf{v_y} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\[4pt] 1 & 0 & \dfrac{∂f}{∂x}(a,b) \\[4pt] 0 & 1 & \dfrac{∂f}{∂y} (a,b) \end{vmatrix} =-\dfrac{∂f}{∂x}(a,b)\textbf{i}-\dfrac{∂f}{∂y}(a,b)\textbf{j}+\textbf{k} \nonumber \]
is normal to the plant \(T\). Thus the equation of \(T\) is
\[-\dfrac{∂f}{∂x}(a,b)(x-a)-\dfrac{∂f}{∂y}(a,b)(y-b)+z-f(a,b)=0 \label{Eq2.5}\]
Multiplying both sides by −1, we have the following result:
The equation of the tangent plane to the surface \(z = f (x, y)\) at the point \((a,b, f (a,b))\) is
\[\dfrac{∂f}{∂x}(a,b)(x-a)+\dfrac{∂f}{∂y}(a,b)(y-b)-z+f(a,b)=0 \label{Eq2.6}\]
Example 2.13
Find the equation of the tangent plane to the surface \(z = x^2 + y^2\) at the point (1,2,5).
Solution
For the function \(f (x, y) = x^2 + y^2\), we have \(\dfrac{∂f}{∂x} = 2x\) and \(\dfrac{∂f}{∂y} = 2y\), so the equation of the tangent plane at the point \((1,2,5)\) is
\[\nonumber 2(1)(x−1)+2(2)(y−2)− z +5 = 0 \text{ , or }\]
\[\nonumber 2x+4y− z −5 = 0\]
In a similar fashion, it can be shown that if a surface is defined implicitly by an equation of the form \(F(x, y, z) = 0\), then the tangent plane to the surface at a point \((a,b, c)\) is given by the equation
\[\dfrac{∂F}{∂x}(a,b, c)(x− a)+ \dfrac{∂F}{∂y}(a,b, c)(y− b)+ \dfrac{∂F}{∂z}(a,b, c)(z − c) = 0 \label{Eq2.7}\]
Note that Equation \ref{Eq2.6} is the special case of Equation \ref{Eq2.7} where \(F(x, y, z) = f (x, y)− z\).
Example 2.14
Find the equation of the tangent plane to the surface \(x^2 + y^2 + z^2 = 9\) at the point (2,2,−1).
For the function \(F(x, y, z) = x^2 + y^2 + z^2 −9\), we have \(\dfrac{∂F}{∂x} = 2x\), \(\dfrac{∂F}{∂y} = 2y\), and \(\dfrac{∂F}{∂z} = 2z\), so the equation of the tangent plane at (2,2,−1) is
\[\nonumber 2(2)(x−2)+2(2)(y−2)+2(−1)(z +1) = 0 \text{, or}\]
\[\nonumber 2x+2y− z −9 = 0\]


1.5: Social Constructionism
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- Miliann Kang, Donovan Lessard, Laura Heston, and Sonny Nordmarken
- University of Massachusetts Amherst via UMass Amherst Libraries
Social constructionism is a theory of knowledge that holds that characteristics typically thought to be immutable and solely biological—such as gender, race, class, ability, and sexuality—are products of human definition and interpretation shaped by cultural and historical contexts (Subramaniam 2010). As such, social constructionism highlights the ways in which cultural categories—like “men,” “women,” “black,” “white”—are concepts created, changed, and reproduced through historical processes within institutions and culture. We do not mean to say that bodily variation among individuals does not exist, but that we construct categories based on certain bodily features, we attach meanings to these categories, and then we place people into the categories by considering their bodies or bodily aspects. For example, by the one-drop rule (see also page 35), regardless of their appearance, individuals with any African ancestor are considered black. In contrast, racial conceptualization and thus racial categories are different in Brazil, where many individuals with African ancestry are considered to be white. This shows how identity categories are not based on strict biological characteristics, but on the social perceptions and meanings that are assumed. Categories are not “natural” or fixed and the boundaries around them are always shifting—they are contested and redefined in different historical periods and across different societies. Therefore , the social constructionist perspective is concerned with the meaning created through defining and categorizing groups of people, experience, and reality in cultural contexts.
The Social Construction of Heterosexuality
What does it mean to be “heterosexual” in contemporary US society? Did it mean the same thing in the late 19th century? As historian of human sexuality Jonathon Ned Katz shows in The Invention of Heterosexuality (1999), the word “heterosexual” was originally coined by Dr. James Kiernan in 1892, but its meaning and usage differed drastically from contemporary understandings of the term. Kiernan thought of “hetero-sexuals” as not defined by their attraction to the opposite sex, but by their “inclinations to both sexes.” Furthermore, Kiernan thought of the heterosexual as someone who “betrayed inclinations to ‘abnormal methods of gratification’” (Katz 1995). In other words, heterosexuals were those who were attracted to both sexes and engaged in sex for pleasure, not for reproduction. Katz further points out that this definition of the heterosexual lasted within middle-class cultures in the United States until the 1920s, and then went through various radical reformulations up to the current usage.
Looking at this historical example makes visible the process of the social construction of heterosexuality. First of all, the example shows how social construction occurs within institutions—in this case, a medical doctor created a new category to describe a particular type of sexuality, based on existing medical knowledge at the time. “Hetero-sexuality” was initially a medical term that defined a deviant type of sexuality. Second, by seeing how Kiernan—and middle class culture, more broadly—defined “hetero-sexuality” in the 19th century, it is possible to see how drastically the meanings of the concept have changed over time. Typically, in the United States in contemporary usage, “heterosexuality” is thought to mean “normal” or “good”—it is usually the invisible term defined by what is thought to be its opposite, homosexuality. However, in its initial usage, “hetero-sexuality” was thought to counter the norm of reproductive sexuality and be, therefore, deviant. This gets to the third aspect of social constructionism. That is, cultural and historical contexts shape our definition and understanding of concepts. In this case, the norm of reproductive sexuality—having sex not for pleasure, but to have children—defines what types of sexuality are regarded as “normal” or “deviant.” Fourth, this case illustrates how categorization shapes human experience, behavior, and interpretation of reality. To be a “heterosexual” in middle class culture in the US in the early 1900s was not something desirable to be—it was not an identity that most people would have wanted to inhabit. The very definition of “hetero-sexual” as deviant, because it violated reproductive sexuality, defined “proper” sexual behavior as that which was reproductive and not pleasure-centered.
Social constructionist approaches to understanding the world challenge the essentialist or biological determinist understandings that typically underpin the “common sense” ways in which we think about race, gender, and sexuality. Essentialism is the idea that the characteristics of persons or groups are significantly influenced by biological factors, and are therefore largely similar in all human cultures and historical periods. A key assumption of essentialism is that “a given truth is a necessary natural part of the individual and object in question” (Gordon and Abbott 2002).  In this example, “sexual orientation” is a given “truth” to individuals—it is thought to be inherent, biologically determined, and essential to their being.
