6.3: Systems of Linear Equations - Three Variables
In this section, you will:
- Solve systems of three equations in three variables.
- Identify inconsistent systems of equations containing three variables.
- Express the solution of a system of dependent equations containing three variables.
John received an inheritance of $12,000 that he divided into three parts and invested in three ways: in a money-market fund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual interest. John invested $4,000 more in municipal funds than in municipal bonds. He earned $670 in interest the first year. How much did John invest in each type of fund?
Understanding the correct approach to setting up problems such as this one makes finding a solution a matter of following a pattern. We will solve this and similar problems involving three equations and three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of visual gymnastics.
Solving Systems of Three Equations in Three Variables
In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination , named after the prolific German mathematician Karl Friedrich Gauss . While there is no definitive order in which operations are to be performed, there are specific guidelines as to what type of moves can be made. We may number the equations to keep track of the steps we apply. The goal is to eliminate one variable at a time to achieve upper triangular form , the ideal form for a three-by-three system because it allows for straightforward back-substitution to find a solution which we call an ordered triple . A system in upper triangular form looks like the following:
The third equation can be solved for and then we back-substitute to find and To write the system in upper triangular form, we can perform the following operations:
- Interchange the order of any two equations.
- Multiply both sides of an equation by a nonzero constant.
- Add a nonzero multiple of one equation to another equation.
The solution set to a three-by-three system is an ordered triple Graphically, the ordered triple defines the point that is the intersection of three planes in space. You can visualize such an intersection by imagining any corner in a rectangular room. A corner is defined by three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.
Number of Possible Solutions
Figure 2 and Figure 3 illustrate possible solution scenarios for three-by-three systems.
- Systems that have a single solution are those which, after elimination, result in a solution set consisting of an ordered triple Graphically, the ordered triple defines a point that is the intersection of three planes in space.
- Systems that have an infinite number of solutions are those which, after elimination, result in an expression that is always true, such as Graphically, an infinite number of solutions represents a line or coincident plane that serves as the intersection of three planes in space.
- Systems that have no solution are those that, after elimination, result in a statement that is a contradiction, such as Graphically, a system with no solution is represented by three planes with no point in common.
Example 1
Determining Whether an Ordered Triple Is a Solution to a System
Determine whether the ordered triple is a solution to the system.
- Answer
-
We will check each equation by substituting in the values of the ordered triple for and
The ordered triple is indeed a solution to the system.
How To
Given a linear system of three equations, solve for three unknowns.
- Pick any pair of equations and solve for one variable.
- Pick another pair of equations and solve for the same variable.
- You have created a system of two equations in two unknowns. Solve the resulting two-by-two system.
- Back-substitute known variables into any one of the original equations and solve for the missing variable.
Example 2
Solving a System of Three Equations in Three Variables by Elimination
Find a solution to the following system:
- Answer
-
There will always be several choices as to where to begin, but the most obvious first step here is to eliminate by adding equations (1) and (2).
The second step is multiplying equation (1) by and adding the result to equation (3). These two steps will eliminate the variable
In equations (4) and (5), we have created a new two-by-two system. We can solve for by adding the two equations.
Choosing one equation from each new system, we obtain the upper triangular form:
Next, we back-substitute into equation (4) and solve for
Finally, we can back-substitute and into equation (1). This will yield the solution for
The solution is the ordered triple See Figure 4 .
Example 3
Solving a Real-World Problem Using a System of Three Equations in Three Variables
In the problem posed at the beginning of the section, John invested his inheritance of $12,000 in three different funds: part in a money-market fund paying 3% interest annually; part in municipal bonds paying 4% annually; and the rest in mutual funds paying 7% annually. John invested $4,000 more in mutual funds than he invested in municipal bonds. The total interest earned in one year was $670. How much did he invest in each type of fund?
- Answer
-
To solve this problem, we use all of the information given and set up three equations. First, we assign a variable to each of the three investment amounts:
The first equation indicates that the sum of the three principal amounts is $12,000.
We form the second equation according to the information that John invested $4,000 more in mutual funds than he invested in municipal bonds.
The third equation shows that the total amount of interest earned from each fund equals $670.
Then, we write the three equations as a system.
To make the calculations simpler, we can multiply the third equation by 100. Thus,
Step 1. Interchange equation (2) and equation (3) so that the two equations with three variables will line up.
Step 2. Multiply equation (1) by and add to equation (2). Write the result as row 2.
Step 3. Add equation (2) to equation (3) and write the result as equation (3).
Step 4. Solve for in equation (3). Back-substitute that value in equation (2) and solve for Then, back-substitute the values for and into equation (1) and solve for
John invested $2,000 in a money-market fund, $3,000 in municipal bonds, and $7,000 in mutual funds.
Try It #1
Solve the system of equations in three variables.
Identifying Inconsistent Systems of Equations Containing Three Variables
Just as with systems of equations in two variables, we may come across an inconsistent system of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as or some other contradiction.
Example 4
Solving an Inconsistent System of Three Equations in Three Variables
Solve the following system.
- Answer
-
Looking at the coefficients of we can see that we can eliminate by adding equation (1) to equation (2).
Next, we multiply equation (1) by and add it to equation (3).
Then, we multiply equation (4) by 2 and add it to equation (5).
The final equation is a contradiction, so we conclude that the system of equations in inconsistent and, therefore, has no solution.
Analysis
In this system, each plane intersects the other two, but not at the same location. Therefore, the system is inconsistent.
Try It #2
Solve the system of three equations in three variables.
Expressing the Solution of a System of Dependent Equations Containing Three Variables
We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.
Example 5
Finding the Solution to a Dependent System of Equations
Find the solution to the given system of three equations in three variables.
- Answer
-
First, we can multiply equation (1) by and add it to equation (2).
We do not need to proceed any further. The result we get is an identity, which tells us that this system has an infinite number of solutions. There are other ways to begin to solve this system, such as multiplying equation (3) by and adding it to equation (1). We then perform the same steps as above and find the same result,
When a system is dependent, we can find general expressions for the solutions. Adding equations (1) and (3), we have
We then solve the resulting equation for
We back-substitute the expression for into one of the equations and solve for
So the general solution is In this solution, can be any real number. The values of and are dependent on the value selected for
Analysis
As shown in Figure 5 , two of the planes are the same and they intersect the third plane on a line. The solution set is infinite, as all points along the intersection line will satisfy all three equations.
Q&A
Does the generic solution to a dependent system always have to be written in terms of
No, you can write the generic solution in terms of any of the variables, but it is common to write it in terms of x and if needed and
Try It #3
Solve the following system.
Media
Access these online resources for additional instruction and practice with systems of equations in three variables.


17.3: Specialized Diets
By the end of this section, you will be able to:
- Examine the standard dietary recommendations for optimal health
- Recognize specific therapeutic diets for diseases or conditions
- Describe consistency modification diets
- Identify what enteral nutrition is and how to administer it
- Understand what parenteral nutrition is and how to administer it
Sometimes dietary intake must be adapted to meet individual nutrient needs. This may include restricting or increasing specific nutrients in the diet. It may also include modifying the consistency of the food when there is impaired swallowing or a need to give rest to the digestive tract. Sometimes nutrition must be administered via alternate means, such as directly into the stomach or via a central line. Nurses must be skilled in identifying, implementing, and evaluating therapeutic diets of all types across a person’s life span and in both acute and chronic settings.
Standard Dietary Recommendations
Good nutrition is essential to healthy living. A healthy diet is key to obtaining optimal nutrition. Holistic nursing care includes routinely assessing a patient’s nutritional status and providing the patient with sound instruction and coaching regarding their dietary needs for optimal nutrition intake. This should occur in all patient care settings. Whether the patient maintains a healthy lifestyle, partakes in unhealthy dietary habits, or has a chronic illness that requires a specialized diet, nutritional health promotion must be made available to all patients. It is essential that nurses understand nutritional principles and be skilled in teaching healthy dietary habits. This section reviews the standards for a healthy diet and discusses MyPlate , an eating strategy designed by the Center for Nutrition Policy and Promotion at the USDA to simplify and standardize dietary eating patterns. Food labels and food safety concerns are also discussed.
Standards for a Healthy Diet
A healthy diet is one that meets the daily recommended intake for macronutrients and micronutrients. Based on the Healthy Eating Index, the USDA recommends the following pattern for dietary intake:
- carbohydrates: 50 to 60 percent of calories per day with less than 10 percent added sugars
- fat: 20 to 30 percent of calories per day with less than 10 percent saturated fat
- protein: 10 to 20 percent of calories per day
- fiber: 25 grams per day
- fluids: 2.5 liters per day
The USDA recommends daily allowances for sodium , calcium, and vitamins D and B12 as well. A healthy diet must take into consideration individual caloric needs. Caloric needs are based on biological sex, age, and activity level. Some diets require strict adherence to the USDA recommendations, including calorie counting; however, the average patient will benefit from teaching related to general principles. Here are some general guidelines for healthy eating:
- Eat a variety of foods, including plenty of vegetables, fruits, and whole grains.
- Choose healthy proteins such as lean meats, fish, poultry, low-fat dairy products, and legumes.
- Drink plenty of water.
- Limit salt, alcohol, saturated and trans fats, and added sugar.
- Eat whole foods as much as possible and avoid highly processed foods.
A healthy diet prioritizes the consumption of sufficient amounts of vegetables and fruits on a daily basis. Vegetables and fruits should be eaten whole whenever possible. There are different categories of vegetables: dark green, red and orange, beans, nuts and seeds (legumes), and starches such as potatoes and cassava. Fruits may be fresh or frozen, canned, or dried. It is best to limit fruit juices, which may be high in calories.
Whole grains are an important part of a healthy diet; they include products made from wheat, rice, or oats that have not been refined. Refined grains, such as white rice and white flour, are grain kernels that have been processed to remove the bran and germ covering of the grain. This refining process removes important nutrients, such as fiber, iron, and vitamins. Whole grains use the entire grain kernel and are more nutritious than refined grains.
