In the previous segment, we concluded with the fact that a curve in may be viewed as an interval *I* together with the following set of data:

- A real-valued function , which will help measure the length,
- A real-valued function , which will help measure the curvature, and,
- A real-valued function , which will help measure the torsion.

In this segment we will see some examples and discuss possible constructions that this allows us.

**Examples:**

The data represents the part of the parabola between . More precisely, this segment of the parabola is the only curve with the above curvature and torsion; any other curve is just the same one with a change in the position of the observer in .

The data represents the helix

**Equivalence of Curves**

Often, the first thing one does after defining/constructing a new object in math is to define an appropriate notion of “sameness” between two of them. To this end, we have the following definition:

*Definition***:** Two curves and are equivalent if there exists a diffeomorphism such that

In calculus language, this would be a “re-parameterization” of a curve. If this happens, then the two curves that they describe coincide completely — they will overlap after a rotation and translation.

The above definition is derived from the plausible requirement for the following to hold:

A change of variable in the second integral leads to the first condition in the definition above.

**Example**:

We saw from our previous example that describes a part of the parabola .

It turns out that the set of data also describes the same curve, with an explicit equivalence being given by the map .

**Intrinsic Integration On Curves**

Suppose that we are given a function which assigns to each point on a curve a real number, say the temperature at that point. We wish to find the average temperature over the curve. To do this, we need to add up (integrate) the temperatures at each point and divide the result by total number of points (length).

The following is an appropriate definition of the integral of a real-valued continuous function on our manifold:

Again, it is adding up values of on , just as in Riemann sum, except that now we have a non-uniform weight applied to each summand.

The reason this definition works is that if we calculate the integral using a different but equivalent set of data, then by the change-of-variables formula, we would get the same answer:

*Observation: *Our definition of integration only depended on the function . Therefore, it suffices to have a way of measuring distances in order to be able to define a parameterization-free, -free notion of integration on a curve.

*Note: *The integration above may be familiar from calculus, or complex functions, as a “line integral”. But it is usually not emphasized there that this is intrinsic to the curve. However, it indeed comes with the curve, independent of the embedding of the curve in

**What’s next …**

Now that we know a way of telling when two parameterizations coincide, in the next installment, we will be able to “glue together” pieces to arrive at a manifold.

The definition of the integration will work globally because different parameterizations agree on it locally.

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Always making me think: Why is this still a thing?

Okay, certainly there are legitimate reasons to collect demographic information, including information regarding gender. Whether to explore the diversity of the field (i.e. graduate admissions or university surveys) or whether it is needed for grant purposes (i.e. conferences) there are times when inquiring about an individual’s gender identity is reasonable. However, why must this information be collected in such biased fashion?

The fact of the matter is that for many individuals gender is a complicated social construct, with which they have a nuanced relationship. As phrased above, the question “Gender?” takes all of these complexities and nuances and *smashes* them; *smashes* them into two extremely narrow little circles. It takes individuals and *smashes *their agency, forcing them to squeeze themselves into a narrow space predefined by someone else.

Moreover, to some who are “gender-expansive”, this question is yet another instance of someone — a colleague, a peer, a fellow mathematician — neither recognizing nor accepting part of who they are. It is yet another instance in which they are in part invisible:

“invisibility may seem like a small price to pay … But invisibility is a dangerous and painful condition. When those who have the power to name and to socially construct reality choose not to see you or hear you … when someone with the authority of a teacher, say, describes the world and you are not in it, there is a moment of psychic disequilibrium, as if you looked into a mirror and saw nothing. Yet you know you exist and others like you, that this is all a game with mirrors.”~Adrienne Rich, “Invisibility in Academe” [1]

In short, this question invalidates the feelings and experiences of those who do not find themselves comfortably within the confines of the two narrow little bubbles provided to them.