Essentialism typically relies on a biological determinist theory of identity. Biological determinism can be defined as a general theory, which holds that a group’s biological or genetic makeup shapes its social, political, and economic destiny (Subramaniam 2014). For example, “sex” is typically thought to be a biological “fact,” where bodies are classified into two categories, male and female. Bodies in these categories are assumed to have “sex”-distinct chromosomes, reproductive systems, hormones, and sex characteristics. However, “sex” has been defined in many different ways, depending on the context within which it is defined. For example, feminist law professor Julie Greenberg (2002) writes that in the late 19th century and early 20th century, “when reproductive function was considered one of a woman’s essential characteristics, the medical community decided that the presence or absence of ovaries was the ultimate criterion of sex” (Greenberg 2002: 113). Thus, sexual difference was produced through the heteronormative assumption that women are defined by their ability to have children. Instead of assigning sex based on the presence or absence of ovaries, medical practitioners in the contemporary US typically assign sex based on the appearance of genitalia.
Differential definitions of sex point to two other primary aspects of the social construction of reality. First, it makes apparent how even the things commonly thought to be “natural” or “essential” in the world are socially constructed. Understandings of “nature” change through history and across place according to systems of human knowledge. Second, the social construction of difference occurs within relations of power and privilege. Sociologist Abby Ferber (2009) argues that these two aspects of the social construction of difference cannot be separated, but must be understood together. Discussing the construction of racial difference, she argues that inequality and oppression actually produce ideas of essential racial difference. Therefore, racial categories that are thought to be “natural” or “essential” are created within the context of racialized power relations—in the case of African-Americans, that includes slavery, laws regulating interracial sexual relationships, lynching, and white supremacist discourse. Social constructionist analyses seek to better understand the processes through which racialized, gendered, or sexualized differentiations occur, in order to untangle the power relations within them.
Notions of disability are similarly socially constructed within the context of ableist power relations. The medical model of disability frames body and mind differences and perceived challenges as flaws that need fixing at the individual level. The social model of disability shifts the focus to the disabling aspects of society for individuals with impairments (physical, sensory or mental differences), where the society disables those with impairments (Shakespeare 2006). Disability, then, refers to a form of oppression where individuals understood as having impairments are imagined to be inferior to those without impairments, and impairments are devalued and unwanted. This perspective manifests in structural arrangements that limit access for those with impairments. A critical disability perspective critiques the idea that nondisability is natural and normal—an ableist sentiment, which frames the person rather than the society as the problem.
What are the implications of a social constructionist approach to understanding the world? Because social constructionist analyses examine categories of difference as fluid, dynamic, and changing according to historical and geographical context, a social constructionist perspective suggests that existing inequalities are neither inevitable nor immutable. This perspective is especially useful for the activist and emancipatory aims of feminist movements and theories. By centering the processes through which inequality and power relations produce racialized, sexualized, and gendered difference, social constructionist analyses challenge the pathologization of minorities who have been thought to be essentially or inherently inferior to privileged groups. Additionally, social constructionist analyses destabilize the categories that organize people into hierarchically ordered groups through uncovering the historical, cultural, and/or institutional origins of the groups under study. In this way, social constructionist analyses challenge the categorical underpinnings of inequalities by revealing their production and reproduction through unequal systems of knowledge and power.


5.8: Relationship Between Financial and Managerial Accounting
Learning Outcomes
- Analyze the relationship between financial and managerial accounting and how they are compartmentalized within modern businesses
Companies need to take a look at their numbers in a variety of ways to be successful. Financial accounting records are required, and are the basis for what is used in managerial accounting. In order to have anything to use for decision making, it is necessary to have historical data, right?
Companies who have been around a while will use historical data for budgeting purposes, while new companies need to prepare budgets and business plans based on what they hope to happen. As mentioned earlier, most smaller businesses will have only a one or a few people in their accounting department who perform the tasks of both financial accounting (historical) and managerial accounting (future). But larger companies will have separate departments that handle these tasks.
The financial and managerial teams need to work together to create the information needed for planning, controlling and decision making within a company. If they don’t work together, neither team will have what they need to create a profitable and successful company! The financial folks need budget numbers from the managerial folks to enter into the accounting software. Then they have the tools needed for comparison to determine if the company is meeting goals.
The managerial team needs historical data from the financial team to prepare their reports. Without that data, it would be impossible to determine which product lines are doing well, when changes need to be made, or how budgets come together. Although the two types of accounting are very different from each other, they work together to create a “strategy” or game plan that fosters this success.
Often falling under one heading of “The Accounting Department,” these two very different tasks are needed to insure success of companies, large or small.
Practice Questions


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Key Concepts
3.1
Prime and Composite Numbers
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The natural numbers can be categorized as 1, prime numbers, and composite numbers.
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Prime numbers have as their only factors 1 and themselves.
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Composite numbers have at least three distinct factors.
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Composite numbers can be written in their prime factorization form, which is found by repeatedly factoring prime factors from the number.
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The greatest common divisor (GCD) of a set of numbers is the largest integer that divides all of the numbers in the set. The prime factorizations of the numbers can be used to identify the greatest common divisor.
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The least common multiple (LCM) of a set of numbers is the smallest integer that is divisible by all of the numbers in the set. The prime factorizations of the numbers can be used to identify the least common multiple.
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There are various ways that the GCD and LCM are applied.
3.2
The Integers
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A set of numbers that can be built from the natural numbers are the integers, which consist of the natural numbers, zero (0), and the negatives of the natural numbers.
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Integers are often graphed on a number line, which helps display the relative positions and values of those numbers.
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The number line can be used to visualize when one integer is larger than or smaller than another integer.
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Arithmetic operations with integers are similar to the operations with natural numbers, except that the sign (positive or negative) of the numbers will determine the sign (positive or negative) of the result.
3.3
Order of Operations
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Establishing shared rules on which arithmetic operations are calculated first is necessary. Without them, different people may find different values for the same expression.
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The highest precedence is with expressions in parentheses. This allows parts of an expression to be calculated in an order different than the basic order of operations.
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The lowest precedence is addition and subtraction, as they are the basis for all other calculations.
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Multiplication and division have precedence over addition and subtraction, as they are representations of repeated addition or subtraction.
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Exponents have precedence over multiplication and division, as they represent repeated multiplication and division.
3.4
Rational Numbers
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Rational numbers are fractions of integers, and can always be written as an integer divided by an integer.
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The numerator and denominator of a fraction may have common factors. In such cases, the fraction can be reduced by canceling common factors. When the numerator and denominator of a fraction have no common factors, the fraction is said to be reduced.