Foods that are high in protein include animal products such as seafood, beef, poultry, and eggs, as well as plant-based foods such as beans, peas, seeds, and nuts. Protein choices should be lean and varied. Dairy is a good source of protein, vitamins, and minerals, but low-fat versions of milk, yogurt, and cheese are typically healthier choices. Fats should be limited to 20 to 30 percent of calories, with strict limits on saturated fats. Oils are liquid fats that provide important nutrients. Oils are classified as unsaturated, polyunsaturated, or saturated. A healthy diet will include unsaturated or polyunsaturated fats and limit saturated fats.
Dietary teaching for everyone should focus on developing consistent eating patterns that include healthy amounts of carbohydrates, proteins, fats, and plenty of fresh water. A healthy diet should limit added sugars, saturated fats, sodium, and alcohol. Due to the variety of foods available, dietary choices can easily be adapted to meet cultural, religious, or budgetary preferences.
Refer back to Unfolding Case Study #3: Part 4 and Unfolding Case Study #3: Part 5 to review the patient data.
| Nursing Notes |
1600:
When asked about diet, patient reports eating out for most meals. She states, “I just don’t have time to cook. It’s much easier to grab a burger on the way home.  No way I could cook after a work shift.” |
MyPlate
MyPlate (Figure 17.5) is a dietary tool developed by the USDA to help individuals maintain healthy eating patterns throughout life. MyPlate uses a 9-inch plate as a visual tool for meeting the USDA’s general guidelines. Using the MyPlate method, vegetables and fruits make up half the plate, grains make up roughly a quarter of the plate, and proteins make up the final quarter. A serving of low- fat dairy is included as well. A personalized MyPlate plan is based on age, sex, height, weight, and activity level. The MyPlate website and MyPlate app are available to identify calorie needs and provide simple planning recommendations based on individual needs.
Food Labeling
Scientists and government agencies have developed food label s to improve decision-making regarding nutrition intake. The USDA Dietary Guidelines provide recommended daily intakes based on DRIs for macronutrients, micronutrients, and fiber . Food labels report the recommended daily allowances for specific nutrients as a percent daily value per serving. A serving is not a regulated amount but rather a typical volume of food consumed. Food labels always identify the number of servings per container, the number of calories per serving, and the percent daily value per serving for specific nutrients.
The food label is a tool used to educate the public about the nutritional value of a product, but what if a person cannot read the label? It is important to remember that people often develop poor vision as they age. As nurses, we should assess our patients’ vision using a Snellen chart to determine the need for a referral for further investigation by an ophthalmologist. We can educate our patients on the use of a handheld magnifier to enhance the print of nutritional labels. We can also encourage them to use the online Myplate.org feature to predetermine which foods to purchase before shopping. Many of the healthier foods such as whole fruits, vegetables, and pure animal protein do not require food labels.
Sometimes, manufacturers will choose to include daily values per container. Labels must include, but are not limited to, the following nutrients: carbohydrates, calcium, iron, potassium, vitamin D, fat , and protein. Fats are broken down into total fat, percent saturated fat, and cholesterol. Carbohydrates are broken down into total carbohydrates, fiber, total sugar, and added sugar. In general, a serving with a percent daily value of 5 percent or less for a nutrient is considered to have a low nutrient value. A serving with a percent daily value of 20 percent or more is considered to have a high nutrient value. A law passed in 2016 required food manufacturers to update their labels. Figure 17.6 details the changes.
Food labels have become more sophisticated over the years, incorporating nutrition science in a way that is more relevant and easily accessible. Information on how to understand and use the Nutrition Facts label is provided by the USDA.
Food Safety
Food safety is an important part of healthy living. The goal of food safety is to prevent foodborne illnesses such as bacterial infections. The following organisms are the cause of most foodborne infections: Campylobacter , Salmonella , and Escherichia coli . Symptoms common to these infections are nausea, vomiting, diarrhea, abdominal pain, and fever. Public health departments have foodborne surveillance programs that include a national reporting system to track outbreaks. Salmonella , Shigella , and Listeria are organisms commonly included in the national reporting system. Shigella is a highly contagious intestinal infection. Listeria is caused by improper processing or pasteurization and is most harmful to individuals who are pregnant, older, or immunocompromised. Surveillance helps track infections and identify outbreaks for public health intervention.
Prevention is key to food safety and reducing the effects of foodborne illness. Whether food is being prepared in the home, a restaurant, or an institution, the same four principles of food safety apply:
- Keep hands, utensils, and surfaces clean.
- Separate raw meat, seafood, and eggs from other foods.
- Cook foods to the appropriate internal temperature.
- Chill cooked foods within two hours.
Healthy food practices are essential to good health. Nurses should reinforce these prevention strategies by encouraging good hygiene and appropriate food handling practices and providing resources for patient education.
Maintaining safe practices for handling and cooking food is vital for good health. Review the recommendations for food safety provided by the CDC. What strategies would you advise for counseling patients about safe eating practices?
Therapeutic Specific Diets
Standard dietary recommendations guide individuals to make nutritious choices, maintain a healthy weight, and reduce the risk of chronic disease. The development of chronic disease affects nutritional goals and complicates nutritional intake. For example, individuals who develop diabetes or heart disease have new nutrient demands depending on the severity of the disease. Someone with diabetes may be advised to maintain stricter calorie or carbohydrate goals, while someone with heart disease will be encouraged to pursue a low- fat , low- sodium dietary pattern. Physiological conditions, such as stroke and reflux disease, can result in difficulty swallowing ( dysphagia ), requiring changes in food texture. Some disease states and medications can impair nutrient absorption, requiring supplementation of nutrients. Many chronic diseases include specialized diets in the treatment plan. It is important for nurses to know the differences between these diets, know why the specialized diets are ordered, and work closely with dietitians and the clinical team to ensure optimal nutrition outcomes.
Consistent Carbohydrate Diet
Carbohydrates are often misunderstood. Many people limit carbohydrates as a means of calorie control, yet both simple and complex carbohydrates are healthy and important nutrients. There are conditions under which carbohydrates must be controlled, however. An example of a diet focused on carbohydrate control is the consistent carbohydrate diet, which is recommended as a treatment for diabetes. Diabetes is a metabolic disorder affecting the utilization of glucose by the body cells. Poor glucose utilization can lead to acute acid-base disturbances and chronic effects including immune dysfunction, cardiovascular disease, and neurovascular changes. The negative effects of diabetes can best be prevented by effective blood sugar control.
The consistent carbohydrate diet maintains the recommended amount of carbohydrates per day (40 to 60 percent of the diet) but is designed to meet metabolic goals through the consumption of consistent carbohydrates at each meal. This consistency spreads out the carbohydrates over all daily meals with the goal of maintaining steady glycemic control throughout the day, avoiding hypoglycemic and hyperglycemic blood glucose swings. This mimics the healthy body’s natural tendency toward glucose control and insulin release. The consistent carbohydrate diet limits the use of added sugars and encourages healthy carbohydrate choices at each meal. The emphasis is on controlling carbohydrates rather than calories. This diet supports healthy dietary patterns for all individuals, encouraging regular consumption of whole grains, whole fruits and vegetables, and low-fat dairy products, and avoiding added sugars and refined and processed foods.
Fat-Restricted Diet
Twenty to 30 percent of a healthy diet should include fat, an essential, energy-dense macronutrient . Unfortunately, not all fat is considered “good” fat. In general, individuals eating a healthy diet should limit saturated fat to 10 percent of calories consumed. Most saturated fat comes from animal proteins. Because fat is associated with increased risk of obesity, heart disease, hypertension, and stroke, individual treatment plans may include fat-restricted diets to limit risk or decrease disease progression.
Fat-restricted diets adhere to the same USDA recommendations for a healthy diet, which is to consume between 20 and 30 percent of calories as fat but limit the types of fat consumed by avoiding saturated and trans fats. Saturated fat is typically solid at room temperature and is high in cholesterol. High-fat meats, butter, whole milk products, and coconut oil are common saturated fats. Trans fats are artificially produced by a process called hydrogenation. Partially hydrogenated fats are found in processed foods such as cookies, crackers, and margarine spreads. For patients with heart disease or at risk for developing heart disease, the American Heart Association recommends eliminating trans fats and limiting saturated fats to 5 to 6 percent of calories consumed, which is significantly lower than the dietary standard of 10 percent. Substituting more monounsaturated and polyunsaturated fats, such as canola, olive, and corn oils, is a healthy way to obtain needed fat in the diet while reducing the risk associated with “bad” fats.
High-Fiber Diet
Dietary fiber is a type of carbohydrate that adds bulk to aid in digestion. Fiber is found in plant foods and is categorized as soluble or insoluble. A high-fiber diet is recommended for most individuals, as it improves digestive health, aids in weight control, and reduces the risk of colon cancer and other intestinal disorders. Foods high in fiber include whole fruits and vegetables, whole grains, and legumes. High-fiber foods are known to decrease risk of gastrointestinal disorders, cardiovascular disease, and many metabolic disorders and should be included in all healthy diet routines.
Low-Fiber Diet
There are a few conditions that contraindicate a high-fiber diet. Limiting the fibrous content of foods eases digestion. Medical conditions that require periods of gastrointestinal rest benefit from a short-term reduction in fiber intake. Gastrointestinal rest allows for limited stimulation and processing in the intestinal tract, and a low-fiber diet is often included in the treatment plan. Eggs, creamy nut butters, low-fiber whole grain cereals, vegetable and fruit juices, applesauce, and canned fruits and vegetables can be included in a low-fiber diet. This diet should avoid all raw or undercooked vegetables, whole grains, nuts, seeds, and fibrous meats. A low-fiber diet is often used following intestinal surgeries or radiation treatments to the abdomen. Fiber should be slowly added back into the diet as digestive health returns.
