8.5: Assisting With Ambulation
Ambulation is the medical term used for walking. Ambulation provides weight-bearing activity that promotes bone health and joint mobility. A physical therapist will determine if a person can safely walk independently, with the assistance of one or two people, or if they require an assistive device such as a cane or walker. This information is documented in their nursing care plan. Similar to when assisting a client to transfer with a gait belt, the nursing assistant should place nonskid footwear on the person and allow them to dangle on the edge of the bed before standing to ambulate. For specific steps, see the “ Ambulation From Wheelchair ” Skills Checklist.
If a resident requires assistance with a cane, the cane should be placed on the resident’s stronger side. The resident should step forward with the strong leg and then use the cane and the weaker leg for the next step.
There are three types of walkers: a standard walker, a two-wheeled walker (2WW), and a four-wheeled walker (4WW). The type of walker a resident should use is recommended by the physical therapist. A 2WW or standard walker allows for more support and a slower gait, whereas a 4WW is used by clients with better balance and mobility.
View a YouTube video [1] of an instructor demonstrating helping clients with a mbulatory assistive devices:
A second staff member can follow a resident who is ambulating with assistance with their wheelchair in case they experience weakness or dizziness. If the client needs to sit while ambulating, the wheelchair brakes should be applied before they sit, or in an emergent situation, the NA should block the back of the wheelchair with their body to ensure stability when the resident sits.
If a resident starts to fall while standing or ambulating, do not attempt to stop their fall or catch the resident because this can cause you to injure your back. Instead, move behind the patient and take one step back with one leg so you have a wide base of support. Support the patient around their waist or hip area or grab the gait belt. Bend one of your legs and place it between the patient’s legs from behind. Slowly slide the patient down your bent leg, lowering yourself to the floor at the same time. Always protect the resident’s head to prevent head injury. After the resident is on the floor, do not move them. For witnessed or unwitnessed falls, notify the nurse immediately for assessment. After the nurse has completed the assessment and met the resident’s immediate needs, use a mechanical lift to transfer the resident back to a wheelchair or bed. An incident report will be completed by the nurse, and the NA will be asked to give a statement on what occurred and their actions in response to the situation. [2] See Figure 8.21 [3] for an image of lowering a resident who is falling to the floor.
- Chippewa Valley Technical College. (2022, December 3). Ambulatory Assistive Devices. [Video]. YouTube. Video licensed under CC BY 4.0 . https://youtu.be/ATn7OOP4Jko ↵
- This work is a derivative of Nursing Fundamentals by Chippewa Valley Technical College and is licensed under CC BY 4.0 ↵
- “Sept-22-2015-132-001.jpg” and “Sept-22-2015-133.jpg” by unknown authors are licensed under CC BY 4.0 . Access for free at https://opentextbc.ca/clinicalskills/chapter/3-7-fall-prevention/ ↵


4.9: Conclusion
The pursuit of pleasure has been a part of the human condition since time began. Desires are something we all have. Some embodied, some not. Culture shapes what we learn to be pleasurable and constrains us from exploring what we might like. Embodied pleasure in the form of sexuality is part of an overall well-being. Sensuality allows us to tune into what we like. Our brain is our largest sex organ, holding the key to our arousal and response. Studies done on pleasure have mostly been framed around white, cis-gendered, heterosexual people with penises so there is a large gap in the field in order to represent totality of the human spectrum. The anatomy of the clitoris was not formally studied and included in the scientific literature until Helen O'Donnell wrote about it in 2005. Masturbation is a great way to stimulate your senses and learn about your body. Alfred Kinsey and work done at the Kinsey Institute brought sexuality studies into the scientific domain. Masters and Johnson’s four phase model of sexual response became the tool for identifying issues in sexual health despite a missing piece. In 1977, Helen Kaplan updated this model to include desire which changed the way in which sexual health issues are treated. Understanding that we are all different and our levels of desire and what inhibits our desires are not fixed can help with self-acceptance. It is time to debunk cultural myths about the right way to be a sexual being and practice radical self-love.


8.10: Reading- Surface Waves
There are two types of surface waves, Rayleigh waves and Love waves. Rayleigh waves are named after Lord Rayleigh (John Strutt), an English aristocrat who, in his work as a scientist and mathematician, developed a detailed mathematical accounting of the type of surface wave named after him. Rayleigh waves are set off by the combined effect of P- and S-waves on the earth’s surface. Rayleigh waves are sometimes called rolling waves. In Rayleigh waves the surface of the earth rises up and sinks down in crests and troughs, similar to waves on the surface of water. People who are outdoors during a major earthquake commonly see Rayleigh waves moving across the surface of the earth, and can feel the ground rising and falling as the waves pass beneath them.
Love waves, sometimes called L-waves, are named after Augustus Love, an English mathematician and physicist who first modeled them mathematically. Love waves involve the surface shearing sideways and then returning to its original form as each wave passes.
All surface waves travel slower than body waves and Rayleigh waves are slower than Love waves.