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An improper fraction is one with a numerator larger than the denominator. Such a fraction can be rewritten as an integer plus a proper fraction. This is called a mixed number.
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Using division and remainder, an improper fraction may be written as a mixed number.
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A mixed number can be converted to an improper fraction by reversing the process for changing an improper fraction to a mixed number.
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The arithmetic operations or addition, subtraction, multiplication and division can all be performed on rational numbers.
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Addition and subtraction of rational numbers can be performed after a common denominator has been identified, and the fractions have been converted to forms having the common denominator.
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Multiplication and division of rational numbers can be performed without regard to common denominators.
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Between any two rational numbers, there is always another rational number. This is the density property of the rational numbers.
3.5
Irrational Numbers
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Irrational numbers are numbers that cannot be written as an integer divided by another integer. One example is pi, denoted
. Another collection of irrational numbers are natural numbers that are not perfect squares.
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Some irrational numbers can be written as a rational part multiplied by an irrational part. If two irrational numbers have the same irrational parts, they can be added or subtracted.
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When irrational numbers are similar, on can multiply and divide the numbers without a calculator.
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Since
, and
, products and quotients of square roots can be determined.
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Because
and
, it is possible to simplify square root expressions so the radicand contains no perfect square factors.
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When a fraction has an irrational number as its denominator, it is possible to convert the denominator into a rational number using its conjugate. Doing so involves multiplying the numerator and denominator by the conjugate of the denominator, and then applying the difference of squares formula.
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With a single square root term
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Using conjugate numbers for two term denominators
3.6
Real Numbers
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Real numbers is the collection of all rational and irrational numbers. Conceptually, it is the collection of all values that can be represented on a number line, or, as a length along with sign.
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The subsets of the real numbers include the natural numbers, integers, rational numbers and irrational numbers. The natural numbers are a subset of the integers, which is a subset of the rational numbers. The rational and irrational numbers are disjoint sets.
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The real numbers, due to order of operation rules and that performing arithmetic operations on real number always results in a real number, have arithmetic properties that apply in all cases. There include the distributive property, the commutative property, and the associative property. Also, every real number has an additive inverse and, except for zero (0), have a multiplicative inverse.
3.7
Clock Arithmetic
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Clock arithmetic uses the idea that after 12 o’clock comes 1 o’clock. For clock arithmetic, this means that every time 12 is passed in an arithmetic process, the next number is 1, not 13.
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To determine the clock result of an arithmetic operation, divide the final result by 12 and keep the remainder. If the remainder is 0, then the time is 12 o’clock.
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Clock arithmetic is technically called modulo 12 arithmetic. To perform modulo 12 arithmetic, calculate the expression, then divide the result by 12. The modulo 12 result is the remainder.
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Days, in our system, pass in groups of seven. To calculate in day arithmetic, modulo 7 is used. To perform modulo 7 arithmetic, calculate the expression, then divide the result by 7. The modulo 7 result is the remainder.
3.8
Exponents
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Exponents are used to express multiplying a number by itself a number of times. The number being multiplied by itself is the base. The number of times it is multiplied by itself is the exponent, which is often referred to as the power.
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Understanding that exponents represent repeated multiplication of a base makes it possible to establish some rules for combining exponential expressions, using the product rule, the quotient rule, and the power rule. Additionally, it allows us to formulate distributive rules for exponents.
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Any non-zero number raised to the 0th power is 1. This makes the definition of the 0th power consistent with the division rule for exponents.
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For consistency, negative exponents represent the reciprocal of the base raised to the power, so that
, provided that
.
3.9
Scientific Notation
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Some numbers are so large or so small that writing the number out is clumsy and make it difficult to determine the true size of the number. Scientific notation makes the number more readable and make the relative size of the number immediately apparent.
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A number written in scientific notation is a number at least 1 and smaller than 10 multiplied by 10 raised to an exponent. Converting between scientific notation and standard notation involves correctly applying multiplication and division by powers of 10, which in practice equates to understanding how moving the decimal point of a number impacts the exponent of 10.
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Adding and subtracting numbers in base 10 requires the exponent of 10 in each number be the same. Once the numbers are converted to have the same exponent with the ten, then the numbers are added or subtracted as indicated, with the power of 10 remaining the same. If the result is not in scientific notation (for instance, the number has exceeded 10), then then number must be converted into scientific notation.
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Multiplying and dividing numbers in scientific notation is done by multiplying or dividing the number parts, then multiplying or dividing the 10 raised to the power parts, then multiplying those two results. If the new number is not in scientific notation, then the result must be converted into scientific notation.
3.10
Arithmetic Sequences
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A sequence is a list of numbers. Any individual number in that list, or sequence, is a term of the sequence. A specific term of a sequence is denoted by the sequence symbol with a subscript indicating where the term in the sequence is.
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A special form of a sequence is an arithmetic sequence. Each arithmetic sequence is determined by its first term and its constant difference. Any term in an arithmetic sequence is determined by adding the constant difference to the preceding term.
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If the first term and the constant difference of an arithmetic sequence are known, then any term of the sequence can be found directly.
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Because arithmetic sequences follow such a strict pattern, the sum of the first
terms of an arithmetic sequence can be determined with the formula
.
3.11
Geometric Sequences
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A special form of a sequence is a geometric sequence. Each geometric sequence is determined by its first term and its constant ratio. Any term in a geometric sequence is determined by multiplying the constant ratio to the preceding term.
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If the first term and the constant ratio of a geometric sequence are known, then any term of the sequence can be found directly.
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Because geometric sequences follow such a strict pattern, the sum of the first
terms of a geometric sequence can be determined with the formula
.
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Finding the sum of a finite geometric sequence
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Applying arithmetic sequences
3.11
Geometric Sequences
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Geometric sequence.
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Finding an arbitrary term in a geometric sequence.
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Constant ratio.
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Finding the sum of a finite geometric sequence.
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Applying arithmetic sequences.


4.2: Linear Momentum and Force
Learning Objectives
- Define linear momentum.
- Explain the relationship between momentum and force.
- State Newton’s second law of motion in terms of momentum.
- Calculate momentum given mass and velocity.
Linear Momentum
The scientific definition of linear momentum is consistent with most people’s intuitive understanding of momentum: a large, fast-moving object has greater momentum than a smaller, slower object. Linear momentum is defined as the product of a system’s mass multiplied by its velocity. In symbols, linear momentum is expressed as
\[\boldsymbol{p}=m \boldsymbol{v} \nonumber\]
Momentum is directly proportional to the object’s mass and also its velocity. Thus the greater an object’s mass or the greater its velocity, the greater its momentum. Momentum \(\boldsymbol{p}\) is a vector having the same direction as the velocity \(\boldsymbol{v}\). The SI unit for momentum is \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\).