Sodium Restriction Diet
The general population should limit sodium intake to less than 2,300 mg/day. Sodium is an essential element needed for many basic physiological processes. Sodium aids in food processing as a flavor enhancer, a curing agent and preservative, and a thickening agent. It also helps regulate the moisture content of food. Because of its versatility, many processed foods are high in sodium content. Unfortunately, high sodium levels increase the risk of cardiovascular disease and hypertension and negatively impact fluid balance within the body. Therefore, sodium-restricted diets are commonly prescribed for individuals with heart disease or at risk for heart disease and chronic fluid retention.
The first step in maintaining a low-sodium diet is to eliminate processed foods from the diet. Eating whole, natural foods keeps the diet naturally low in sodium. Next, all food label s should be carefully monitored for sodium content. Products with less than 200 mg per serving are healthy choices for this diet. Eating a diet high in fresh fruits and vegetables and fresh lean meats, reading food labels, and substituting herbs and spices for added salt are important ways to maintain a low-sodium diet and manage health risks.
Kidney Diet
The kidneys regulate fluid and electrolyte balance in the body.  A kidney diet is prescribed when kidney function becomes chronically impaired and restricting certain nutrients is necessary to preserve kidney function. Foods high in protein , sodium, and potassium are limited in a kidney diet. Protein metabolism leads to waste products that must be filtered by the kidneys; therefore, a kidney diet will limit protein (animal or plant based) to small portions (2–3 oz). Because water follows sodium in the body, high sodium in the blood can lead to excess fluid volume and increased blood pressure. Because high blood pressure adversely impacts the function of delicate nephrons, sodium restrictions are placed to help preserve kidney function. With advanced kidney disease, phosphorous and potassium are also restricted. While most people on a kidney diet should have 2,000 mL intake of fluid daily, in advanced disease fluid may be restricted even further based on individual need. Routine lab work is required to monitor kidney function, fluid status, and electrolytes.
High-Calorie, High-Protein Diet
A high- calorie , high-protein diet may be prescribed when there is a need for extra nutrition to aid healing. Individuals with extensive burns or poor healing wounds may be placed on a diet rich in quality proteins, providing needed energy and nutrients to speed healing and tissue repair. This diet minimizes carbohydrate intake and increases protein intake. Proteins with high amounts of saturated fats should be avoided. Egg whites, fish, fowl, dairy, and legumes are excellent protein choices. Avocados, cheese, nuts, and dried fruit make excellent calorie-dense snack choices. Weight trends must be closely monitored; also, because high protein levels can negatively impact kidney function, kidney labs must be routinely followed.
Consistency Modification Diets
Some diets are designed to alter the texture and consistency of foods rather than the nutritional content. Anatomical abnormalities, such as cleft palates, may prevent normal mastication. Neurological dysfunction can impede swallow reflexes. Gastroesophageal reflux disease can cause esophageal strictures that can cause dysphagia (difficulty swallowing). These conditions often require changes in food texture to improve nutritional access and decrease the risk of aspiration of food particles into the lungs. Typically, a speech pathologist will make recommendations to the clinical team regarding the need for modified diets based on the results of swallow studies. Nurses must collaborate with the whole clinical team, as they are responsible for managing the dietary intake of patients on modified diets. It is essential to understand, maintain, and monitor outcomes for all patients on modified diets.
Liquid Diet
Liquid diets are used to meet short-term clinical goals. Liquids are classified by the consistency of the fluid. Clear liquids melt to a transparent fluid that contains no pulp. A full liquid is a fluid that contains sufficient residue but still maintains a fluid form. Sometimes the consistency of liquids is restricted based on viscosity. Liquid diets are implemented for a variety of reasons and are often an essential part of the treatment plan.
Clear Liquid
A clear liquid diet is often used as short-term treatment with specific clinical goals. Clear liquids are necessary before and after certain medical procedures (including after surgery) and for patients who need to rest their bowels or replace fluid and electrolytes, for example, due to severe diarrhea. This is not a diet that can be maintained long term because it has little nutritional value. A clear liquid diet includes transparent liquids with minimal residue. This includes broths, tea, clear juices, clear gelatin, popsicles, and sport drinks. The liquids cannot contain any pulp or substance.
Clear liquids can be thickened to improve the consistency for individuals who aspirate on thin liquids. A thickening agent is added to the clear liquid to increase the viscosity, allowing for the thickened liquids to move more slowly and improve swallowing. The viscosity can be varied as needed. Common consistencies include nectar thick, honey thick, or pudding thick. Nurses often collaborate with speech pathologists when thickened liquids are being considered.
Full Liquid
A full liquid diet contains residue and is not transparent. It includes foods that become liquid at room temperature. While it contains a sufficient amount of nutrition, it can be difficult to meet daily nutritional needs and should only be used short term. Full liquid diets include cream soups, ice cream, pudding, and juices with pulp. A full liquid diet is part of the treatment plan when restarting oral feeding following a period of enteral or parenteral feeding or for short-term care when a patient is unable to tolerate a mechanical soft diet.
Soft Diets
Soft diets are diets in which the consistency and texture of the food are manipulated to improve swallowing or digestion. These diets typically contain lower fiber (less than 2 grams per serving) and a soft texture. They exclude fibrous meats, raw vegetables and fruits, beans, shellfish, and whole grains. The texture is dependent on the specific impairment. Soft diets are often used for patients with some degree of dysphagia or for postsurgical care.
Mechanical Soft
A mechanical soft diet is most often used with individuals who have a chewing or swallowing deficit. This is commonly seen in patients with dysphagia or dental impairment. The food is processed to a soft texture. Vegetables and meats should be tender, well cooked, and chopped or ground to the texture of mashed potatoes. Eggs, white rice and pasta, applesauce, and dairy products work well in a mechanical soft diet.
Pureed Diet
Pureed diets are processed to a smooth liquid consistency. Unlike a full liquid diet, pureed foods can come from a variety of sources, though it is best to avoid nuts and raw fruits and vegetables. It is best to puree foods separately. A pureed diet is prescribed for patients with an inability to chew and for some types of dysphagia. It is typically used for a short-term period following acute illness or injury but may be used for the long term when dysphagia cannot be resolved or when frailty or dementia limit nutritional intake at the end of life.
See the competency checklist for Assisting a Patient with Eating. You can find the checklists on the Student resources tab of your book page on openstax.org.
Enteral Nutrition
An alternate form of nutrition, enteral nutrition , utilizes a tube or catheter for administration. Enteral feedings are liquid and come in a variety of formulas to meet specific needs. These feedings are prescribed for individuals who have lost the ability to ingest food orally but who otherwise have a functioning digestive tract, capable of absorption and digestion of nutrients. Enteral nutrition typically has two routes of administration: nasogastric (NG) tube and percutaneous enteral gastrostomy (PEG) tube. Care of the patient on enteral feedings requires the expertise of an interdisciplinary team. Nurses are integral members of the enteral nutrition team and must be skilled in the administration, management, and evaluation of nutritional status when enteral nutrition is prescribed. Figure 17.7 shows the different access points for enteral tubes.
Nasogastric and Orogastric Tubes
NG tubes are sometimes placed when food cannot be safely ingested. They are utilized for the short term only, typically less than four weeks. The NG tube is a single- or double-lumen tube that is passed through the nares, pharynx, and esophagus and into the stomach; it can be used for suctioning stomach contents and administering medications in addition to providing enteral nutrition. Orogastric (OG) tubes work in the same manner except they are inserted through the mouth into the esophagus and then into the stomach. OG tubes are typically used with mechanically intubated and sedated patients and should never be used in conscious patients because they can induce a gag reflex and cause vomiting.
See the competency checklist for Administering a Tube Feeding. You can find the checklists on the Student resources tab of your book page on openstax.org.
Percutaneous Enteral Gastrostomy Tube
A PEG tube is placed directly through the abdominal wall via endoscopy. The PEG tube is placed in the stomach for direct administration of formula feeds and medications. In some situations, the tube can be advanced through the gastric pouch into the proximal part of the small intestine (jejunostomy). PEG tubes and jejunostomies are utilized when alternative nutrition is needed long term. PEG tubes may also be used when there is an obstruction to the esophagus, or the esophagus has been removed. Percutaneous endoscopic jejunostomy (PEJ) tubes are inserted through the abdominal wall directly into the jejunum, bypassing the esophagus and stomach. PEJ tubes are used when all or part of the stomach has been removed or if the provider determines PEJ placement would best suit the patient’s needs.
See the competency checklist for Caring for a Gastrostomy Tube. You can find the checklists on the Student resources tab of your book page on openstax.org.
Nursing Considerations for Administration
Nursing considerations for enteral nutrition involve management of care concerning placement and maintenance of the tube, administration of formula feeds and medications, and prevention and monitoring of complications. Care is dependent on the type of tube being used and the underlying medical conditions of the patient.
Communicating about NG Tube Insertion
Nurse:
Rosa, BSN
Clinical setting:
ICU
Years in practice:
1
Facility location:
Vermont
I was caring for a patient who had been in an automobile accident where he sustained multiple injuries all over his body, including an injury to his mouth and jaw. His eyes were swollen shut, and he was rarely conscious. The last nurse attempted to insert the NG tube twice, but both times, Mr. Zhu resisted by twisting and turning his head and pulling it out with his hands. During report, my supervisor told me that in order to insert the tube, I might need to increase Mr. Zhu’s sedation or request for an order for wrist restraints. I considered all of these options and decided to try another way. I gathered the supplies and approached Mr. Zhu’s room. I knocked on the door and began talking to Mr. Zhu, even though he seemed unconscious and unaware of my presence. I told him who I was and why I was there, and I let him know where I was going to touch him to do his assessment and care.  These injuries meant that he could not chew his food or have anything in his mouth for a while. In order to give him nutrition, he needed a tube from his nose to his stomach. This was better for him than any other option right now. I told him it would be uncomfortable but quick. Mr. Zhu did not move or indicate understanding of what I told him. However, when I began to insert the NG tube, it went in smoothly, and Mr. Zhu did not resist. Once I was finished, I thanked Mr. Zhu and told him that I had secured the tube so that it would not be pulled out and he would not have to have it reinserted again. I thanked him again and exited the room.