Contributors and Attributions
- Seismic waves. Authored by : Ralph L. Dawes and Cheryl D. Dawes. Provided by : Wenatchee Valley College. Located at : http://commons.wvc.edu/rdawes/G101OCL/Basics/earthquakes.html#waves . Project : Geology 101 - Introduction to Physical Geology. License : CC BY: Attribution


2.21: Dividing Radical Expressions
Dividing Radical Expressions
We have used the Quotient Property of Radical Expressions to simplify roots of fractions. We will need to use this property ‘in reverse’ to simplify a fraction with radicals. We give the Quotient Property of Radical Expressions again for easy reference. Remember, we assume all variables are greater than or equal to zero so that no absolute value bars are needed.
If \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\) are real numbers, \(b≠0\), and for any integer \(n≥2\) then,
\(\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} \quad \text { and } \quad \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}}\)
We will use the Quotient Property of Radical Expressions when the fraction we start with is the quotient of two radicals, and neither radicand is a perfect power of the index. When we write the fraction in a single radical, we may find common factors in the numerator and denominator.
Simplify:
- \(\dfrac{\sqrt{72 x^{3}}}{\sqrt{162 x}}\)
- \(\dfrac{\sqrt[3]{32 x^{2}}}{\sqrt[3]{4 x^{5}}}\)
Solution :
a.
\(\dfrac{\sqrt{72 x^{3}}}{\sqrt{162 x}}\)
Rewrite using the quotient property,
\(\sqrt{\dfrac{72 x^{3}}{162 x}}\)
Remove common factors.
\(\sqrt{\dfrac{\cancel{18} \cdot 4 \cdot x^{2} \cdot \cancel{x}}{\cancel{18} \cdot 9 \cdot \cancel{x}}}\)
Simplify.
\(\sqrt{\dfrac{4 x^{2}}{9}}\)
Simplify the radical.
\(\dfrac{2 x}{3}\)
b.
\(\dfrac{\sqrt[3]{32 x^{2}}}{\sqrt[3]{4 x^{5}}}\)
Rewrite using the quotient property, \(\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}}\).
\(\sqrt[3]{\dfrac{32 x^{2}}{4 x^{5}}}\)
Simplify the fraction under the radical.
\(\sqrt[3]{\dfrac{8}{x^{3}}}\)
Simplify the radical.
\(\dfrac{2}{x}\)
Simplify:
- \(\dfrac{\sqrt{75 q^{5}}}{\sqrt{108 q}}\)
- \(\dfrac{\sqrt[3]{72 b^{2}}}{\sqrt[3]{9 b^{5}}}\)
- Answer
-
- \(\dfrac{5 q^{2}}{6}\)
- \(\dfrac{2}{b}\)
Simplify:
- \(\dfrac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}\)
- \(\dfrac{\sqrt[3]{-250 m n^{-2}}}{\sqrt[3]{2 m^{-2} n^{4}}}\)
Solution :
a.
\(\dfrac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}\)
Rewrite using the quotient property.
\(\sqrt{\dfrac{147 a b^{8}}{3 a^{3} b^{4}}}\)
Remove common factors in the fraction.
\(\sqrt{\dfrac{49 b^{4}}{a^{2}}}\)
Simplify the radical.
\(\dfrac{7 b^{2}}{a}\)
b.
\(\dfrac{\sqrt[3]{-250 m n^{-2}}}{\sqrt[3]{2 m^{-2} n^{4}}}\)
Rewrite using the quotient property.
\(\sqrt[3]{\dfrac{-250 m n^{-2}}{2 m^{-2} n^{4}}}\)
Simplify the fraction under the radical.
\(\sqrt[3]{\dfrac{-125 m^{3}}{n^{6}}}\)
Simplify the radical.
\(-\dfrac{5 m}{n^{2}}\)
Simplify:
- \(\dfrac{\sqrt{162 x^{10} y^{2}}}{\sqrt{2 x^{6} y^{6}}}\)
- \(\dfrac{\sqrt[3]{-128 x^{2} y^{-1}}}{\sqrt[3]{2 x^{-1} y^{2}}}\)
- Answer
-
- \(\dfrac{9 x^{2}}{y^{2}}\)
- \(\dfrac{-4 x}{y}\)
Simplify: \(\dfrac{\sqrt{54 x^{5} y^{3}}}{\sqrt{3 x^{2} y}}\)
Solution :
\(\dfrac{\sqrt{54 x^{5} y^{3}}}{\sqrt{3 x^{2} y}}\)
Rewrite using the quotient property.
\(\sqrt{\dfrac{54 x^{5} y^{3}}{3 x^{2} y}}\)
Remove common factors in the fraction.
\(\sqrt{18 x^{3} y^{2}}\)
Rewrite the radicand as a product using the largest perfect square factor.
\(\sqrt{9 x^{2} y^{2} \cdot 2 x}\)
Rewrite the radical as the product of two radicals.
\(\sqrt{9 x^{2} y^{2}} \cdot \sqrt{2 x}\)
Simplify.
\(3 x y \sqrt{2 x}\)
Simplify: \(\dfrac{\sqrt{96 a^{5} b^{4}}}{\sqrt{2 a^{3} b}}\)
- Answer
-
\(4 a b \sqrt{3 b}\)
Rationalizing a Denominator Containing One Term
Before the calculator became a tool of everyday life, approximating the value of a fraction with a radical in the denominator was a very cumbersome process!
For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator. This process is still used today, and is useful in other areas of mathematics, too.
Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.
Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still should be rationalized. It is not considered simplified if the denominator contains a radical.
Similarly, a radical expression is not considered simplified if the radicand contains a fraction.
Simplified Radical Expressions
A radical expression is considered simplified if there are
- no factors in the radicand have perfect powers of the index
- no fractions in the radicand
- no radicals in the denominator of a fraction
To rationalize a denominator with a square root, we use the property that \((\sqrt{a})^{2}=a\). If we square an irrational square root, we get a rational number.
We will use this property to rationalize the denominator in the next example.
Simplify:
- \(\dfrac{4}{\sqrt{3}}\)
- \(\sqrt{\dfrac{3}{20}}\)
- \(\dfrac{3}{\sqrt{6 x}}\)
Solution :
To rationalize a denominator with one term, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.
a.
| Multiply both the numerator and denominator by \(\sqrt{3}\). | |
| Simplify. |
b. We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.
| The fraction is not a perfect square, so rewrite using the Quotient Property. | |
| Simplify the denominator. | |
| Multiply the numerator and denominator by \(\sqrt{5}\). | |
| Simplify. | |
| Simplify. |
c.
| Multiply the numerator and denominator by \(\sqrt{6x}\). | |
| Simplify. | |
| Simplify. |
Simplify:
- \(\dfrac{5}{\sqrt{3}}\)
- \(\sqrt{\dfrac{3}{32}}\)
- \(\dfrac{2}{\sqrt{2 x}}\)
- Answer
-
- \(\dfrac{5 \sqrt{3}}{3}\)
- \(\dfrac{\sqrt{6}}{8}\)
- \(\dfrac{\sqrt{2 x}}{x}\)
When we rationalized a square root, we multiplied the numerator and denominator by a square root that would give us a perfect square under the radical in the denominator. When we took the square root, the denominator no longer had a radical.
We will follow a similar process to rationalize higher roots. To rationalize a denominator with a higher index radical, we multiply the numerator and denominator by a radical that would give us a radicand that is a perfect power of the index. When we simplify the new radical, the denominator will no longer have a radical.
For example,
We will use this technique in the next examples.
Simplify:
- \(\dfrac{1}{\sqrt[3]{6}}\)
- \(\sqrt[3]{\dfrac{7}{24}}\)
- \(\dfrac{3}{\sqrt[3]{4 x}}\)
Solution :
To rationalize a denominator with a cube root, we can multiply by a cube root that will give us a perfect cube in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.
a.
| The radical in the denominator has one factor of \(6\). Multiply both the numerator and denominator by \(\sqrt[3]{6^{2}}\), which gives us \(2\) more factors of \(6\). | |
| Multiply. Notice the radicand in the denominator has \(3\) powers of \(6\). | |
| Simplify the cube root in the denominator. |
b. We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.
| The fraction is not a perfect cube, so rewrite using the Quotient Property. | |
| Simplify the denominator. | |
| Multiply the numerator and denominator by \(\sqrt[3]{3^{2}}\). This will give us \(3\) factors of \(3\). | |
| Simplify. | |
| Remember, \(\sqrt[3]{3^{3}}=3\). | |
| Simplify. |
c.
| Rewrite the radicand to show the factors. | |
| Multiple the numerator and denominator by \(\sqrt[3]{2 \cdot x^{2}}\). This will get us \(3\) factors of \(2\) and \(3\) factors of \(x\). | |
| Simplify. | |
| Simplify the radical in the denominator. |
Simplify:
- \(\dfrac{1}{\sqrt[3]{2}}\)
- \(\sqrt[3]{\dfrac{3}{20}}\)
- \(\dfrac{2}{\sqrt[3]{25 n}}\)
- Answer
-
- \(\dfrac{\sqrt[3]{4}}{2}\)
- \(\dfrac{\sqrt[3]{150}}{10}\)
- \(\dfrac{2 \sqrt[3]{5 n^{2}}}{5 n}\)
Simplify:
- \(\dfrac{1}{\sqrt[4]{2}}\)
- \(\sqrt[4]{\dfrac{5}{64}}\)
- \(\dfrac{2}{\sqrt[4]{8 x}}\)
Solution :
To rationalize a denominator with a fourth root, we can multiply by a fourth root that will give us a perfect fourth power in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.
a.
|
The radical in the denominator has one factor of \(2\).
Multiply both the numerator and denominator by \(\sqrt[4]{2^{3}}\), which gives us \(3\) more factors of \(2\). |
|
| Multiply. Notice the radicand in the denominator has \(4\) powers of \(2\). | |
| Simplify the fourth root in the denominator. |
b. We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.
| The fraction is not a perfect fourth power, so rewrite using the Quotient Property. | |
| Rewrite the radicand in the denominator to show the factors. | |
| Simplify the denominator. | |
| Multiply the numerator and denominator by \(\sqrt[4]{2^{2}}\). This will give us \(4\) factors of \(2\). | |
| Simplify. | |
| Remember, \(\sqrt[4]{2^{4}}=2\). | |
| Simplify. |
c.
| Rewrite the radicand to show the factors. | |
| Multiply the numerator and denominator by \(\sqrt[4]{2 \cdot x^{3}}\). This will get us \(4\) factors of \(2\) and \(4\) factors of \(x\). | |
| Simplify. | |
| Simplify the radical in the denominator. | |
| Simplify the fraction. |
Simplify:
- \(\dfrac{1}{\sqrt[4]{3}}\)
- \(\sqrt[4]{\dfrac{3}{64}}\)
- \(\dfrac{3}{\sqrt[4]{125 x}}\)
- Answer
-
- \(\dfrac{\sqrt[4]{27}}{3}\)
- \(\dfrac{\sqrt[4]{12}}{4}\)
- \(\dfrac{3 \sqrt[4]{5 x^{3}}}{5 x}\)
Rationalizing a Denominator Containing Two Terms
When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates Pattern to rationalize the denominator .