Definition: LINEAR MOMENTUM
Linear momentum is defined as the product of a system’s mass multiplied by its velocity:
\[\boldsymbol{p}=m \boldsymbol{v}. \nonumber \]
Example \(\PageIndex{1}\): Calculating Momentum: A Football Player and a Football
(a) Calculate the momentum of a 110-kg football player running at 8.00 m/s. (b) Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s.
Strategy
No information is given regarding direction, and so we can calculate only the magnitude of the momentum, \(p\). (As usual, a symbol that is in italics is a magnitude, whereas one that is italicized and is boldfaced or has an arrow is a vector.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum given in the equation, which becomes
\[p=m v \nonumber\]
when only magnitudes are considered.
Solution for (a)
To determine the momentum of the player, substitute the known values for the player’s mass and speed into the equation.
\[p_{\text {player }}=(110 \mathrm{~kg})(8.00 \mathrm{~m} / \mathrm{s})=880 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s} \nonumber\]
Solution for (b)
To determine the momentum of the ball, substitute the known values for the ball’s mass and speed into the equation.
\[p_{\text {ball }}=(0.410 \mathrm{~kg})(25.0 \mathrm{~m} / \mathrm{s})=10.3 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s} \nonumber\]
The ratio of the player’s momentum to that of the ball is
\[\frac{p_{\text {player }}}{p_{\text {ball }}}=\frac{880}{10.3}=85.9. \nonumber\]
Discussion
Although the ball has greater velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the momentum of the football, as you might guess. As a result, the player’s motion is only slightly affected if he catches the ball. We shall quantify what happens in such collisions in terms of momentum in later sections.
Momentum and Newton’s Second Law
The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the “quantity of motion.” Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is
\[\boldsymbol{F}_{\mathrm{net}}=\frac{\Delta \boldsymbol{p}}{\Delta t}, \nonumber \]
where \(\boldsymbol{F}_{\text {net }}\) is the net external force, \(\Delta \boldsymbol{p}\) is the change in momentum, and \(\Delta t\) is the change in time.
NEWTON’S SECOND LAW OF MOTION IN TERMS OF MOMENTUM
The net external force equals the change in momentum of a system divided by the time over which it changes: \(\boldsymbol{F}_{\mathrm{net}}=\Delta \boldsymbol{p} / \Delta t\).
MAKING CONNECTIONS: FORCE AND MOMENTUM
Force and momentum are intimately related. Force acting over time can change momentum, and Newton’s second law of motion, can be stated in its most broadly applicable form in terms of momentum. Momentum continues to be a key concept in the study of atomic and subatomic particles in quantum mechanics.
This statement of Newton’s second law of motion includes the more familiar \(\boldsymbol{F}_{\mathrm{net}}=m \boldsymbol{a}\) as a special case. We can derive this form as follows. First, note that the change in momentum \(\Delta \boldsymbol{p}\) is given by
\[\Delta \boldsymbol{p}=\Delta(m \boldsymbol{v}). \nonumber \]
If the mass of the system is constant, then
\[\Delta(m \boldsymbol{v})=m \Delta \boldsymbol{v}. \nonumber \]
So that for constant mass, Newton’s second law of motion becomes
\[\boldsymbol{F}_{\mathrm{net}}=\frac{\Delta \boldsymbol{p}}{\Delta t}=m \frac{\Delta \boldsymbol{v}}{\Delta t} \nonumber \]
Because \(\frac{\Delta v}{\Delta t}=\boldsymbol{a}\), we get the familiar equation
\[\boldsymbol{F}_{\mathrm{net}}=m \boldsymbol{a} \nonumber \]
when the mass of the system is constant .
Newton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail ; however, the relationship between momentum and force remains useful when mass is constant, such as in the following example.
Example \(\PageIndex{2}\): Calculating Force: Venus Williams’ Racquet
During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that the ball’s speed just after impact is 58 m/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)?
Strategy
This problem involves only one dimension because the ball starts from having no horizontal velocity component before impact. Newton’s second law stated in terms of momentum is then written as \(\boldsymbol{F}_{\text {net }}=\Delta \boldsymbol{p} / \Delta t\).
As noted above, when mass is constant, the change in momentum is given by
\[\Delta p=m \Delta v=m\left(v_{\mathrm{f}}-v_{\mathrm{i}}\right). \nonumber\]
In this example, the velocity just after impact and the change in time are given; thus, once \(\Delta p\) is calculated, \(F_{\text {net }}=\frac{\Delta p}{\Delta t}\) can be used to find the force.
Solution
To determine the change in momentum, substitute the values for the initial and final velocities into the equation above.
\[\begin{aligned}
\Delta p &=m\left(v_{\mathrm{f}}-v_{\mathrm{i}}\right) \\
&=(0.057 \mathrm{~kg})(58 \mathrm{~m} / \mathrm{s}-0 \mathrm{~m} / \mathrm{s}) \\
&=3.306 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s} \approx 3.3 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}
\end{aligned} \nonumber\]
Now the magnitude of the net external force can determined by using \(F_{\text {net }}=\frac{\Delta p}{\Delta t}\):
\[\begin{aligned}
F_{\text {net }} &=\frac{\Delta p}{\Delta t}=\frac{3.306 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}}{5.0 \times 10^{-3} \mathrm{~s}} \\
&=661 \mathrm{~N} \approx 660 \mathrm{~N},
\end{aligned} \nonumber\]
where we have retained only two significant figures in the final step.
Discussion
This quantity was the average force exerted by Venus Williams’ racquet on the tennis ball during its brief impact (note that the ball also experienced the 0.56-N force of gravity, but that force was not due to the racquet). This problem could also be solved by first finding the acceleration and then using \(F_{\text {net }}=m a\), but one additional step would be required compared with the strategy used in this example.
Section Summary
- Linear momentum ( momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity.
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In symbols, linear momentum \(\boldsymbol{p}\) is defined to be
\[\boldsymbol{p}=m \boldsymbol{v}, \nonumber\]
where mm is the mass of the system and \(\boldsymbol{v}\) is its velocity. - The SI unit for momentum is \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\).
- Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
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In symbols, Newton’s second law of motion is defined to be
\[F_{\mathrm{net}}=\frac{\Delta \boldsymbol{p}}{\Delta t}, \nonumber\]
where \(\boldsymbol{F}_{\text {net }}\) is the net external force, \(\Delta \boldsymbol{p}\) is the change in momentum, and \(\Delta t\) is the change time.
Glossary
- linear momentum
- the product of mass and velocity
- second law of motion
- physical law that states that the net external force equals the change in momentum of a system divided by the time over which it changes


1.1.1: What does Health and Wellness mean?