An NG tube can be placed by a nurse but requires a clinician’s order. Once placed, the NG tube is anchored to the nose or cheek, and the length of the external tubing is documented. Placement must be verified via x-ray imaging prior to use. A PEG tube or jejunostomy is placed by a physician via endoscopic procedure. Placement is also verified by x-ray imaging prior to use. The placement of NG and PEG tubes must be verified prior to every subsequent use, per facility protocol. The tubing is monitored for migration every shift and with every use. If placement of the tube is in question, gastric contents can be aspirated and their pH tested for appropriate acidity. Follow-up x-rays can be ordered for reevaluation of placement if needed. Incorrect placement can lead to aspiration or perforation.
The registered dietitian determines the type of formula needed following patient assessment. The nurse administers the enteral feeding as prescribed. Enteral nutrition can be administered via bolus or pump with a set volume and rate of administration. Free water flushes are commonly administered prior to and following administration or in tandem with an enteral feeding pump. Tubing should be assessed for migration and obstructions with every formula feed. The nurse should monitor for leakage and skin breakdown. The tubing should be cleansed routinely according to facility protocol, typically using water or saline and soft gauze. Any redness or purulent drainage should be documented and reported.
Enteral feedings are associated with complications. It is the nurse’s responsibility to mitigate these types of risk:
-
To reduce the risk of aspiration,
- maintain slow, steady administration rates;
- keep the patient’s head of bed elevated 30° to 45° as tolerated;
- monitor for migration of visible NG/OG tubing; and
- assess for residual gastric contents and follow facility policy for rate reduction protocols.
-
To reduce the risk of clogged tubing,
- flush the tube with water every shift and prn (as needed),
- flush tubes before and after administration of feedings or medication,
- assess for any residual gastric contents and follow facility policy for rate reduction protocol, and
- irrigate tubing per facility protocol.
-
To reduce the risk of gastric distress,
- administer feedings at room temperature;
- maintain prescribed rate of administration;
- assess placement every shift and prior to feeding or medication administration; and
- assess for signs of gastric distress including nausea, diarrhea, abdominal pain, and distension.
Parenteral Nutrition
A form of alternative nutrition is parenteral nutrition (PN) , in which a nutrient formula is administered intravenously. PN has a high osmolality and causes irritation to veins, so it should always be administered through a central venous catheter. A central venous catheter provides intravenous access directly to the vena cava, which empties into the heart. PN is reserved for individuals who do not have a functional gastrointestinal tract. This includes children with congenital gastrointestinal malformations, individuals with severe ulcerative colitis or bowel obstruction, and those who are critically ill with organ failure. The formula is composed of macronutrients and micronutrients. The three major macronutrients are dextrose (a form of glucose), proteins, and fat (lipid) emulsions. Vitamins, trace minerals, and electrolytes are also added. There are two types of PN: peripheral parenteral nutrition (PPN) and total parenteral nutrition (TPN).
Types of Parenteral Nutrition
PN requires central venous access. A central venous catheter is inserted in the jugular or subclavian vein and terminates in the vena cava. Central lines can be tunneled for long-term use (months to years) or nontunneled for intermittent use. Peripherally inserted central catheter (PICC) lines are long catheters inserted through a vein in the arm. Commonly, the basilic vein is used, terminating in the superior vena cava. PICC lines are used for intermediate use, typically weeks to months. Standards of care must be followed to reduce complication rates and improve patient outcomes. Policies regarding the care and use of central lines should follow facility protocols.
Peripheral Parenteral Nutrition
PPN is administered through a PICC line using a pump. PPN is intended for short-term use, typically less than ten days. The peripheral insertion reduces risk and irritation by limiting osmolality, which can lead to higher volume per feeding. Care of the PICC line should follow standard guidelines and facility protocols.
Total Parenteral Nutrition
TPN is administered through a central venous catheter. TPN is administered when the gastrointestinal tract is not functional or when the patient needs to rest their bowels. Energy and nutrient needs are calculated based on individual need and body weight. Nutrients include amino acids, fatty acids, vitamins, minerals, and electrolytes. Carbohydrates are provided as dextrose. Lipid emulsions are often administered separately. In recent years, the use of three-in-one admixtures, which are composed of dextrose, proteins, and lipid emulsions, has improved patient care by streamlining TPN administration and reducing complications.
TPN is administered through a central line into the vena cava of the heart for long-term access. Close monitoring is required whenever a patient is receiving TPN. Weight, intake and output, and blood glucose must be monitored carefully. Bloodwork must be evaluated and documented to follow trends involving complete blood count, electrolytes and blood urea nitrogen (BUN), prealbumin, and albumin . Complications associated with TPN include central line infections, fluid volume overload, glucose and electrolyte disturbances, and liver and gallbladder dysfunction.
Nursing Considerations for Administration
Nursing considerations related to PN include catheter care, administration of parenteral nutrition (PN) , preventing and responding to complications, and ongoing assessment of nutritional status. Nurses should follow these guidelines for central line care and administration of PN:
- PICC dressings should be changed every seventy-two hours using sterile technique.
- PN should be administered through a dedicated lumen.
- Parenteral tubing should be changed every twenty-four hours.
- The PN label should be verified against the documented order.
Complications associated with PN include central line infections, sepsis, poor blood sugar control, hepatic dysfunction, and fluid and electrolyte imbalance. Nursing interventions should be geared toward prevention:
- Maintain strict sterile technique for central line care.
- Monitor vital signs routinely.
- Monitor blood sugar and administer insulin per protocol.
- Assess for fluid imbalance.
- Monitor labs for trends involving electrolytes, kidney, and liver function.


9.1: Development and Geography- An Introduction
If you could choose anywhere on the planet, where would you like to live? Would you choose a place with mountains or with beaches? A place with high taxes or few regulations? Do you love your community/state/country? Could you make more money somewhere else or might you be happier with warmer/colder weather? Do you think that you are able to realize your full personal potential in the place where you live now? Why or why not?  Nonetheless, there are certain indicators that geographers can use to categorize places according to how developed they are in terms of technology, infrastructure, wealth, and opportunity. As you might guess, the differences between places can be quite stark, but it’s important to understand the dynamics and geography of the patterns and processes associated with income, well-being, and opportunity. This chapter explains how those distinctions are made and how they vary across time and place using the concept of development (the processes related to improving people’s lives through access to resources, technology, education, wealth, opportunity, and choice).
Which places on earth are the most developed? There is no simple answer to this question. San Francisco is often voted as the most beautiful city in the U.S. but the cost of living makes this place unaffordable for all but the wealthiest of residents. Ancient cities like Jerusalem, Athens, and Baghdad contain amazing architecture and the roots of western civilization, but streets also tend to be narrow and housing crowded. China has experienced the fastest economic growth of any other country in the past 30 years, but it is accompanied by catastrophic levels of air pollution, poor working conditions, and severe limits on personal liberty. Within the United States, people in Colorado tend to be the healthiest while those in Utah have the largest houses and people in Texas are the most loyal to their home state. There are multiple ways to measure development, and geographers spend a great deal of time and effort studying, measuring, and quantifying the differences and commonalities.
A few general truths about development and wealth in our 21 st century world can simplify the complexity:
- The world continues to be divided into the Global North and Global South by the Brandt Line (See Figure 9.1 ). Levels of wealth, well- being, access to technology, and health tend to be higher in northern countries than in southern ones. The line is problematic because it over-generalizes, but it remains a meaningful starting point to understanding development from a global scale.
- The majority of people living in the 21 st century have a higher standard of living, earn more money, are healthier, and live longer than was the case for people 50 years ago.
- In spite of #2, the wealth disparity between those at the bottom and those at the top remains greater than ever before. According to Credit Suisse’s global wealth report, the globe’s richest 1% (of people) control more than half of the world’s wealth, up from 42.5% in 2008.


3.3: Instantaneous Velocity and Speed
- Explain the difference between average velocity and instantaneous velocity.
- Describe the difference between velocity and speed.
- Calculate the instantaneous velocity given the mathematical equation for the velocity.
- Calculate the speed given the instantaneous velocity.
We have now seen how to calculate the average velocity between two positions. However, since objects in the real world move continuously through space and time, we would like to find the velocity of an object at any single point. We can find the velocity of the object anywhere along its path by using some fundamental principles of calculus. This section gives us better insight into the physics of motion and will be useful in later chapters.
Instantaneous Velocity
The quantity that tells us how fast an object is moving anywhere along its path is the instantaneous velocity , usually called simply velocity . It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero. To illustrate this idea mathematically, we need to express position x as a continuous function of t denoted by x(t). The expression for the average velocity between two points using this notation is \(\bar{v} = \frac{x(t_{2}) - x(t_{1})}{t_{2} - t_{1}}\). To find the instantaneous velocity at any position, we let t 1 = t and t 2 = t + \(\Delta\)t. After inserting these expressions into the equation for the average velocity and taking the limit as \(\Delta\)t → 0, we find the expression for the instantaneous velocity:
\[v(t) = \lim_{\Delta t \to 0} \frac{x(t + \Delta t) - x(t)}{\Delta t} = \frac{dx(t)}{dt} \ldotp\]
The instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x with respect to t:
\[v(t) = \frac{d}{dt} x(t) \ldotp \label{3.4}\]
Like average velocity, instantaneous velocity is a vector with dimension of length per time. The instantaneous velocity at a specific time point t 0 is the rate of change of the position function, which is the slope of the position function x(t) at t 0 . Figure \(\PageIndex{1}\) shows how the average velocity \(\bar{v} = \frac{\Delta x}{\Delta t}\) between two times approaches the instantaneous velocity at t 0 . The instantaneous velocity is shown at time t 0 , which happens to be at the maximum of the position function. The slope of the position graph is zero at this point, and thus the instantaneous velocity is zero. At other times, t 1 , t 2 , and so on, the instantaneous velocity is not zero because the slope of the position graph would be positive or negative. If the position function had a minimum, the slope of the position graph would also be zero, giving an instantaneous velocity of zero there as well. Thus, the zeros of the velocity function give the minimum and maximum of the position function.