\(\begin{array}{c c}{(a-b)(a+b)} & {(2-\sqrt{5})(2+\sqrt{5})} \\ {a^{2}-b^{2}} &{ 2^{2}-(\sqrt{5})^{2}} \\ {}&{4-5} \\ {}&{-1}\end{array}\)
When we multiple a binomial that includes a square root by its conjugate, the product has no square roots.
Simplify: \(\dfrac{5}{2-\sqrt{3}}\)
Solution :
| Multiply the numerator and denominator by the conjugate of the denominator. | |
| Multiply the conjugates in the denominator. | |
| Simplify the denominator. | |
| Simplify the denominator. | |
| Simplify. |
Simplify: \(\dfrac{2}{4-\sqrt{6}}\).
- Answer
-
\(\dfrac{4+\sqrt{6}}{5}\)
Notice we did not distribute the \(5\) in the answer of the last example. By leaving the result factored we can see if there are any factors that may be common to both the numerator and denominator.
Simplify: \(\dfrac{\sqrt{3}}{\sqrt{u}-\sqrt{6}}\).
Solution :
| Multiply the numerator and denominator by the conjugate of the denominator. | |
| Multiply the conjugates in the denominator. | |
| Simplify the denominator. |
Simplify: \(\dfrac{\sqrt{5}}{\sqrt{x}+\sqrt{2}}\).
- Answer
-
\(\dfrac{\sqrt{5}(\sqrt{x}-\sqrt{2})}{x-2}\)
Be careful of the signs when multiplying. The numerator and denominator look very similar when you multiply by the conjugate.
Simplify: \(\dfrac{\sqrt{x}+\sqrt{7}}{\sqrt{x}-\sqrt{7}}\).
Solution :
| Multiply the numerator and denominator by the conjugate of the denominator. | |
| Multiply the conjugates in the denominator. | |
| Simplify the denominator. |
We do not square the numerator. Leaving it in factored form, we can see there are no common factors to remove from the numerator and denominator.
Simplify: \(\dfrac{\sqrt{q}-\sqrt{10}}{\sqrt{q}+\sqrt{10}}\)
- Answer
-
\(\dfrac{(\sqrt{q}-\sqrt{10})^{2}}{q-10}\)


26.5D: Female Duct System
The Fallopian tubes, or oviducts, connect the ovaries to the uterus.
- Describe the structures of the female duct system
Key Points
- The Fallopian tube allows passage of the egg from the ovary to the uterus.
- The lining of the Fallopian tubes are ciliated and have several segments, including the infundibulum, ampullary, isthmus, and interstitial regions.
- Interspersed between the ciliated cells are peg cells, which contain apical granules and produce the tubular fluid that contains nutrients for spermatozoa, oocytes, and zygotes.
- Occasionally, the embryo implants into the Fallopian tube instead of the uterus, creating an ectopic pregnancy.
Key Terms
- oviduct : A duct through which an ovum passes from an ovary to the uterus or to the exterior.
- fallopian tubes : Also known as oviducts, uterine tubes, and salpinges (singular salpinx), two very fine tubes lined with ciliated epithelia, leading from the ovaries of female mammals into the uterus via the uterotubal junction.
- ovarian follicle : The basic units of female reproductive biology, each composed of roughly spherical aggregations of cells found in the ovary.
The Fallopian tubes, also known as oviducts, uterine tubes, and salpinges (singular salpinx), are two very fine tubes lined with ciliated epithelia, leading from the ovaries of female mammals into the uterus via the uterotubal junction. In non-mammalian vertebrates, the equivalent structures are the oviducts. These tubes allows passage of the egg from the ovary to the uterus.
The different segments of the fallopian tube are ( lateral to medial):
- The infundibulum with associated fimbriae near the ovary
- The ampullary region that represents the major portion of the lateral tube
- The isthmus, which is the narrower part of the tube that links to the uterus
- The interstitial (intramural) part that transverses the uterine musculature
The tubal ostium is the point at which the tubal canal meets the peritoneal cavity, while the uterine opening of the Fallopian tube is the entrance into the uterine cavity, the uterotubal junction.
Uterine Segments : Illustrative drawing of the anterior view of the uterus showing the uterine segments
There are two types of cells within the simple columnar epithelium of the Fallopian tube. Ciliated cells predominate throughout the tube, but are most numerous in the infundibulum and ampulla. Estrogen increases the production of cilia on these cells.