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How have you used the word Health or Wellness in your communication with others? The terms Health and Wellness are often used interchangeably. What do they mean to you?
Activity: Healthy versus Unhealthy
What do you mean by healthy and unhealthy?
- Take a piece of paper and fold it in half.
- On one side of the paper write “healthy” and the other side write “unhealthy”
- Either make a list or draw pictures of what you consider healthy and unhealthy.
As you review the following information about the wellness, see if you have an opportunity to expand your list or drawings for aspects of health that may be missing from your list.
The World Health Organization (WHO) defines health as “a state of complete physical, mental, and social well-being and not merely the absence of disease or infirmity (illness)” and defines wellness as “the optimal state of health of individuals and groups, [1] ” which may be expressed as “a positive approach to living.” The National Wellness Institute [2] , explains wellness as “an active process through which people become aware of, and make choices toward, a more successful existence.”
The National Wellness Institute states that there is general agreement that:
- Wellness is considered a conscious, self-directed and evolving process of achieving full potential.
- Wellness is multidimensional and holistic, encompassing lifestyle, mental and spiritual well-being, and the environment.
- Wellness is positive and affirming.
The primary difference between health and wellness is that health is your state of being, or a goal to achieve, and wellness is the active process of achieving it through growth and change. Through wellness we hope to reach our fullest potential of health and well-being.
The Nine Dimensions of Wellness
Both health and wellness are very broad terms. To help identify ways we can grow and change to reach our optimal health, it is helpful to view wellness in terms of the Nine Dimensions of Wellness. The Nine Dimensions of Wellness can help to identify ways you can make wellness a part of your everyday life.
The Nine Dimensions of Wellness include: Physical, Emotional, Social, Environmental, Financial, Intellectual, Occupational, Spiritual, and Cultural. These dimensions of wellness are not isolated, but rather are interconnected. Problems or challenges in one dimension can impact the other, while the opposite can also be true, that by improving your health and wellness in one area can help to improve other dimensions. For example, your emotional wellness can be positively impacted through exercise (physical wellness) but negatively impacted by financial struggles that often lead to high stress.
Reflection Opportunity: My Wellness Ratings and Actions
As you read through the following Nine Dimensions of Wellness, give yourself a score from 1 (low) to 10 (high) for your level of wellness in each dimension. Along with your score, also record a list of possible behaviors you could change or actions you could take that would increase your wellness score for each dimension.
For example:
“For Physical Wellness I would give myself a score of [insert your score 1 being low] out of 10. I could increase this score by either changing or implementing the following two behaviors: [insert a behavior you could change or an action you could take] and [insert a behavior you could change or an action you could take]”
Physical Wellness
People who are focused on improving their physical health recognize the need for regular physical activity, they choose healthy foods, go to the doctor, and get adequate sleep. People who are physically well actively make healthy decisions on a daily basis. This may include making a habit of exercising three to five times per week or recording their dietary intake to ensure they are receiving proper nutrition for optional health.
How would you rate your physical wellness and what behaviors could you change, or actions could you take, to positively improve your physical wellness?
Emotional Wellness
Coping effectively with life and expressing emotions in an appropriate manner are keys to emotional wellness. An emotionally well person successfully expresses and manages an entire range of feelings, including anger, doubt, hope, joy, desire, fear, and many others. People who are emotionally well maintain a high level of self-esteem. They have a positive body-image and the ability to regulate their feelings. They know where to seek support and help regarding their mental health, including but not limited to, seeking professional counseling services.
How would you rate your emotional wellness and what behaviors could you change, or actions could you take, to positively improve your emotional wellness?
Social Wellness
People who are focused on their social wellness are striving for positive relationships, developing a sense of connection, belonging, and sustained support system. A socially well person builds healthy relationships based on interdependence, trust, and respect. Those who are socially well have a keen awareness of the feelings of others. They develop a network of friends and co-workers who share a common purpose, and who provide support and validation.
How would you rate your social wellness and what behaviors could you change, or actions could you take, to positively improve your social wellness?
Environmental Wellness
Environmental wellness includes a desire to positively impact our planet Earth and also our local community by striving to occupy pleasant, healthy, and safe environments that support well-being and positively impact the quality of our surroundings (including protecting and preserving nature). An environmentally well person appreciates the external cues and stimuli that an environment can provide . People who have achieved environmental wellness recognize the limits to controlling an environment and seek to understand the role an individual plays in the environment.
How would you rate your environmental wellness and what behaviors could you change, or actions could you take, to positively improve your environmental wellness?
Financial Wellness
Financial wellness refers to achieving satisfaction with current and future financial situations by handling finances wisely. Those who are financially well are fully aware of their current financial state. They set long- and short-term goals regarding finances that will allow them to reach their personal financial goals.
How would you rate your financial wellness and what behaviors could you change, or actions could you take, to positively improve your financial wellness?
Intellectual Wellness
Intellectual wellness includes being open-minded, recognizing creative abilities, and/or finding ways to expand knowledge and skills. Those who enjoy intellectual wellness engage in lifelong learning. They seek knowledge and activities that further develop their critical thinking and heighten global awareness. They engage in activities associated with the arts, philosophy, and reasoning.
How would you rate your intellectual wellness and what behaviors could you change, or actions could you take, to positively improve your intellectual wellness?
Occupational Wellness
Occupational wellness refers to personal fulfillment and enrichment from one’s work and/or responsibilities. An occupationally well person enjoys the pursuit of a career which is fulfilling on a variety of levels. This person finds satisfaction and enrichment in work, while always in pursuit of opportunities to reach the next level of professional success.
How would you rate your occupational wellness and what behaviors could you change, or actions could you take, to positively improve your occupational wellness?
Spiritual Wellness
Spiritual wellness refers to having a sense of purpose and meaning in life, this may come from establishing peace, harmony, and balance in our lives. People who can be described as spiritually well have identified a core set of beliefs that guide their decision making, and other faith- based endeavors. While firm in their spiritual beliefs, they understand others may have a distinctly different set of guiding principles. They recognize the relationship between spirituality and identity in all individuals.
How would you rate your spiritual wellness and what behaviors could you change, or actions could you take, to positively improve your spiritual wellness?
Cultural Wellness
Cultural wellness refers to the way you interact with others who are different from you. This includes understanding and celebrating our differences. Culturally well people are aware of their own cultural background, as well as the diversity and richness present in other cultural backgrounds. Cultural wellness implies understanding, awareness and intrinsic respect for aspects of diversity. A culturally well person acknowledges and accepts the impact of these aspects of diversity on sexual orientation, religion, gender, racial and ethnic backgrounds, age groups, and disabilities
How would you rate your cultural wellness and what behaviors could you change, or actions could you take, to positively improve your cultural wellness?