Given the position-versus-time graph of Figure \(\PageIndex{2}\), find the velocity-versus-time graph.
Strategy
The graph contains three straight lines during three time intervals. We find the velocity during each time interval by taking the slope of the line using the grid.
Solution
Time interval 0 s to 0.5 s: \(\bar{v} = \frac{\Delta x}{\Delta t}=\frac{0.5\; m − 0.0\; m}{0.5\; s − 0.0\; s} = 1.0\; m/s\)
Time interval 0.5 s to 1.0 s: \(\bar{v} = \frac{\Delta x}{\Delta t}=\frac{0.0\; m − 0.0\; m}{1.0\; s − 0.5\; s} = 0.0\; m/s\)
Time interval 1.0 s to 2.0 s: \(\bar{v} = \frac{\Delta x}{\Delta t}=\frac{0.0\; m − 0.5\; m}{2.0\; s − 1.0\; s} = -0.5\; m/s\)
The graph of these values of velocity versus time is shown in Figure \(\PageIndex{3}\).
Significance
During the time interval between 0 s and 0.5 s, the object’s position is moving away from the origin and the position-versus-time curve has a positive slope. At any point along the curve during this time interval, we can find the instantaneous velocity by taking its slope, which is +1 m/s, as shown in Figure \(\PageIndex{3}\). In the subsequent time interval, between 0.5 s and 1.0 s, the position doesn’t change and we see the slope is zero. From 1.0 s to 2.0 s, the object is moving back toward the origin and the slope is −0.5 m/s. The object has reversed direction and has a negative velocity.
Speed
In everyday language, most people use the terms speed and velocity interchangeably. In physics, however, they do not have the same meaning and are distinct concepts. One major difference is that speed has no direction; that is, speed is a scalar.
We can calculate the average speed by finding the total distance traveled divided by the elapsed time:
\[Average\; speed = \bar{s} = \frac{Total\; distance}{Elapsed\; time} \ldotp \label{3.5}\]
Average speed is not necessarily the same as the magnitude of the average velocity, which is found by dividing the magnitude of the total displacement by the elapsed time. For example, if a trip starts and ends at the same location, the total displacement is zero, and therefore the average velocity is zero. The average speed, however, is not zero, because the total distance traveled is greater than zero. If we take a road trip of 300 km and need to be at our destination at a certain time, then we would be interested in our average speed.
However, we can calculate the instantaneous speed from the magnitude of the instantaneous velocity:
\[Instantaneous\; speed = |v(t)| \ldotp \label{3.6}\]
If a particle is moving along the x-axis at +7.0 m/s and another particle is moving along the same axis at −7.0 m/s, they have different velocities, but both have the same speed of 7.0 m/s. Some typical speeds are shown in the following table.
| Speed | m/s | mi/h |
|---|---|---|
| Continental drift | 10 -7 | 2 x 10 -7 |
| Brisk walk | 1.7 | 3.9 |
| Cyclist | 4.4 | 10 |
| Sprint runner | 12.2 | 27 |
| Rural speed limit | 24.6 | 56 |
| Official land speed record | 341.1 | 763 |
| Speed of sound at sea level | 343 | 768 |
| Space shuttle on reetry | 7800 | 17,500 |
| Escape velocity of Earth* | 11,200 | 25,000 |
| Orbital speed of Earth around the Sun | 29,783 | 66,623 |
| Speed of light in a vacuum | 299,792,458 | 670,616,629 |
| *Escape velocity is the velocity at which an object must be launched so that it overcomes Earth’s gravity and is not pulled back toward Earth. |
Calculating Instantaneous Velocity
When calculating instantaneous velocity, we need to specify the explicit form of the position function x(t). If each term in the x(t) equation has the form of At n where A is a constant and n is an integer, this can be differentiated using the power rule to be:
\[\frac{d\left(A t^{n}\right)}{d t}=A n t^{n-1} \ldotp \label{3.7}\]
Note that if there are additional terms added together, this power rule of differentiation can be done multiple times and the solution is the sum of those terms. The following example illustrates the use of Equation \ref{3.7}.
The position of a particle is given by x(t) = 3.0t + 0.5t 3 m.
- Using Equation \ref{3.4} and Equation \ref{3.7}, find the instantaneous velocity at t = 2.0 s.
- Calculate the average velocity between 1.0 s and 3.0 s.
Strategy
Equation \ref{3.4} give the instantaneous velocity of the particle as the derivative of the position function. Looking at the form of the position function given, we see that it is a polynomial in t. Therefore, we can use Equation \ref{3.7}, the power rule from calculus, to find the solution. We use Equation \ref{3.6} to calculate the average velocity of the particle.
Solution
- v(t) = \(\frac{dx(t)}{dt}\) = 3.0 + 1.5t 2 m/s. Substituting t = 2.0 s into this equation gives v(2.0 s) = [3.0 + 1.5(2.0) 2 ] m/s = 9.0 m/s.
- To determine the average velocity of the particle between 1.0 s and 3.0 s, we calculate the values of x(1.0 s) and x(3.0 s):
\[x(1.0 s) = \big[(3.0)(1.0) + 0.5(1.0)^{3} \big]m = 3.5\; m\]
\[x(3.0 s) =\big[(3.0)(3.0) + 0.5(3.0)^{3}\big] m = 22.5\; m\]
Then the average velocity is
\[\bar{v} = \frac{x(3.0\; s) - x(1.0\; s)}{t(3.0\; s) - t(1.0\; s)} = \frac{22.5 - 3.5\; m}{3.0 - 1.0\; s} = 9.5\; m/s \ldotp\]
Significance
In the limit that the time interval used to calculate \(\bar{v}\) goes to zero, the value obtained for \(\bar{v}\) converges to the value of v.
Consider the motion of a particle in which the position is x(t) = 3.0t − 3t 2 m.
- What is the instantaneous velocity at t = 0.25 s, t = 0.50 s, and t = 1.0 s?
- What is the speed of the particle at these times?
Strategy
The instantaneous velocity is the derivative of the position function and the speed is the magnitude of the instantaneous velocity. We use Equation \ref{3.4} and Equation \ref{3.7} to solve for instantaneous velocity.
Solution
- v(t) = \(\frac{dx(t)}{dt}\) = 3.0 − 6.0t m/s
- v(0.25 s) = 1.50 m/s, v(0.5 s) = 0 m/s, v(1.0 s) = −3.0 m/s
- Speed = |v(t)| = 1.50 m/s, 0.0 m/s, and 3.0 m/s
Significance
The velocity of the particle gives us direction information, indicating the particle is moving to the left (west) or right (east). The speed gives the magnitude of the velocity. By graphing the position, velocity, and speed as functions of time, we can understand these concepts visually Figure \(\PageIndex{4}\). In (a), the graph shows the particle moving in the positive direction until t = 0.5 s, when it reverses direction. The reversal of direction can also be seen in (b) at 0.5 s where the velocity is zero and then turns negative. At 1.0 s it is back at the origin where it started. The particle’s velocity at 1.0 s in (b) is negative, because it is traveling in the negative direction. But in (c), however, its speed is positive and remains positive throughout the travel time. We can also interpret velocity as the slope of the position-versus-time graph. The slope of x(t) is decreasing toward zero, becoming zero at 0.5 s and increasingly negative thereafter. This analysis of comparing the graphs of position, velocity, and speed helps catch errors in calculations. The graphs must be consistent with each other and help interpret the calculations.
The position of an object as a function of time is x(t) = −3t 2 m. (a) What is the velocity of the object as a function of time? (b) Is the velocity ever positive? (c) What are the velocity and speed at t = 1.0 s?


9.8.5: Inferring Recombination From Genetic Data
-
- Last updated
- Save as PDF
- Ying Liu
- City College of San Francisco
In the preceding examples, we had the advantage of knowing the approximate chromosomal positions of each allele involved, before we calculated the recombination frequencies. Knowing this information beforehand made it relatively easy to define the parental and recombinant genotypes, and to calculate recombination frequencies. However, in most experiments, we cannot directly examine the chromosomes, or even the gametes, so we must infer the arrangement of alleles from the phenotypes over two or more generations. Importantly, it is generally not sufficient to know the genotype of individuals in just one generation; for example, given an individual with the genotype AaBb , we do not know from the genotype alone whether the loci are located on the same chromosome, and if so, whether the arrangement of alleles on each chromosome is AB and ab or Ab and aB (Figure \(\PageIndex{6}\)) . The top cell has the two dominant alleles together and the two recessive alleles together and is said to have the genes in the coupling (or cis ) configuration . The alternative shown in the cell below is that the genes are in the repulsion (or trans ) configuration .
Fortunately for geneticists, the arrangement of alleles can sometimes be inferred if the genotypes of a previous generation are known. For example, if the parents of AaBb had genotypes AAB B and aabb respectively, then the parental gametes that fused to produce AaBb would have been genotype AB and genotype ab . Therefore, prior to meiosis in the dihybrid, the arrangement of alleles would likewise be AB and ab (Figure \(\PageIndex{7}\)). Conversely, if the parents of AaBb had genotypes aaBB and AAbb , then the arrangement of alleles on the chromosomes of the dihybrid would be aB and Ab . Thus, the genotype of the previous generation can determine which of an individual’s gametes are considered recombinant, and which are considered parental.