Interspersed between the ciliated cells are peg cells, which contain apical granules and produce the tubular fluid. This fluid contains nutrients for spermatozoa, oocytes, and zygotes. The secretions also promote capacitation of the sperm by removing glycoproteins and other molecules from the plasma membrane of the sperm. Progesterone increases the number of peg cells, while estrogen increases their height and secretory activity. Tubal fluid flows against the action of the ciliae, toward the fimbrated end.
When an ovum is developing in an ovary, it is encapsulated in a sac known as an ovarian follicle. On maturation, the follicle and the ovary’s wall rupture, allowing the ovum to escape. The egg is caught by the fimbriated end and travels to the ampulla where typically the sperm are met and fertilization occurs. The fertilized ovum, now a zygote, travels towards the uterus aided by the tubal cilia and tubal muscle. After about five days, the new embryo enters the uterine cavity and implants about a day later. Occasionally, the embryo implants into the Fallopian tube instead of the uterus, creating an ectopic pregnancy.


9.13: A Summary of Reactivity - SN1, SN2, E1, E1cB, and E2
-
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Objectives
After completing this section, you should be able to
- determine whether a specified substrate is most likely to undergo an E1, E2, S N 1 or S N 2 reaction under a given set of conditions.
- describe the conditions under which a given substrate is most likely to react by a specified mechanism (E1, E2, S N 1 or S N 2).
Study Notes
This section summarizes much of what has been discussed in the chapter. It focuses on how a given substrate will behave under certain conditions, but does not deal with the stereochemistry of the products.
Having discussed the many factors that influence nucleophilic substitution and elimination reactions of alkyl halides, we must now consider the practical problem of predicting the most likely outcome when a given alkyl halide is reacted with a given nucleophile. As we noted earlier, there are multiple variables to be considered, the most important being the substitution of the alkyl halide .  In general, in order for an S N 1 or E1 reaction to occur, the relevant carbocation intermediate must be relatively stable such as with tertiary halides, secondary allylic halides, and secondary benzylic halides.
The next most import variable for predicting the outcome of a reaction is the nature of the nucleophilic reactant .  Weak nucleophiles that are also weak bases tend to favor S N 1 and E1 reactions.
- Good Nucleophiles Which are Weak Bases: I – , Br – , SCN – , N 3 – , CH 3 CO 2 – , RS – , CN – , Amines, etc.
- Good Nucleophiles Which are Strong Bases: HO – , RO – .
- Poor Nucleophiles which are Weak Bases: H 2 O, ROH, RSH.
General Rules For Predicting a Reaction
Consider these in the order listed.
Methyl Alkyl Halides
- An S N 2 substitution occurs regardless if a good or poor nucleophile is used.
-
An E2 elimination occurs if a strong, sterically hindered base is used. - An E1cB elimination occurs if a strong base is used and the leaving group is two carbons away from a carbonyl group.
-
An S N 2 substitution occurs if a good nucleophile is used.
Secondary Alkyl Halides
-
An E1cB elimination occurs if a strong base is used and the leaving group is two carbons away from a carbonyl group. - An E2 elimination occurs if a strong base is used.
- An S N 2 reaction occurs if a good nucleophile that is a weak bases is used in a polar aprotic solvent.
- An S N 1 reaction along with an E1 reaction occurs if a poor nucleophile that is a weak bases is used in a protic solvent.
Tertiary Alkyl Halides
- An E1cB elimination occurs if a strong base is used and the leaving group is two carbons away from a carbonyl group.
- An E2 elimination occurs if a strong base is used.
- An S N 1 reaction along with an E1 reaction occurs if a poor nucleophile that is a weak bases is used in a protic solvent.
Worked Example\(\PageIndex{1}\)
1) For the following, please determine what kind of reaction is occurring and predict the product(s).
a)
b)
c)
- Answer
-
a) The substrate is a secondary halide so the product is determined by the nature of the nucleophile used in the reaction. Cyanide ( - CN) is a good nucleophile which is a weak base. The fact that the nucleophile is a weak base means that an E2 reaction is not favored. Also, the fact that cyanide is a good nucleophile means that S N 2 substitutions are favored over S N 1. When a secondary halide is reacted with a good nucleophile which is a weak base, the preferred reaction is S N 2. After S N 2 substitution the product is a nitrile.
b) The substrate is a secondary halide so the product is determined by the nature of the nucleophile used in the reaction. Methoxide ( - OCH 3 ) is a strong base so it could prefer to remove a hydrogen from the substrate. This makes an E2 elimination the preferred reaction and an alkene the product.
c) The substrate is a secondary halide so the product is determined by the nature of the nucleophile used in the reaction. Methanol (HOCH 3 ) is a weak nucleophile that is a weak base. Being a weak base means that methanol is not capable of actively removing a hydrogen to cause an E2 reaction. Also, because methanol is a weak nucleophile is not capable of attacking the substrate and causing an S N 2 reaction. For the methanol nucleophile to react the substrate must first eject it leaving group to form the highly reactive carbocation intermediate. Formation of the carbocation is the rate determine step for both the S N 1 and E1 reactions so they each form a separate product.