2.1: Introduction to Two-Person Zero-Sum Games
In all of the examples from the last section, whatever one player won, the other player lost.
A two-player game is called a zero-sum game if the sum of the payoffs to each player is constant for all possible outcomes of the game. More specifically, the terms (or coordinates) in each payoff vector must add up to the same value for each payoff vector. Such games are sometimes called constant-sum games instead.
We can always think of zero-sum games as being games in which one player's win is the other player's loss.
Consider a poker game in which each player comes to the game with \($100\). If there are five players, then the sum of money for all five players is always \($500\). At any given time during the game, a particular player may have more than \($100\), but then another player must have less than \($100\). One player's win is another player's loss.
Consider the cake division game. Determine the payoff matrix for this game. It is important to determine what each player's options are first: how can the “cutter” cut the cake? How can the “chooser” pick her piece? The payoff matrix is given in Table \(2.1.1\).
| Table 2.1.1: Payoff Matrix for Cake Cutting Game | |||
|---|---|---|---|
| Chooser | |||
| Larger Piece | Smaller Piece | ||
| Cutter | Cut Evenly | (half, half) | (half, half) |
| Cut Unevenly | (small piece, large piece) | (large piece, small piece) |
In order to better see that this game is zero-sum (or constant-sum), we could give values for the amount of cake each player gets. For example, half the cake would be \(50 \%\), a small piece might be \(40 \%\). Then we can rewrite the matrix with the percentage values in Table \(2.1.2\).
| Table 2.1.2: Payoff Matrix, in Percent of Cake, for Cake Cutting Game. | |||
|---|---|---|---|
| Chooser | |||
| Larger Piece | Smaller Piece | ||
| Cutter | Cut Evenly | \((50, 50)\) | \((50, 50)\) |
| Cut Uenvenly | \((40, 60)\) | \((60, 40)\) |
In each outcome, the payoffs to each player add up to \(100\) (or \(100 \%\)). In more mathematical terms, the coordinates of each payoff vector add up to \(100\). Thus the sum is the same, or constant, for each outcome.
It is probably simple to see from the matrix in Table \(2.1.2\) that Player 2 will always choose the large piece, thus Player 1 does best to cut the cake evenly. The outcome of the game is the strategy pair denoted [Cut Evenly, Larger Piece], with resulting payoff vector \((50, 50)\text{.}\)
But why are we going to call these games “zero-sum” rather than “constant-sum”? We can convert any zero-sum game to a game where the payoffs actually sum to zero.
Consider the above poker game where each player begins the game with \($100\). Suppose at some point in the game the five players have the following amounts of money: \($50\), \($200\), \($140\), \($100\), \($10\). Then we could think of their gain as \(-$50\), \($100\), \($40\), \($0\), \(-$90\). What do these five numbers add up to?
Convert the cake division payoffs so that the payoff vectors sum to zero (rather than \(100\)).
The solution is given in Table \(2.1.3\).
| Table 2.1.3: Zero-sum Payoff Matrix for Cake Cutting Game. | |||
|---|---|---|---|
| Chooser | |||
| Larger Piece | Smaller Piece | ||
| Cutter | Cut Evenly | \((0, 0)\) | \((0, 0)\) |
| Cut Unvenly | \((-10, 10)\) | \((10, -10)\) |
But let's make sure we understand what these numbers mean. For example, a payoff of \((0,0)\) does not mean each player gets no cake, it means they don't get any more cake than the other player. In this example, each player gets half the cake (\(50 \%\)) plus the payoff.
In the form of Example \(2.1.4\), it is easy to recognize a zero-sum game since each payoff vector has the form \((a, -a)\) (or \((-a, a)\)).
2.1.1: Example—An Election Campaign Game
Two candidates, Arnold and Bainbridge, are facing each other in a state election. They have three choices regarding the issue of the speed limit on I-\(5\): They can support raising the speed limit to \(70\) MPH, they can support keeping the current speed limit, or they can dodge the issue entirely. The next three examples present three different payoff matrices for Arnold and Bainbridge.
The candidates have the information given in Table \(2.1.4\) about how they would likely fare in the election based on how they stand on the speed limit.
| Table 2.1.4: Percentage of the Vote for Example \(2.1.5\). | ||||
|---|---|---|---|---|
| Bainbridge | ||||
| Raise Limit | Keep Limit | Dodge | ||
| Arnold | Raise Limit | \((45, 55)\) | \((50, 50)\) | \((40, 60)\) |
| Keep Limit | \((60, 40)\) | \((55, 45)\) | \((50, 50)\) | |
| Dodge | \((45, 55)\) | \((55, 45)\) | \((40, 60)\) |
For the following questions, assume Arnold and Bainbridge have the payoff matrix given in Example \(2.1.5\).
- Explain why Example \(2.1.5\) is a zero-sum game.
- What should Arnold choose to do? What should Bainbridge choose to do? Be sure to explain each candidate's choice. And remember, a player doesn't just want to win, he wants to get THE MOST votes– for example, you could assume these are polling numbers and that there is some margin of error, thus a candidate prefers to have a larger margin over his opponent!
- What is the outcome of the election?
- Does Arnold need to consider Bainbridge's strategies is in order to decide on his own strategy? Does Bainbridge need to consider Arnold's strategies is in order to decide on his own strategy? Explain your answer.
Bainbridge's mother is injured in a highway accident caused by speeding. The new payoff matrix is given in Table \(2.1.6\).
| Table 2.1.5: Percentage of the Vote for Example \(2.1.6\). | ||||
|---|---|---|---|---|
| Bainbridge | ||||
| Raise Limit | Keep Limit | Dodge | ||
| Arnold | Raise Limit | \((45, 55)\) | \((10, 90)\) | \((40, 60)\) |
| Keep Limit | \((60, 40)\) | \((55, 45)\) | \((50, 50)\) | |
| Dodge | \((45, 55)\) | \((10, 90)\) | \((40, 60)\) |
For the following questions, assume Arnold and Bainbridge have the payoff matrix given in Example \(2.1.6\).
- Explain why Example \(2.1.6\) is a zero-sum game.
- What should Arnold choose to do? What should Bainbridge choose to do? Be sure to explain each candidate's choice.
- What is the outcome of the election?
- Does Arnold need to consider Bainbridge's strategies is in order to decide on his own strategy? Does Bainbridge need to consider Arnold's strategies is in order to decide on his own strategy? Explain your answer.