Let us now consider a complete experiment in which our objective is to measure recombination frequency (Figure \(\PageIndex{8}\)). We need at least two alleles for each of two genes, and we must know which combinations of alleles were present in the parental gametes. The simplest way to do this is to start with pure-breeding lines that have contrasting alleles at two loci. For example, we could cross short-tailed mice, brown mice ( aaBB ) with long-tailed, white mice ( AAbb ). Based on the genotypes of the parents, we know that the parental gametes will be aB or Ab (but not ab or AB ), and all of the progeny will be dihybrids, AaBb . We do not know at this point whether the two loci are on different pairs of homologous chromosomes, or whether they are on the same chromosome, and if so, how close together they are.
The recombination events that may be detected will occur during meiosis in the dihybrid individual. If the loci are completely or partially linked, then prior to meiosis, alleles aB will be located on one chromosome, and alleles Ab will be on the other chromosome (based on our knowledge of the genotypes of the gametes that produced the dihybrid). Thus, recombinant gametes produced by the dihybrid will have the genotypes ab or AB , and non-recombinant (i.e. parental) gametes will have the genotypes aB or Ab .
How do we determine the genotype of the gametes produced by the dihybrid individual? The most practical method is to use a testcross (Figure \(\PageIndex{8}\)), in other words to mate AaBb to an individual that has only recessive alleles at both loci ( aabb ). This will give a different phenotype in the F 2 generation for each of the four possible combinations of alleles in the gametes of the dihybrid. We can then infer unambiguously the genotype of the gametes produced by the dihybrid individual, and therefore calculate the recombination frequency between these two loci. For example, if only two phenotypic classes were observed in the F 2 (i.e. short tails and brown fur ( aaBb ), and white fur with long tails ( Aabb ) we would know that the only gametes produced following meiosis of the dihybrid individual were of the parental type: aB and Ab , and the recombination frequency would therefore be 0%. Alternatively, we may observe multiple classes of phenotypes in the F 2 in ratios such as shown in Table \(\PageIndex{1}\):
|
tail phenotype |
fur phenotype |
number of progeny |
gamete from dihybrid |
genotype of F 2 from test cross |
(P)arental or (R)ecombinant |
|
short |
brown |
48 |
aB |
aaBb |
P |
|
long |
white |
42 |
Ab |
Aabb |
P |
|
short |
white |
13 |
ab |
aabb |
R |
|
long |
brown |
17 |
AB |
AaBb |
R |
Given the data in Table \(\PageIndex{1}\), the calculation of recombination frequency is straightforward:
\[\begin{align} \textrm{recombination frequency} &= \mathrm{\dfrac{number\: of\: recombinant\: gametes}{total\: number\: of\: gametes\: scored}}\\ \textrm{R.F.} &= \dfrac{13+17}{48+42+13+17}\\ &=25\% \end{align}\]


2.5.1: Rules and Derivations
Natural deduction systems are meant to closely parallel the informal reasoning used in mathematical proof (hence it is somewhat “natural”). Natural deduction proofs begin with assumptions. Inference rules are then applied. Assumptions are “discharged” by the \(\Intro{\lnot}\) , \(\Intro{\lif}\) , \(\Elim{\lor}\) and \(\Elim{\lexists{}{}}\) inference rules, and the label of the discharged assumption is placed beside the inference for clarity.
An assumption is any sentence in the topmost position of any branch.
Derivations in natural deduction are certain trees of sentences, where the topmost sentences are assumptions, and if a sentence stands below one, two, or three other sequents, it must follow correctly by a rule of inference. The sentences at the top of the inference are called the premises and the sentence below the conclusion of the inference. The rules come in pairs, an introduction and an elimination rule for each logical operator. They introduce a logical operator in the conclusion or remove a logical operator from a premise of the rule. Some of the rules allow an assumption of a certain type to be discharged . To indicate which assumption is discharged by which inference, we also assign labels to both the assumption and the inference. This is indicated by writing the assumption as “ \(\Discharge{A}{n}\) .”
It is customary to consider rules for all the logical operators \(\land\) , \(\lor\) , \(\lif\) , \(\lnot\) , and \(\lfalse\) , even if some of those are considered as defined.


1.4.1: Analysing Texts- Media and Theory
-
- Last updated
- Save as PDF
- Media Hack Team
- University of Otago and others via Creative Commons
Communication and media permeate our society. At its most basic level, we can think of communication as the exchange of information or meaning – but what does that mean? When are you not exchanging information or meaning? To try and help make sense of the wealth of encounters in which communication might be said to be occurring, we try and categorize communication into different types. Some of these types include:
- interpersonal communication, or one-on-one talking and rhetoric, where we analyse things like tone, body language, and speech
- mass communication, which includes one-to-many and many-to-many communication acts, with a particular interest in mediated communication, such as with the press.
- Organizational communication, where we look at how people organize their information exchanges to maintain and facilitate group behaviour
- Intercultural communication, which looks at the exchange of information and ideas across different cultural groups or subgroups.
But is communication purely a functional and pragmatic tool for the exchange of information, and serves no other function or plays no other role within society? Or is there more than one level to consider when studying the act and role of communication in a social context?
Building on work started by Robert T. Craig, we can generally talk about seven models or traditions of communication – we can look at any communication practice through the lens of one or more of these models to develop a more nuanced understanding of communication in everyday life.
The seven models (adapted from Miller) are:
- Rhetorical – this model is concerned primarily with communication as a discourse, and tends to concern itself primarily with interpersonal, one-to-one or one-to-few communication acts, such as speech. Post the linguistic turn of the mid- to late-twentieth century, rhetoric has expanded its area of focus to include mass communication that attempts to persuade, such as political communication and advertising. A rhetorical approach to communication might look at who was speaking to whom, in what context, and to what end or purpose (i.e.: to persuade or to change an opinion or belief).
- Semiotic – this model sees communication primarily as an exchange of signs within a meaning-making system. We will be going into much further depth into the idea of semiotics in a later module, but for now it is just worth noting that semiotics approaches communication in itself, seeing the communication as a sign within a sign system, which employs signs in culturally contextualized combinations to convey meaning.
- Phenomenological – this model is primarily concerned with communication as an experience. A phenomenological approach would see communication as both a representation and a reinforcement of what the communicators see to be self-evident. A phenomenological approach can take on both interpersonal and mass communications, and may also take under its purvey objects or ideas as sites of meaning-making.
- Cybernetic – this model views communication as a flow of information. This is not just the pragmatic A sends a message to B type of flow, but also tries to take into account factors which influence and constraint the flow of information, including social factors such as mores and etiquette, technological factors such as channel access and availability, and political factors such as regulation.
- Psychological or Socio-psychological – this model, as the name suggests, is concerned with the impact of communication acts on the individual, particularly their sense of self in society. This model sees communication as representing certain individual choices made in order to maximize benefit to the individual or group. We’ll come back to these ideas when we discuss symbolic interactionism in the next module.
- Sociocultural – this model sees communication as a way of replicating and reinforcing (and challenging) the social order. This approach assumes that people in societies have models of how that society should operate; communication acts to build, reinforce, and propagate these models.
- Critical – this model views communication as a set of assumptions that are open to challenge and negotiation; as you might have guessed, it has strong links to the socio-cultural tradition. Approaches such as Marxist critique are representative of this model.
We can look at any act of communication through any of these models, and uncover something different about the communication itself, the meaning it helps construct and reconstruct, and how that communication and the communicators involved fit within a socio-cultural context.
It’s worth stepping back even further for a moment, and turning your attention to the wider role of communication in everyday life. To do this, it is worth starting with a question: can you imagine a world without communication?
What does it look like?
What might be some of the cultural roles of communication in everyday life? In no particular order, we can talk about the following:
1 Communication tells stories, and perhaps more importantly, they retell stories. Communication, particularly mass communication and the mass media, tends to fall into repetitive and recursive patterns of representation and ideology, ones that often reinforce the dominant hegemony of a culture or society. For example, the narratives around masculinity and sports such as rugby — think of the way stories about All Blacks are constructed and reconstructed in the media.
By using these narrative strategies in our communication patterns, we are engaging in a kind of shorthand that facilitates communication behaviours, signals a shared culture or values system, and indicates an expectation of communicative and ideological symmetry. However, such myths also propagate a particular set of ideas about what is valued within a culture, and what or who is marginalized.
Let’s take a look at the New England Patriots for example – think of the language that is used to describe them , and the thoughts and feelings it evokes.
These sports stars and their games, which are a form of communication, reinforce particular, in this case modern and Western, views about masculinity, which then has an impact on how the readers of these texts understand the masculine in their culture.
We’ll be coming back to sports stars and cultural mythology when we touch on gender in a later module. In the meantime see this link :
2 A second role that communication plays in everyday life is to help maintain social order by taking on a kind of surveillance operation, in that communication demonstrates to a society what behaviors are considered acceptable and unacceptable. This may be through object representation, or it may be through commentary on active behaviors. An example of the former might be soaps. Soaps like Gossip Girl play out, in a mass media, the narratives of types of private lives and decision processes of different groups of people in a palatable, easy to understand and follow manner. It is not by accident that reality TV and soap operas follow the same structural beats. Both are structured to play out long-form ‘life’ narratives, with different characters taking on the role of hero or villain. An example of commentary might be gossip, whether about a sports star, a pop singer, or your neighbors.
Facebook is a great example of this, in that it runs on social communication – on gossip and networking – and people very deliberately structure their behavior on these sites to present the best ‘face’ and thus receive the most desirable commentary and feedback from others. For example, think about how long you spent choosing and setting up your profile picture to use for your Facebook page? Along similar lines, new communication channels and technologies now allow us to extend our interpersonal surveillance, both across space and through time. Whereas once students who went off to college in a way escaped the surveillance of their parents, now their parents can (unless they’ve locked their account) follow their children’s antics on Facebook. Those same antics may then form part of a person’s online profile which is seen by potential employers as they use those same tools of surveillance to get a sense of who you are before they call you in for a job interview.
Either way, this role of communication serves to demonstrate correct social behavior, and the consequences of incorrect social behavior, and to allow social sanctions and feedbacks on our own behavior and decisions.