Exercises \(\PageIndex{1}\)
1. Identify the dominant reaction mechanism (S N 1, S N 2, E1, or E2) and predict the major product for the following reactions.
a)
b)
c)
d)
e)
-
2)
Identify the function of the following reagents. The reagents will be a strong/weak nucleophile and/or a strong/weak base.
a) Cl -
b) NaH
c) t-BuO -
d) OH -
e) H 2 O
f) HS -
g) MeOH
3) Identify which mechanism the following reactions would undergo. - 4) Identify all the products of the following reactions and specify the major product.
-
a)
b)
c)
d)
e)
- 5) The following reaction yields five different products. Give the mechanisms for how each is formed.
- 6) Label the following reactions as S N 1, S N 2, E1, or E2.
- Answer
-
1)
2)
a) Cl - ; strong nucleophile
b) NaH ; strong base
c) t-BuO - ; strong base
d) OH - ; strong nucleophile ; strong base
e) H 2 O ; weak nucleophile ; weak base
f) HS - ; strong nucleophile
g) MeOH ; weak nucleophile ; weak base
3)
a) E2, S N 1
b) S N 2, E2
c) S N 2
d) S N 1, E1
e) E2, S N 1
4)
a)
b)
c)
d)
e)
5)
6)
A – S N 2
B – E1
C – S N 1
D – E2


5.11: Manifolds (Part 1)
This section can be omitted on a first reading.
Why We Need Manifolds
General relativity doesn’t assume a predefined background metric, and this creates a chicken-and-egg problem. We want to define a metric on some space, but how do we even specify the set of points that make up that space? The usual way to define a set of points would be by their coordinates. For example, in two dimensions we could define the space as the set of all ordered pairs of real numbers (x, y). But this doesn’t work in general relativity, because space is not guaranteed to have this structure. For example, in the classic 1979 computer game Asteroids, space “wraps around,” so that if your spaceship flies off the right edge of the screen, it reappears on the left, and similarly at the top and bottom (Figure \(\PageIndex{1}\)).
Even before we impose a metric on this space, it has topological properties that differ from those of the Euclidean plane. By “topological” we mean properties that are preserved if the space is thought of as a sheet of rubber that can be stretched in any way, but not cut or glued back together. Topologically, the space in Asteroids is equivalent to a torus (surface of a doughnut), but not to the Euclidean plane.
Another useful example is the surface of a sphere. In example 10 , we calculated \(\Gamma^{\theta}_{\phi \phi}\). A similar calculation gives \(\Gamma^{\theta}_{\phi \phi} = \cot \frac{\theta}{R}\). Now consider what happens as we drive our dogsled north along the line of longitude \(\phi\) = 0, cross the north pole at \(\theta\) = 0, and continue along the same geodesic. As we cross the pole, our longitude changes discontinuously from 0 to \(\pi\). Consulting the geodesic equation, we see that this happens because \(\Gamma^{\theta}_{\phi \phi}\) blows up at \(\theta\) = 0. Of course nothing really special happens at the pole. The bad behavior isn’t the fault of the sphere, it’s the fault of the \((\theta, \phi)\) coordinates we’ve chosen, that happen to misbehave at the pole. Unfortunately, it is impossible to define a pair of coordinates on a two-sphere without having them misbehave somewhere. (This follows from Brouwer’s famous 1912 “Hairy ball theorem,” which states that it is impossible to comb the hair on a sphere without creating a cowlick somewhere.)
Topological Definition of a Manifold
This motivates us to try to define a “bare-bones” geometrical space in which there is no predefined metric or even any predefined set of coordinates.
There is a general notion of a topological space, which is too general for our purposes. In such a space, the only structure we are guaranteed is that certain sets are defined as “open,” in the same sense that an interval like 0 < x < 1 is called “open.” A point in an open set can be moved in any direction without leaving the set. An open set is essentially a set without a boundary, for in a set like 0 ≤ x ≤ 1, the boundary points 0 and 1 can only be moved in one direction without taking them outside.
A topological space is too general for us because it can include spaces like fractals, infinite-dimensional spaces, and spaces that have different numbers of dimensions in different regions. It is nevertheless useful to recognize certain concepts that can be defined using only the generic apparatus of a topological space, so that we know they do not depend in any way on the presence of a metric. An open set surrounding a point is called a neighborhood of that point. In a topological space we have a notion of getting arbitrarily close to a certain point, which means to take smaller and smaller neighborhoods, each of which is a subset of the last. But since there is no metric, we do not have any concept of comparing distances of distant points, e.g., that P is closer to Q than R is to S. A continuous function is a purely topological idea; a continuous function is one such that for any open subset U of its range, the set V of points in its domain that are mapped to points in U is also open. Although some definitions of continuous functions talk about real numbers like \(\epsilon\) and \(\delta\), the notion of continuity doesn’t depend on the existence of any structure such as the real number system. A homeomorphism is a function that is invertible and continuous in both directions. Homeomorphisms formalize the informal notion of “rubber-sheet geometry without cutting or gluing.” If a homeomorphism exists between two topological spaces, we say that they are homeomorphic; they have the same structure and are in some sense the same space.
The more specific type of topological space we want is called a manifold. Without attempting any high level of mathematical rigor, we define an n-dimensional manifold M according to the following informal principles: 16
Note
For those with knowledge of topology, these can be formalized a little more: we want a completely normal, second-countable, locally connected topological space that has Lebesgue covering dimension n, is a homogeneous space under its own homeomorphism group, and is a complete uniform space. I don’t know whether this is sufficient to characterize a manifold completely, but it suffices to rule out all the counterexamples of which I know.