Bainbridge begins giving election speeches at college campuses and monster truck rallies. The new payoff matrix is given in Table \(2.1.6\).
| Table 2.1.6: Percentage of the Vote for Example \(2.1.7\). | ||||
|---|---|---|---|---|
| Bainbridge | ||||
| Raise Limit | Keep Limit | Dodge | ||
| Arnold | Raise Limit | \((35, 65)\) | \((10, 90)\) | \((60, 40)\) |
| Keep Limit | \((45, 55)\) | \((55, 45)\) | \((50, 50)\) | |
| Dodge | \((40, 60)\) | \((10, 90)\) | \((65, 35)\) |
For the following questions, assume Arnold and Bainbridge have the payoff matrix given in Example \(2.1.7\).
- Explain why Example \(2.1.7\) is a zero-sum game.
- What should Arnold choose to do? What should Bainbridge choose to do? Be sure to explain each candidate's choice.
- What is the outcome of the election?
- Does Arnold need to consider Bainbridge's strategies is in order to decide on his own strategy? Does Bainbridge need to consider Arnold's strategies is in order to decide on his own strategy? Explain your answer.
In each of the above scenarios, is there any reason for Arnold or Bainbridge to change his strategy? If there is, explain under what circumstances it makes sense to change strategy. If not, explain why it never makes sense to change strategy.
2.1.2: Equilibrium Pairs
Chances are, in each of the exercises above, you were able to determine what each player should do. In particular, if both players play your suggested strategies, there is no reason for either player to change to a different strategy.
A pair of strategies is an equilibrium pair if neither player gains by changing strategies.
For example, consider the game matrix from Example \(1.2.1\), Table \(1.2.3\).
| Table \(2.1.7\): Payoff Matrix for Example \(1.2.1\) | |||
|---|---|---|---|
| Player 2 | |||
| X | Y | ||
| Player 1 | A | \((100, -100)\) | \((-10, 10)\) |
| B | \((0, 0)\) | \((-1, 11)\) |
You determined that Player 2 should choose to play Y, and thus, Player 1 should play B (i.e., we have the strategy pair [B, Y]). Why is this an equilibrium pair? If Player 2 plays Y, does Player 1 have any reason to change to strategy A? No, she would lose \(10\) instead of \(1\)! If Player 1 plays B, does Player 2 have any reason to change to strategy X? No, she would gain \(0\) instead of \(1\)! Thus neither player benefits from changing strategy, and so we say [B, Y] is an equilibrium pair.
For now, we can use a “guess and check” method for finding equilibrium pairs. Take each outcome and decide whether either player would prefer to switch. Remember, Player 1 can only choose a different row, and Player 2 can only choose a different column. In our above example there are four outcomes to check: [A, X], [A, Y], [B, X], and [B, Y]. We already know [B, Y] is an equilibrium pair, but let's check the rest. Suppose the players play [A, X]. Does Player 1 want to switch to B? No, she'd rather get \(100\) than \(0\). Does player 2 want to switch to Y? Yes! She'd rather get \(10\) than \(-100\). So [A, X] is NOT an equilibrium pair since a player wants to switch. Now check that for [A, Y] Player 1 would want to switch, and for [B, X] both players would want to switch. Thus [A, Y] and [B, X] are NOT equilibrium pairs. Now you can try to find equilibrium pairs in any matrix game by just checking each payoff vector to see if one of the players would have wanted to switch to a different strategy.
Are the strategy pairs you determined in the three election scenarios equilibrium pairs? In other words, would either player prefer to change strategies? (You don't need to check whether any other strategies are equilibrium pairs.)
Use the “guess and check” method to determine any equilibrium pairs for the following payoff matrices.
-
\(\begin{bmatrix}(2,-2) & (2,-2) \\(1,-1) & (3,-3) \end{bmatrix}\)
After trying the above examples, do you think every game has an equilibrium pair? Can games have multiple equilibrium pairs?
Do all games have equilibrium pairs?
Can a game have more than one equilibrium pair?
The last three exercises give you a few more games to practice with.
Consider the game ROCK, PAPER, SCISSORS (Rock beats Scissors, Scissors beat Paper, Paper beats Rock). Construct the payoff matrix for this game. Does it have an equilibrium pair? Explain your answer.
Two television networks are battling for viewers for \(7\) pm Monday night. They each need to decide if they are going to show a sitcom or a sporting event. Table \(2.1.8\) gives the payoffs as percent of viewers.
| Table \(2.1.8\): Payoff matrix for Battle of the Networks | |||
|---|---|---|---|
| Network 2 | |||
| Sitcom | Sports | ||
| Network 1 | Sitcom | \((55, 45)\) | \((52, 48)\) |
| Sports | \((50, 50)\) | \((45, 55)\) |
- Explain why this is a zero-sum game.
- Does this game have an equilibrium pair? If so, find it and explain what each network should do.
- Convert this game to one in which the payoffs actually sum to zero. Hint: if a network wins \(60 \%\) of the viewers, how much more than 50% of the viewers does it have?
This game is an example of what economists call Competitive Advantage. Two competing firms need to decide whether or not to adopt a new type of technology. The payoff matrix is in Table \(2.1.9\). The variable \(a\) is a positive number representing the economic advantage a firm will gain if it is the first to adopt the new technology.
| Table \(2.1.9\): Payoff matrix for Competitve Advantage | |||
|---|---|---|---|
| Firm B | |||
| Adopt New Tech | Stay Put | ||
| Firm A | Adopt New Tech | \((0, 0)\) | \((a, -a)\) |
| Stay Put | \((-a, a)\) | \((0, 0)\) |
- Explain the payoff vector for each strategy pair. For example, why should the pair [Adopt New Tech, Stay Put] have the payoff \((a, -a)\text{?}\)
- Explain what each firm should do.
- Give a real life example of Competitive Advantage.
We've seen how to describe a zero-sum game and how to find equilibrium pairs. We've tried to decide what each player's strategy should be. Each player may need to consider the strategy of the other player in order to determine his or her best strategy. But we need to be careful, although our intuition can be useful in deciding the best strategy, we'd like to be able to be more precise about finding strategies for each player. We'll learn some of these tools in the next section.


5: Numeration Systems
In the following chapter, we will try to focus on two main ideas. The first will be an examination of basic number and counting systems and the symbols that we use for numbers. We will look at our own modern (Western) number system as well those of a couple of selected civilizations to see the differences and diversity that is possible when humans start counting. The second idea we will look at will be base systems. By comparing our own base-ten (decimal) system with other bases, we will quickly become aware that the system that we are so used to, when slightly changed, will challenge our notions about numbers and what symbols for those numbers actually mean.
Thumbnail: Roman Numerals. (Public Domain; Monaneko via Wikipedia )


11.6: Musique Concrete
This page has a little more information on the style of music exemplified by our listening example, Poeme Electronique.