If I agree there are legitimate reasons to collect gender demographics, and if I find the above format highly problematic, what do I think the answer is? Well I think a better approach is to give agency to the respondents. As the Human Rights Campaign notes the least restrictive and most preferable option is to allow individuals to self-identify. For example, by re-phrasing the question in the introduction as follows:

Doing this empowers the respondent to define their gender identity as they see fit. I should also point out – not that it should matter – that phrasing the question in this way changes little how the question writer interacts with the responses. (Note other parts of academia have already begun moving to eliminate such biased and discriminatory questions.)

Some of you might be wonder why I have bothered spending ~500 words on a topic so “insignificant” as to how to ask someone their gender. The fact of the matter is that these things are not “insignificant” – they matter, and they have substantial effects on the lives and careers of many people. (I am reminded, in part, of the following tweet.) Sure, this might not have an effect on your life, but that does not mean the opening question does not make others feel uncomfortable, outcast, and discriminated against. And to say this is “insignificant” is to say these people are, well, insignificant …

Which is unacceptable. No person is insignificant. No one should ever be made to feel insignificant. No one should ever be … invisible.

For us to create a more inclusive, supportive, and welcoming mathematical community, these are the sorts of things we need to begin to think and care about. We need to stop seeing things like this as “insignificant”. We need to begin to see the world and individuals as they are: complexly. Most importantly, we need to step up and change the ways we act. Changing how we phrase this question is a small first step.

[1] Roberts, T. et al. *The Broadview Anthology of Expository Prose 2 ^{nd} Ed. *Broadview Press. 2011.

There are competing theories online about possible interpretations of John von Neumann’s quote, but *manifolds *are definitely some mathematics that “you don’t understand … you just get used to them,” — at least for a while.

In a series of posts reflecting on my own experience, I will try to motivate the conceptualization of manifolds, and the implications such an abstraction has/had on our understanding of, basically, shapes. I hope to point to some beautiful geometry in low dimensions that you may have passed by too quickly to take notice of.

I must underline the subjective nature of my articles, and that by no means are they meant to narrate the history of the subject, or depict a current fashion in the community. This is simply “another article.”

The first three articles will be dedicated to converting the conventional calculus of curves into manifold language. We will see that a curve can be replaced by an interval endowed with some structure. This will pave the way for an exposition of the theory of surfaces in subsequent articles. The reason for such an extended sequence is to include as much detail and as many examples as possible.

** The Question**

**What We Know from Calculus**

From calculus we fix a Cartesian coordinate system for three-space and then parameterize our curve by a map from a subset of ,

If is a smooth parameterization, then the length of the object between two of its points and is given by

The curvature is given by

and the torsion is given by

where differentiation is taken coordinate-wise.

Notice that the integral above involves *only the* *length *of . Thus, even if we don’t have itself , but only some , we will again be able to measure the length of any segment of our curve.

This observation invites a search for a representation of our curves independent of the three-space.

**The Answer to the Question**

In a search for a 1-dimensional embodiment of a curve, the following theorem is the best we could hope for.

**Theorem:** Given differentiable real-valued functions , and , with ranging in an interval , there is a curve with being its curvature and being its torsion and giving its length between points. **Moreover**, any other curve with this description is obtained by changing our origin and rotating the axes rigidly. (See Chapter 1 of do Carmo’s book *Differential Geometry of Curves and Surfaces.*)

Thanks to this theorem, an interval along with the following set of data can be thought of as a curve in :

- A real-valued function , which will help measure the length,
- A real-valued function , which will help measure the curvature, and,
- A real-valued function , which will help measure the torsion.

Note that this data is independent of the specific positioning of our curve in 3-space.

**So far…**

A curve in is nothing more than an interval in along with three real-valued functions defined on it! Once we have this set of data, we can forget about our original curve as a subset of , and work in 1-dimension. After all, we can reconstruct an exact copy of our curve in 3-space from the data whenever we wish.

**In the Next Installment…**

- We will discuss some constructions, e.g integration on curves, that this description makes possible;
- We will see examples of curves described by a set of data,
- We will find out a way to tell when two sets of data determine the same curve, which will allow for the definition of 1-manifolds in the following article.