3 A third role that communication plays is to help us interpret and make sense of information. This has become a particularly important role in the modern communication landscape, as information overload has become an increasingly common part of our everyday lives. Again, new media and electronic communication channels have raised the profile of this role of communication in everyday life. We can see this most clearly in news media, and the idea of agenda setting, but even the use of particular language or even images can also frame a topic or give subtle hints and clues as to how we are ‘meant’ to interpret information according to our culture.
4 This act of interpreting, of governing behavior, and even of storytelling, is of course at its heart a representation of a particular set of values and ideologies that emerge from and are part of a culture. It is very important to remember that no act of communication is ideologically neutral! Ideologies are part of our lifeworlds, part of the assumptions we make and encode into and decode out of our communication practices and behaviors, and so if we are to study communication, we need to address these ideological underpinnings.
5 That’s not to say that we always communicate thinking about these things – they are often covert, assumed and unchallenged parts of our everyday lives. But they exist, even when we think about communication on a basic uses and gratifications level. Uses and Gratifications Theory is a very simple theory you may already be familiar with at some level. It hypothesizes that audiences, and communicators in general, are active in why they seek out media and communicative exchanges. These four reasons are diversion, socialization, identity, and surveillance. We may have multiple simultaneous reasons, or we may just be bored and channel surfing. But even those decisions and positions – to be bored, to seek our diversion through television rather than another way, or to settle on a particular channel or show – reflect ideologies and social, technological, and political pressures that are so every day that we don’t even think about them anymore.
So, to recap. Communication is pervasive and an integral part of our everyday lives – we cannot imagine our world without communication. Communication serves a number of concurrent functions in society – pragmatic, normative, ideological and informational. We can approach an analysis of communication at a number of different levels – rhetorical, semiotic, phenomenological, cybernetic, socio-psychological, socio-cultural or critical. We can move between these different levels to uncover different aspects of communicative processes. But in general, it is important to remember that communication serves more than just a functional process of getting information from point A to point B. Communication also propagates what a society considers to be normal or normative behavior. It reinforces the dominant ideologies of a society, and may also create space for new ideologies to come in and challenge that dominant hegemony. We can see these ideological debates being played out as people challenge and critique existing stories and narratives that permeate and propagate through our communicative culture. We can break communication down to the level of the individual sign, or scale it up to see it play out over time across an entire culture, but communication is never ideologically neutral.
In the next section, we’ll discuss how communication fits and is a part of cultural systems.
Discussion Questions
- Can you identify a time or situation when you are not exchanging information or meaning, and in a wider context, discuss what a world without communication might be like?
- Find examples of each of the seven models of communication. Give reasons for your choices.
- In the media or elsewhere, identify examples of competing ideologies that challenge an existing hegemony. Describe their context – political, commercial, social and academic etc – and give a short outline of each, including the media role in communicating the information.
References
Craig, Robert T. and Heidi L Muller. Theorizing Communication: Readings Across Traditions . Los Angeles: Sage Publications, 2007.
Miller, K. Communication Theories: Perspectives, Processes and Contexts . 2 nd Ed. McGaw Hill: New York, 2005. Pp 1-16


8: Probability and Sampling Last updated Save as PDF Page ID 35025 Illustrative Mathematics OpenUp Resources 8.1: Probabilities of Single Step Events 8.1.1: Mystery Bags 8.1.2: Chance Experiments 8.1.3: What are Probabilities? 8.1.4: Estimating Probabilities Through Repeated Experiments 8.1.5: More Estimating Probabilities 8.1.6: Estimating Probabilities Using Simulation 8.2: Probabilites of Multi-step Events 8.2.1: Simulating Multi-step Experiments 8.2.2: Keeping Track of All Possible Outcomes 8.2.3: Multi-step Experiments 8.2.4: Designing Simulations 8.3: Sampling 8.3.1: Comparing Groups 8.3.2: Larger Populations 8.3.3: What Makes a Good Sample? 8.3.4: Sampling in a Fair Way 8.4: Using Samples 8.4.1: Estimating Population Measures of Center 8.4.2: Estimating Population Proportions 8.4.3: More about Sampling Variability 8.4.4: Comparing Populations Using Samples 8.4.5: Comparing Populations With Friends 8.5: Let's Put it to Work 8.5.1: Memory Test


Chapter 17: Revolutionary Creative Labor
-
-
Last updated
-
-
Save as PDF
-
This chapter elaborates the concept of revolutionary creative labor. The Arab uprisings, particularly the conflict in Syria, have given rise to a notion of creative resistance. Various activists, journalists, academics, and curators have used that phrase to celebrate a gamut of expressive practices and forms encompassing graffiti, digital memes and mash-ups, handheld banners, political rap, and others.¹ The wording combines two terms with overwhelmingly positive connotations that evoke human ingenuity and agency.
-
-
17.1: Introduction
-
Introduction of the goals of this chapter: to explore revolutionary creative labor and its impact on understanding precarity in cultural production, with a focus on the Arab uprisings, especially the conflict in Syria.
-
-
17.2: Creativity and Labor in Social Movement and Production Studies – A Snapshot
-
Examining concepts related to creativity and labor as covered by social movement and cultural production studies. How identifying something as “creative” is a strategic and discriminatory move.
-
-
17.3: Industrial and Revolutionary – Two Types of Creative Labor?
-
Differences between industrial creative labor and revolutionary creative labor, in terms of confrontationality, centrality of the body, position on the sanctioned-unsanctioned divison, visibility, and status as either waged or unwaged.
-
-
17.4: Subjectivity and Revolutionary Creative Labor
-
The concepts of expression, production, and revolution in creativity as developed by Hans Joas, and their convergence in revolutionary creative labor. Subjectivity as created by revolutionary creative labor, changing the relationship between ruler and ruled.
-
-
17.5: Notes
-


21.11B: Adjustments at High Altitude
The human body can adapt to high altitude through immediate and long-term acclimatization processes.
- Evaluate the respiratory adjustments to high altitude
Key Points
- At high altitude, in the short term, the lack of oxygen is sensed by the peripheral chemoreceptors, which causes an increase in ventilation. An increase in heart rate and decrease in stroke volume also occurs.
- During acclimatization over a few days to weeks, the body produces more red blood cells to counteract the lower oxygen saturation in blood in high altitudes.
- Full adaptation to high altitude is achieved when the increase of red blood cells reaches a plateau and stops.
- Increased red blood cell levels remain for about two weeks after acclimatization, which makes it a popular training regimen for athletes.
Key Terms
- acclimatization : Long-term adjustment to high altitude, which is primarily due to increased red blood cell production and capillary tissue perfusion.
- hematocrit : This is the amount of red blood cells in a given volume of blood.
The human body can adapt to high altitude through immediate and long-term acclimatization. At high altitude there is lower air pressure compared to a lower altitude or sea-level altitude.
Due to Boyle’s law, at higher altitude the partial pressure of oxygen in the air is lower, and less oxygen is breathed in with every breath. The partial pressure gradients for gas exchange are also decreased, along with the percentage of oxygen saturation in hemoglobin.
Humans can survive at high altitudes with impaired short-term functions that eventually adjust in the long term. Some altitudes are too high for acclimatization to work, and can cause death if people stay there for too long.
Short-Term Adjustments
At high altitude, in the short term, the lack of oxygen is sensed by the peripheral chemoreceptors, which causes an increase in breathing rate ( hyperventilation ). However, hyperventilation also causes the adverse effect of alkalosis due to increasing the rate by which carbon dioxide is removed from the body, which inhibits the respiratory center from enhancing the respiratory rate to meet the oxygen demands.
Additionally, the peripheral chemoreceptors cause sympathetic nervous system stimulation, which causes the heart rate to increase while stroke volume decreases, and digestion is impaired. Shortness of breath is common, and urination increases.
Along with alkalosis, these effects make up the symptoms of altitude sickness, which become worse during exercise at high altitudes (which involves more anaerobic respiration than at lower altitudes), but falls off during acclimatization.
Acclimatization
Acclimatization to high altitude requires days, or even weeks. Gradually, the body compensates for the respiratory alkalosis by kidney excretion of bicarbonate, which allows adequate respiration to provide oxygen without risking alkalosis. It takes about four days at any given altitude, and can be enhanced by drugs such as acetazolamide (which decreases fluid retention).
Staying hydrated during acclimatization is important to minimize altitude sickness symptoms and to counteract increased urination. The heart rate and ventilation rate at rest both remain elevated despite the acclimatization, while heart rate at maximum activity level will be reduced.
The main difference brought about by acclimatization that explain why it makes high altitudes more comfortable for the body is increased levels of circulating red blood cells, which improve the carrying capacity of oxygen by hemoglobin in the body. This is an adaptive response due to erythropotein secretions in the kidneys (from lack of oxygen in the tissues ) that act on the liver to increase erythrocyte (red blood cell) production.
Blood volume decreases, which also increases the hematocrit, which is the concentration of hemoglobin in blood. This increase in red blood cells remains for a few weeks after one returns to lower altitude, so those who acclimatize to high altitude will experience improved athletic performance at lower altitudes. Capillary density and tissue perfusion also increase.
These physiological changes make high-altitude athletic training popular for athletes, such as Olympic athletes. Full hematological adaptation to high altitude is achieved when the increase of red blood cells reaches a plateau and stops.
The length of full hematological adaptation can be approximated by multiplying the altitude in kilometers by 11.4 days. For example, to adapt to 4,000 meters (13,000 ft.) of altitude would require 45.6 days.
The upper altitude limit of this linear relationship has not been fully established, in part because extremely high altitudes have such little oxygen content that they would be fatal regardless of acclimatization.