- M1 Dimension: M’s dimension is n.
- M2 Homogeneity: No point has any locally definable property that distinguishes it from any other point.
- M3 Completeness: M is complete, in the sense that specifying an arbitrarily small neighborhood gives a unique definition of a point.
Example 12: Lines
The set of all real numbers is a 1-manifold. Similarly, any line with the properties specified in Euclid’s Elements is a 1-manifold. All such lines are homeomorphic to one another, and we can therefore speak of “the line.”
Example 13: a circle
A circle (not including its interior) is a 1-manifold, and it is not homeomorphic to the line. To see this, note that deleting a point from a circle leaves it in one connected piece, but deleting a point from a line makes two. Here we use the fact that a homeomorphism is guaranteed to preserve “rubber-sheet” properties like the number of pieces.
Example 14: No changes of dimension
A “lollipop” formed by gluing an open 2-circle (i.e., a circle not including its boundary) to an open line segment is not a manifold, because there is no n for which it satisfies M1.
It also violates M2, because points in this set fall into three distinct classes: classes that live in 2-dimensional neighborhoods, those that live in 1-dimensional neighborhoods, and the point where the line segment intersects the boundary of the circle.
Example 15: No manifolds made from the rational numbers
The rational numbers are not a manifold, because specifying an arbitrarily small neighborhood around \(\sqrt{2}\) excludes every rational number, violating M3.
Similarly, the rational plane defined by rational-number coordinate pairs (x, y) is not a 2-manifold. It’s good that we’ve excluded this space, because it has the unphysical property that curves can cross without having a point in common. For example, the curve y = x 2 crosses from one side of the line y = 2 to the other, but never intersects it. This is physically undesirable because it doesn’t match up with what we have in mind when we talk about collisions between particles as intersections of their world-lines, or when we say that electric field lines aren’t supposed to intersect.
Example 16: no boundary
The open half-plane y > 0 in the Cartesian plane is a 2-manifold. The closed half-plane y ≥ 0 is not, because it violates M2; the boundary points have different properties than the ones on the interior.
Example 17: Disconnected manifolds
Two nonintersecting lines are a 1-manifold. Physically, disconnected manifolds of this type would represent a universe in which an observer in one region would never be able to find out about the existence of the other region.
Example 18: No bad glue jobs
Hold your hands like you’re pretending you know karate, and then use one hand to karate-chop the other. Suppose we want to join two open half-planes in this way. As long as they’re separate, then we have a perfectly legitimate disconnected manifold. But if we want to join them by adding the point P where their boundaries coincide, then we violate M2, because this point has special properties not possessed by any others. An example of such a property is that there exist points Q and R such that every continuous curve joining them passes through P. (Cf. problem 5 )
Hausdorff Property
Pioneering topologist Felix Hausdorff defined the following property of a topological space.
Definition: Hausdorff property
Given any two points, it is possible to find disjoint neighborhoods of them.
A joke/mnemonic, which probably works best for people with a certain type of British accent, is that in a Hausdorff space, any two points can be “housed off” from one another inside their own nonintersecting open sets. The notion appeals strongly to our intuitive ideas about how space and time behave, and the standard definition of a manifold implies that it is Hausdorff. When we model Minkowski space using real-number coordinates, it is Hausdorff. Since the equivalence principle says that spacetime is locally Minkowski, we could also say that it implies spacetime is Hausdorff. However, general relativity allows spacetime to behave badly in cases such as singularities, so it is imaginable that our universe contains points that violate the Hausdorff property. There are interesting and physically well-motivated spacetimes, such as some versions of the Taub-NUT space, that are non-Hausdorff. Since we have no empirical data on the behavior of spacetime under the most extreme conditions, we cannot say whether spacetime is really Hausdorff. One should maintain some skepticism about whether such an idealized category is even meaningful in science, since it refers to phenomena at arbitrarily small scales, whereas theories and measurements are limited in the scales they can deal with. A good discussion of the Hausdorff property as applied to relativity is given by Earman. 17
Example 19: Branching universes
Figure \(\PageIndex{4}\) shows spacetime diagrams of 1 + 1-dimensional universes that branch like a tree. These are meant to be pictures of classical general relativity, although some of the strongest motivation for considering such possibilities comes from attempts to construct a theory of quantum gravity. In such theories, it is commonly expected that spacetime will have a structure at the Planck scale that is a kind of “quantum foam.”
The example in Figure \(\PageIndex{4A}\) is a manifold, and is Hausdorff. This is an example of topology change, meaning that the spacelike section at one time has a different topology than the section at another. 18 Although such a branching can occur without the existence of any singularities, theorems by Tipler and Geroch show that other types of misbehavior must occur, including causality violations and the need for forms of matter that violate energy conditions.
Figure \(\PageIndex{4B}\) is qualitatively different. Here we have formed a spacetime by gluing together three pieces. No curvature is implied; these could be three pieces of Minkowski space. The spacetime is not a manifold, since the points at the join have different local properties than points elsewhere. The machinery of general relativity breaks down in a case like this, but for example we could imagine that a geodesic in this spacetime could fork off into two different geodesics after the split.