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If you’re interested in writing some of your own mathematically-oriented posts, be sure to let us know by contacting someone on the editorial board directly or using the Contact page on our website!

]]>Imagine that you are at the center of a circular pool of unit radius. At the very edge of the pool, there is a monster who is trying to catch you. The monster cannot swim, so it can only run around the edge of the pool. You know that you can run faster than it while on the ground, so if you can manage to reach any point on the edge of the pool before the monster can get there then you will escape. However, the monster runs four times faster than you can swim. Is it possible for you to escape from the pool and the monster?

In order to clarify the riddle, suppose that the monster instead ran three times faster than you could swim. Then the obvious thing to do would be to swim in a straight line to the edge of the pool directly opposite from the monster’s starting point. To reach this point, the monster would have to travel π times as far as you (i.e., a distance of π*r* versus *r*). Since π > 3, you would thus be safe.

However, since the monster runs four times as fast as you, the simple strategy above will result in you getting eaten. Can you think of a more complicated strategy that does work in this case?

If that’s too easy, then can you escape if the monster runs five times faster than you? What is the maximum speed (e.g., five times, six times) that the monster can run such that you can still escape?

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According to the *An Inquiry-Oriented Approach to Undergraduate Mathematics *research paper in the *Journal of Mathematical Behavior*, Kwon and Rasmussen (2007) talk about the Inquiry-Oriented Differential Equations project as a collaborative effort to improve the undergraduate math education and to study how undergraduate math can draw on the theoretical and instructional advances initiated at the K-12 level as well as to create and sustain learning environment for powerful student learning. While there are several reasons for the initiation of the Inquiry-Oriented Differential Equations project such as the huge number of math departments at universities and colleges, the increase of student body diversity, and the decline in the number of math majors as mentioned by Kwon and Rasmussen, universities and community colleges need to implement this project as a fundamental way of improving the undergraduate math curriculum in general and differential equations curriculum in particular because this project can encourage the useful interaction between teacher and student through class activities and discussions in order to enable the teacher to measure the student mathematical thinking and how the teacher can form a new way of thinking based on what the student thinks.

In addition, Kwon and Rasmussen (2007) discuss several various characterizations of inquiry process in different research communities, for example, in general, inquiry can be identified as a set of assumptions using critical and logical mathematical thinking and considering alternative mathematical explanations, while in the philosophy of mathematics education, inquiry can be defined as the ability to learn how to speak and act mathematically, participate in mathematical discussions, pose conjectures, and solve new or unfamiliar math problems. I totally agree with all those different characterizations of inquiry process because they are located under the title of how teacher inquiry into student mathematical thinking.

For example, when I taught *Calculus II *in Fall 2015, I implemented this inquiry process using tools in my math book: *A Friendly Introduction to Differential Equations *because in my *Calculus II *class, there were some topics that are also relevant to differential equations such as finding the general solution of differential equations using the separable method. When an abstract topic in *Calculus II *such as partial fraction decomposition came up, I asked my students to think about a method that can save time and might solve fifty percent of partial fraction decomposition problems. Some students told me that there are no other methods to avoid solving systems of linear equations to find the required constants while others started giving me assumptions and suggestions about possible methods of doing partial fractions decompositions. In this situation, the students are learning how to mathematically investigate other methods built on their assumptions and suggestions and I encourage them to creatively think about that mathematical problems in search of solutions.

There are several advantages of applying the inquiry approach in our math classes such as differential equations and its positive effect on the improvement of teaching and learning is one of the fundamental goals that we are looking for in our math classes. We want to provide our students with the tools for successful advanced mathematical thinking processes and the methods of reinventing mathematical ideas and implementing IO and AMT has been a successful way for me to do just that.

References

Kaabar, M. (2015, January 5). A Friendly Introduction to Differential Equations. *Printed by CreateSpace, San Bernardino, CA*, http://www.mohammed-kaabar.net/#!differential-equations-book/cuvt. Accessed on August 29, 2016.

Kwon, O., & Rasmussen, C. (2007). An Inquiry-Oriented Approach to Undergraduate Mathematics. *Journal of Mathematical Behavior, 26*(1), 189-194.