An athletic training camp in the Swiss Alps
LICENSES AND ATTRIBUTIONS
CC LICENSED CONTENT, SHARED PREVIOUSLY
- Curation and Revision. Authored by : Boundless.com. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
CC LICENSED CONTENT, SPECIFIC ATTRIBUTION
- Physical exercise. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Physical_exercise%23Cardiovascular_system . License : CC BY-SA: Attribution-ShareAlike
- Physical exercise. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Physical_exercise%23Cardiovascular_system . License : CC BY-SA: Attribution-ShareAlike
- Asthma. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Asthma . License : CC BY-SA: Attribution-ShareAlike
- Physical exercise. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Physical_exercise%23Cardiovascular_system . License : CC BY-SA: Attribution-ShareAlike
- asthma. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/asthma . License : CC BY-SA: Attribution-ShareAlike
- Boundless. Provided by : Boundless Learning. Located at : www.boundless.com//physiology/definition/anaerobic-exercise . License : CC BY-SA: Attribution-ShareAlike
- Asthma. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Asthma_.jpg . License : CC BY-SA: Attribution-ShareAlike
- Effects of high altitude on humans. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Effects_of_high_altitude_on_humans . License : CC BY-SA: Attribution-ShareAlike
- Lung volume. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Lung_volume . License : CC BY-SA: Attribution-ShareAlike
- alkalosis. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/alkalosis . License : CC BY-SA: Attribution-ShareAlike
- hematocrit. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/hematocrit . License : CC BY-SA: Attribution-ShareAlike
- Asthma. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Asthma_.jpg . License : CC BY-SA: Attribution-ShareAlike
- Swiss_Olympic_training_base.jpg. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Effects_of_high_altitude_on_humans#/media/File:Swiss_Olympic_training_base.jpg . License : CC BY-SA: Attribution-ShareAlike


21.3: Common Operators in Quantum Mechanics
Some common operators occurring in quantum mechanics are collected in the table below:
| Observable Name | Symbol | Operator | Operation |
|---|---|---|---|
| Position | \({\bf r}\) | \(\hat{\bf r}\) | Multiply by \({\bf r}\) |
| Momentum | \({\bf p}\) | \(\hat{\bf p}\) | \(-i \hbar \left(\hat{i}\dfrac{\partial}{\partial x} +\hat{j} \dfrac{\partial}{\partial y}+\hat{k} \dfrac{\partial}{\partial z} \right)\) |
| Kinetic energy | \(K\) | \(\hat{K}\) | \(- \dfrac{\hbar^2}{2m} \left(\dfrac{\partial^2}{\partial x^2} +\dfrac{\partial^2}{\partial y^2} +\dfrac{\partial^2}{\partial z^2} \right)\) |
| Potential energy | \(V({\bf r})\) | \(\hat{V}({\bf r})\) | Multiply by \(V({\bf r})\) |
| Total energy | \(E\) | \(\hat{H}\) | \(-\dfrac{\hbar^2}{2m} \left(\dfrac{\partial^2}{\partial x^2} +\dfrac{\partial^2}{\partial y^2} +\dfrac{\partial^2}{\partial z^2} \right) +V({\bf r})\) |
| Angular momentum | \(L\) | \(\hat{L}^2\) | \(\hat{L}_x^2+\hat{L}_y^2+\hat{L}_z^2\) |
| \(L_x\) | \(\hat{L}_x\) | \(-i\hbar\left(y\dfrac{\partial}{\partial z} - z \dfrac{\partial}{\partial y} \right)\) | |
| \(L_y\) | \(\hat{L}_y\) | \(-i \hbar \left(z\dfrac{\partial}{\partial x} - x \dfrac{\partial}{\partial z} \right)\) | |
| \(L_z\) | \(\hat{L}_z\) | \(-i \hbar \left(x\dfrac{\partial}{\partial y} - y \dfrac{\partial}{\partial x} \right)\) |
In the sections below we analyze in details two main operators for the energy and the angular momentum.
Hamiltonian Operator
The main quantity that quantum mechanics is interested in is the total energy of the system, \(E\). The operator corresponding to this quantity is called Hamiltonian :
\[ \hat{H} = - \dfrac{\hbar^2}{2} \sum_i \dfrac{1}{m_i} \nabla_i^2 + V, \label{22.3.1} \]
where \(i\) is an index over all the particles of the system. Using the formalism of operators in conjunction with Equation \ref{22.3.1}, we can write the TISEq just simply as:
\[ \hat{H} \psi = E\psi. \label{22.3.2} \]
Comparing Equation \ref{22.3.1} to the classical analog in Equation 18.3.2 , we notice how the first term in the Hamiltonian operator represents the corresponding kinetic energy operator, \(\hat{K}\), while the second term represents the potential energy operator, \(\hat{V}\). For a one-electron system—such as the ones we studied in chapter 20 —we can write:
\[ \hat{K}=- \dfrac{\hbar^2}{2m} \left(\dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2} + \dfrac{\partial^2}{\partial z^2} \right) = \dfrac{\hbar^2}{2m} \nabla^2, \label{22.3.3} \]
which is universal and applies to all systems. The potential energy operator \(\hat{V}\) is what differentiate each system. Using Equation \ref{22.3.2}, we can then simply obtain the TISEq for each of the first three models discussed in chapter 20 by simply using:
\[\begin{equation} \begin{aligned} \text{Free particle:}\qquad \hat{V} &= 0, \\ \text{Particle in a box:}\qquad \hat{V} &= 0 \; \text{inside the box, } \hat{V} = \infty \; \text{outside the box},\\ \text{Harmonic oscillator:}\qquad \hat{V} &= \dfrac{1}{2}kx^2. \\ \end{aligned} \end{equation} \label{22.3.4} \]
While these three cases are trivial to solve, the case of the rigid rotor is more complicated to solve, since the kinetic energy operator needs to be solved in spherical polar coordinates, as we will show in the next section.
Angular Momentum Operator
To write the kinetic energy operator \(\hat{K}\) for the rigid rotor, we need to express the Laplacian, \(\nabla^2\), in spherical polar coordinates:
\[ \nabla^2=\nabla^2_r - \dfrac{\hat{L}^2}{r^2}, \label{22.3.5} \]
where \(\nabla_r^2 = \dfrac{1}{r^2}\dfrac{\partial}{\partial r} \left( r^2\dfrac{\partial}{\partial r} \right)\) is the radial Laplacian, and \(\hat{L}^2\) is the square of the total angular momentum operator, which is:
\[\begin{equation} \begin{aligned} \hat{L}^2 &=\hat{L}\cdot\hat{L}=\left(\mathbf{i}\hat{L}_x+\mathbf{j}\hat{L}_y+\mathbf{k}\hat{L}_z\right)\cdot\left(\mathbf{i}\hat{L}_x+\mathbf{j}\hat{L}_y+\mathbf{k}\hat{L}_z \right) \\ &=\hat{L}_x^2+\hat{L}_y^2+\hat{L}_z^2, \end{aligned} \end{equation} \label{22.3.6} \]
with \(\left\{\mathbf{i},\mathbf{j},\mathbf{k}\right\}\) the unitary vectors in three-dimensional space. The component along each direction, \(\left\{\hat{L}_x,\hat{L}_y,\hat{L}_z\right\}\), are then expressed in cartesian coordinates using to the following formulas:
\[\begin{equation} \begin{aligned} \hat{L}_x &= -i\hbar\left(y\dfrac{\partial}{\partial z} - z \dfrac{\partial}{\partial y} \right), \\ \hat{L}_y &= -i \hbar \left(z\dfrac{\partial}{\partial x} - x \dfrac{\partial}{\partial z} \right), \\ \hat{L}_z &= -i \hbar \left(x\dfrac{\partial}{\partial y} - y \dfrac{\partial}{\partial x} \right). \end{aligned} \end{equation} \label{22.3.7} \]
The eigenvalues equation corresponding to the total angular momentum is:
\[ \hat{L}^2 Y(\theta, \varphi) = \hbar^2 \ell(\ell+1) Y_{\ell}^{m_{\ell}}(\theta, \varphi), \label{22.3.8} \]
where \(\ell\) is the azimuthal quantum number and \(Y_{\ell}^m(\theta, \varphi)\) are the spherical harmonics, both of which we already encountered in chapter 20 . Recall once again that each energy level \(E_{\ell}\) is \((2\ell+1)\)-fold degenerate in \(m_{\ell}\), since \(m_{\ell}\) can have values \(-\ell, -\ell+1, \ldots, \ell-1, \ell\). This means that there are \((2\ell+1)\) states with the same energy \(E_{\ell}\), each characterized by the magnetic magnetic quantum number \(m_{\ell}\). This quantum number can be determined using the following eigenvalues equation:
\[ \hat{L}_z Y(\theta, \varphi) = \hbar m_{\ell} Y_{\ell}^{m_{\ell}}(\theta, \varphi). \label{22.3.9} \]
The interpretation of these results is rather complicated, since the angular momenta are quantum operators and they cannot be drawn as vectors like in classical mechanics. Nevertheless, it is common to depict them heuristically as in figure \(\PageIndex{1}\), 1 where a set of states with quantum numbers \(\ell =2\), and \(m_{\ell}=-2,-1,0,1,2\) are reported. Since \(|L|={\sqrt {L^{2}}}=\hbar {\sqrt {6}}\), the vectors are all shown with length \(\hbar \sqrt{6}\). The rings represent the fact that \(L_{z}\) is known with certainty, but \(L_{x}\) and \(L_{y}\) are unknown; therefore every classical vector with the appropriate length and \(z\)-component is drawn, forming a cone. The expected value of the angular momentum for a given ensemble of systems in the quantum state characterized by \(\ell\) and \(m_{\ell}\), could be somewhere on this cone but it cannot be defined for a single system.
- This diagram is taken from Wikipedia by user Maschen, and is of public domain


21: Organic Chemistry
Organic chemistry involving the scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms. Study of structure includes many physical and chemical methods to determine the chemical composition and the chemical constitution of organic compounds and materials. Study of properties includes both physical properties and chemical properties, and uses similar methods as well as methods to evaluate chemical reactivity, with the aim to understand the behavior of the organic matter.
Contributors and Attributions
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).