Yet a third possibility is to reinterpret Figure \(\PageIndex{4B}\) so that there are two different copies of the seam. For example, we could let the portion of the diagram extending into the past be represented by points with t < 0, while the two branches continuing into the future could each have t ≥ 0, so that for a given x we would have two different events with coordinates (t = 0, x). It would not be possible to put these two points into disjoint neighborhoods, so this version of the space is not Hausdorff.
18 For a recent treatment, see Borde, 1994, “Topology change in classical general relativity,” arxiv.org/abs/gr-qc/9406053.
References
17 John Earman, ‘Pruning some branches from “branching spacetimes”,’ pitt.edu/~jearman/Earman2008a.pdf


10.9: References
Buehlman, K. T., Gottman, J. M., & Katz, L. F. (1992). How a couple views their past predicts their future: Predicting divorce from an oral history interview. Journal of Family Psychology, 5 (3-4), 295-318.
Chapman, G. D. (1992). The Five Love Languages. Northfield Publishing.
Chen, J., Merrick, M. T., Smith, S. G., Kresnow, M.-jo, Wang, J., & Zhang, X. (2018). National intimate partner and sexual violence survey: 2015 Data Brief Updated Report. Centers for Disease Control and Prevention, National Center for Injury Prevention and Control, Division of Violence Prevention.
Fetters A. (2018, November 15). How friendsgiving took over millennial culture. The Atlantic . https://www.theatlantic.com/family/archive/2018/11/millennials-friendsgiving-history/575941/
Greene, G. J., Fisher, K. A., Kuper, L., Andrews, R., & Mustanski, B. (2015). Is this normal? Is this not normal? There's no set example. Sexual health intervention preferences of LGBT youth in romantic relationships. Sexuality research & social policy: Journal of NSRC, 12 (1), 1-14.
Holt, B. (2020, July 2). For interracial couples, advocacy is a love language. New York Times. www.nytimes.com/2020/07/02/fashion/weddings/for-interracial-couples-advocacy-is-a-love-language.html
Johnson, S. M. (2008). Hold me tight: Seven conversations for a lifetime of love . Little, Brown Spark.
Johnson, S. (2014, February 14). What is emotionally focused therapy (or EFT)? [Video]. YouTube. https://www.youtube.com/watch?v=xQCg-jC25fo
Knapp, M. L. & Vangelisti, A. L. (2009) Interpersonal communication and human relationships. Pearson.
Leopold, T., & Kalmijn, M. (2016). Is divorce more painful when couples have children? Evidence from long-term panel data on multiple domains of well-being. Demography , 53 (6), 1717-1742. https://doi.org/10.1007/s13524-016-0518-2
Livingston, G. & Brown, A. (2018). Intermarriage in the U.S. 50 Years after Loving v. Virginia. Pew Research Center. https://www.pewresearch.org/social-trends/2017/05/18/intermarriage-in-the-u-s-50-years-after-loving-v-virginia/
Metts, S., & Cupach, W. R. (1989). The role of communication in human sexuality. In K. McKinney and S. Sprecher (Eds.), Human sexuality: The societal and interpersonal context (pp. 139–161). Ablex.
Montesi, J., Fauber, R., Gordon, E., & Heimberg, R. (2011).  Journal of Social and Personal Relationships. 28, 591-609.
Moore, P. (2014, October 28). Poll Results: Ghosting. YouGovAmerica. https://today.yougov.com/topics/lifestyle/articles-reports/2014/10/28/poll-results-ghosting
Schutz, W.C. (1966). The interpersonal underworld. Science & Behavior Books.
Weston, K. (1991). Families we choose: Lesbians, gays, kinship . Columbia University Press.
Wheeless, L. R., Wheeless, V. E., & Baus, R. (1984). Sexual communication, communication satisfaction, and solidarity in the developmental stages of intimate relationships. Western Journal of Communication, 48, 217-230.


4.5: Headers and Footers
Learning Objectives
- Insert text into header and footer.
Headers and footers are the top and bottom sections of the document respectively. They are separate sections from the main document, and are often used to hold footnotes, page numbers, titles, and other information.
The header and footer controls are in the Header & Footer group of the Insert tab. (Insert>Header & Footer)
The Header and Footer options each contain a dropdown menu. Click one of the Built-in options to insert it, or select Edit Header (or Edit Footer) to edit the header (or footer).
The header/footer will be represented by a dotted line. While in header/footer mode (when the dotted lines are visible), you cannot edit the main text. You can go back to editing the main text by selecting the Close Header and Footer option on the right side of the ribbon.
The information in the header or footer is repeated in every header or footer in the document.
Practice Question
Page Numbers
You can number the pages in your document automatically in the header/footer options with the Page Number option. Choose a location on the page by going to the Top of Page, Bottom of Page, Page Margins, or Current Position options and selecting an alignment. (Current Position refers to where the cursor is located.)
The page number will be repeated on every page in the header or footer, but the number will change to reflect the appropriate page number. You can also change the way the pages are numbered by going to the Format Page Numbers option.
Contributors and Attributions
- Headers and Footers. Authored by : Lumen Learning. License : CC BY: Attribution