Have you ever been asked to hang out with a new friend, but maybe you didn’t know where they lived? Maybe you were going to an interview for your new job but needed directions. The solution to both of these problems is using a GPS system. We are very fortunate to be able to input an address and be given step by step directions to our destination. One rainy afternoon I began thinking about this and asked myself, “What was the world like before GPS? How did people navigate?” I began to think about how we can easily make it to the other side of the city, but how we now can make it across to the other side of the world. How did people navigate the sea without any technology? Thankfully George Christoph (view his talk https://youtu.be/AGCUm_jWtt4) and others give us somewhat of an insight on how ships were sailed way back when.

When sailors are sailing ships the vessels can be thrown off course very easily whether it was a human error, change in current, or a nasty storm. The slightest miscue could potentially cause the sailors to miss their destination. Since there was ambiguity in whether or not the ship was on the right course the sailors came up with an idea that would help them know if land was near. In the early days of sailing the Vikings would release a bird to fly above the ship. If the bird flew up above the ship and circled randomly then there wasn’t any land near. If the bird flew with a purpose away from the ship then the ship would follow in hopes of the bird finding land.

Sailing with the help of birds was a creative idea, but didn’t always lead to the results that sailors wanted. Another idea was discovered that included more mathematical reasoning. If the sailors knew the speed of the boat, then they would be able to figure out how far away they were form their port, which would give them their location at sea. The sailors would count the number of seconds that it took for a piece of rubbish in the water to get from the front of the vessel to the rear. They took the length of the boat and divided it by the time to get an estimation for their speed. This technique was clever, but proved to be too unreliable when attempting to navigate an ocean.

If sailors knew the longitude and latitude of their ship then they would know their location at sea which would greatly improve their chances of landing where they desire. Latitude wasn’t very difficult for sailors to determine. All that was needed was a compass, a calendar, and how high the sun should be at noon. However, what was more complicated was determining the longitude of the ship. The way that sailors determined the longitude included the sun and an accurate clock. The sun was important because of the rotation it appears to make in the sky. The earth rotates and makes it appear that the sun is making a 360 degree rotation around the earth. This means that each hour the sun would move fifteen degrees. Every four minutes is would move one degree. Every one minute it moves 15 arc degrees and so on. This information can be used if you have an accurate clock set to Greenwich mean time. For example, if the GMT time was 18:00 and the local time was 10:00 then you have a difference of minus eight hours, which means your angle measure would be fifteen times eight which equals 120 degrees. This means you are 120 degrees west. These calculations would prove to be very helpful for sailors who were navigating the sea. The only problem was the fact that the clocks only kept accurate time while on land. The movement, temperature difference, and humidity all disrupted the clocks while at sea. It wasn’t until 1775 when a man named Larcum Kendall made a clock based off the clocks made by John Harrison. The clock was taken on a three year voyage and they found that the clock had kept accurate time and only had an error of about eight seconds per day.

What the sailors were doing in the late 1700s and beyond is what we call using a GPS system. What is amazing is that our technology has developed over time to the point where it is taking the sailors from ships and keeping them on land while the ship takes its course. Technology is being created that would allow an unmanned ship to sail across the ocean to the proper destination by remote control and a GPS location. The technology is said to be a handful of years away from being put into action, but would be an amazing feat if this unmanned ship could actually sail across the ocean with no problems. The technology and systems that were created have been expanded upon to great lengths.

Works Cited:

*How Did Early Sailors Navigate the Oceans?* *YouTube*. N.p., n.d. Web. 23 Apr. 2016. <https://www.youtube.com/watch?v=4DlNhbkPiYY>.

*How does math guide our ships at sea?* By George Christoph. *YouTube*. N.p., n.d. Web. 23 Apr. 2016. <https://www.youtube.com/watch?v=AGCUm_jWtt4>.

Mahoney, Donna. “Underwriters get ready for crewless ships.” *Business Insurance* 15 Feb. 2016: 4. *Wilson OmniFile Full Text Select Edition*. Web. 4 Apr. 2016.