Introduction
Musique concrète (meaning “concrete music”) is a genre of electroacoustic music that is made in part from acousmatic sound. It can feature sounds derived from recordings of musical instruments, voice, and the natural environment as well as those created using synthesizers and computer-based digital signal processing. Also, compositions in this idiom are not restricted to the normal musical rules of melody, harmony, rhythm, metre, and so on. Originally contrasted with “pure” elektronische Musik (based solely on the production and manipulation of electronically produced sounds rather than recorded sounds), the theoretical basis of musique concrète as a compositional practice was developed by Pierre Schaeffer, beginning in the early 1940s.
Musique Concrete
By 1949 Schaeffer’s compositional work was known publicly as musique concrète . Schaeffer stated: “when I proposed the term ‘musique concrète,’ I intended … to point out an opposition with the way musical work usually goes. Instead of notating musical ideas on paper with the symbols of solfege and entrusting their realization to well-known instruments, the question was to collect concrete sounds, wherever they came from, and to abstract the musical values they were potentially containing.” According to Pierre Henry, “musique concrète was not a study of timbre, it is focused on envelopes, forms. It must be presented by means of non-traditional characteristics, you see . . . one might say that the origin of this music is also found in the interest in “plastifying” music, of rendering it plastic like sculpture…musique concrète, in my opinion . . . led to a manner of composing, indeed, a new mental framework of composing.” Schaeffer had developed an aesthetic that was centred upon the use of sound as a primary compositional resource. The aesthetic also emphasised the importance of play ( jeu ) in the practice of sound based composition. Schaeffer’s use of the word jeu , from the verb jouer , carries the same double meaning as the English verb play: “to enjoy oneself by interacting with one’s surroundings,” as well as “to operate a musical instrument.”
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Musique concrete. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Musique_concr%C3%A8te . License : CC BY-SA: Attribution-ShareAlike


6.5: Putting It All Together
Patient Scenario
Mrs. Vang is an 83-year-old resident that was recently admitted to a long-term memory care facility. She was diagnosed with Alzheimer’s disease last year. She is alert to self but often has periods where she is uncooperative and is unable to follow commands. She has experienced a decline in the ability to provide self-care and wanders and paces at night. She recently fell when wandering outside of her room at night.
Applying the Nursing Process
Assessment: Mrs. Vang is alert to self only and does not follow commands during the assessment. She is unable to provide self-care despite cueing.
Based on the assessment information that has been gathered, the following nursing care plan is created for Mrs. Vang:
Nursing Diagnosis: Wandering related to separation from familiar environment as manifested by frequent movement from place to place and pacing.
Overall Goal: The patient will remain safe and free from falls.
SMART Expected Outcome: Mrs. Vang will experience reduced episodes of wandering within 72 hours.
Planning and Implementing Nursing Interventions:
The nurse will provide orientation cues, such as family pictures in the patient room, as appropriate. The nurse will encourage a daily routine by all caregivers to prevent discomfort issues related to thirst, hunger, or lack of sleep. The nurse will encourage patient autonomy and provide choices in decisions as appropriate. The nurse will provide opportunities for reminiscence and cultivate therapeutic communication using touch and validation of emotional communication. The nurse will place a bed alarm to alert staff at night when the patient is getting out of bed. The nurse will implement a wander guard ankle bracelet to notify staff if the patient wanders near an exit door.
Sample Documentation:
Mrs. Vang has impaired thought processes as a result of her Alzheimer’s disease. A care routine has been established. The patient receives appropriate visual cues and reorientation to the environment. Safety interventions have been implemented, and the patient is being monitored for signs of increasing confusion or mental decline.
Evaluation:
Mrs. Vang has remained safe within the care environment and demonstrated no additional decline in thought processes. Her wandering at night has decreased and the bed alarm alerts staff when she gets out of bed. SMART outcome “met.”


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2.9: Collaborative Group Activities
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With your group, consider the question with which we began this chapter. How many ways can you think of to prove to a member of the “Flat Earth Society” that our planet is, indeed, round?
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Make a list of ways in which a belief in astrology (the notion that your life path or personality is controlled by the position of the Sun, Moon, and planets at the time of your birth) might be harmful to an individual or to society at large.
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Have members of the group compare their experiences with the night sky. Did you see the Milky Way? Can you identify any constellations? Make a list of reasons why you think so many fewer people know the night sky today than at the time of the ancient Greeks. Discuss reasons for why a person, today, may want to be acquainted with the night sky.
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Constellations commemorate great heroes, dangers, or events in the legends of the people who name them. Suppose we had to start from scratch today, naming the patterns of stars in the sky. Whom or what would you choose to commemorate by naming a constellation after it, him, or her and why (begin with people from history; then if you have time, include living people as well)? Can the members of your group agree on any choices?
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Although astronomical mythology no longer holds a powerful sway over the modern imagination, we still find proof of the power of astronomical images in the number of products in the marketplace that have astronomical names. How many can your group come up with? (Think of things like Milky Way candy bars, Eclipse and Orbit gum, or Comet cleanser.)


3.11: Antimicrobials Introduction
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- Ernstmeyer & Christman (Eds.)
- Chippewa Valley Technical College via OpenRN
Learning Objectives
- Identify the classifications and actions of antimicrobial medications
- Give examples of when, how, and to whom antimicrobial drugs may be administered
- Identify the side effects and special considerations associated with antimicrobial therapy
- Include considerations and implications of using antimicrobial medications across the lifespan
- Include evidence-based concepts when using the nursing process
- Identify and interpret related laboratory tests
Have you ever been prescribed an antibiotic for an infection and asked, “Why do I have to finish taking all these pills when I already feel better”? Or, perhaps you wondered why the healthcare provider chose a certain medication over another or why the pharmacist told you to avoid certain foods when taking a certain antibiotic.
You may have had these questions in your own healthcare experiences. It is important to remember that if you have these questions, many of your patients will as well. Learning about the various types of antimicrobials and how they work will help you provide better health education to your patients.
Did you know that the use of antimicrobial agents dates back to ancient times?
Although the discovery of antimicrobials and their subsequent widespread use is commonly associated with modern medicine, there is evidence that humans have been exposed to antimicrobial compounds for millennia. Chemical analyses of the skeletal remains of people from between 350 and 550 AD of people living near the Nile River have shown residue of the antimicrobial agent tetracycline in high enough quantities to suggest the purposeful fermentation of tetracycline-producing Streptomyces during the beer-making process. The resulting beer, which was thick and gruel-like, was used to treat a variety of ailments in both adults and children, including gum disease and wounds.
Additionally, the antimicrobial properties of plants and honey have been recognized by various cultures around the world, including Indian and Chinese herbalists who have long used plants for a wide variety of medical purposes. Healers of many cultures understood the antimicrobial properties of fungi, and their use of moldy bread or other mold-containing products to treat wounds have been well documented for centuries. [1]
- This work is a derivative of Microbiology by OpenStax and is licensed under CC BY 4.0 ↵