“Unmanned Ship Technology.” *Business Insurance* 15 Feb. 2016: 27. *Wilson OmniFile Full Text Select Edition*. Web. 4 Apr. 2016.

Ship with bird link- Included in the photo.

Clock link- http://yeoldecybershoppe.com/malahide/uncategorized/patrick-malahide-john-harrison/

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Anyone involved in the discipline of math can most likely recall one, if not multiple, instances of being questioned on the usefulness of math. Eduardo Saenz de Cabezon addresses this question in his TED talk “Math is forever” (which can be found here). He claims there are three types of responses. First, the attacking one, which states math has a meanings all its own without the need for application. Next is the defensive one, which replies math is behind everything from bridge building to credit card numbers. The third response is where Eduardo claims math’s utility stems from its ability to control intuition, thus making it eternal.

Is math forever? Eduardo seems to think so stating diamonds aren’t forever, a theorem is. Mathematicians spend their lives generating conjectures and searching for ways to prove them. Once a conjecture is proven true though, it becomes a theorem, which is a truth that will remain so forever. Therefore, concepts such as the Pythagorean Theorem and the Honeycomb Theorem will forever be true, regardless of whether or not we are here to acknowledge it. This idea is rooted from Platonism, which is the philosophical view that there are abstract math objects that exist independently from our thoughts. Thus, all math truths are waiting to be discovered and not invented.

There are two main contributors in the world of mathematical philosophy. The first is German mathematician David Hilbert (pictured to the left), creator of Hilbert’s Program. He claimed that all math is formulized in axiomatic form with a proof to accompany it; it is done so by using finitary methods only which gives proper justification for classical mathematic problems. Hilbert believed theories could be developed without the need for intuition and would generate a set of rules and axioms that are consistent so one cannot prove an assertion as well as its opposite. Hilbert, like Eduardo, believed the capabilities of math were limitless.

Hilbert’s work, in turn, inspired the work of Kurt Gӧdel (pictured right) and his Incompleteness Theorems. Gӧdel proved that Hilbert’s concept of a decision procedure that generates axioms cannot be possible; there will always be conjectures that need a proof that may not actually exist. Gӧdel’s first incompleteness theorem proved that math knowledge cannot be specifically summed up and identified. Even the soundest basic rules will have statements about numbers that can’t be verified. It is important to note however, that Gӧdel never had the intention of disproving Hilbert’s program but rather to offer a new view.

So this leaves the math community open to explore if math is created or exists regardless of human recognition. If a tree falls in the woods when no one is around, does it make a sound? If no one has been able to prove a conjecture, does that theorem still exist? Like many schools of thought, there is ambiguity and uncertainty. As an individual in the math community, we all are responsible for looking into the information and opinions and coming to our own conclusions. Yet one thing remains certain, intuition and creativity

are absolutely essential in mathematics.

Sources:

de Cabezon, Eduardo Saenz. “Math is Forever.” *TED.* TED, Oct. 2014. Web. 05 Apr. 2016.

Elwes, Richard. “Ultimate Logic. (Cover Story).” *New Scientist. *211.2823 (2011): 30-33. *Academic Search Complete. *Web. 4 Apr. 2016

Linnebo, Oystein. “Platonism in the Philosophy of Mathematics.” *Stanford University. *Stanford University, 18 July 2009. Web. 05 Apr. 2016.

Peterson, Ivars. “The Limits of Mathematics.” *Science News. *Society for Science & the Public, 2 Mar. 2006. Web. 5 Apr. 2016.

Zach, Richard. “Hilbert’s Program.” *Stanford University. *Stanford University, 31 July 2003. Web. 05 Apr. 2016.

Image 1 retrieved from: https://www.bing.com/images/search?q=Incompleteness+Theorems

Image 2 retrieved from: https://www.bing.com/images/search?q=david+hilbert

Image 3 retrieved from: https://www.bing.com/images/search?q=Incompleteness+Theorems