*A brief note on form:*
NOTE: I apologize if there is some bizarre behavior of the more links when they are not clicked in order and otherwise. I am working on an algorithm that will behave properly while also letting me keep the current typesetting. For an optimal experience, I suggest you click the more links in order. If you would like to collapse the embedded content, hit refresh (cmd+R in Safari). If you are good at Javascript and you think you can help, please view the source code and contact me with suggestions. I thank you kindly for you understanding and patience during this construction period. MTP 12/10/10
Throughout this course, we attempted to communicate the core content of the course via the **Equations of the Week.**
To bring them into your conscious mind, listed below are the **equations of the week** in their original form. Take a minute to remember what these **equations** mean to you: How do they relate to your *macroscopic observations,* both during the *labs* and your *final project*, as well as during your own *cooking experiences*? What do you remember from *lecture* about the *microscopic phenomena* that explained the *macroscopic observations* of the materials described by these **equations**? How did you use these **equations** in the *homework* to relate your *macroscopic observations* to various *microscopic phenomena* in order to say something useful about the world? Finally, bring to mind what it is about these **equations** that is still confusing you (i.e. What questions can't you answer?). My goal by the end of this document is to replace these confusions with clarity; please feel free to use the have a question? form to the right if anywhere along the way you feel I have not succeeded. (Below, I purposely do not define the symbols used in the **equations of the week**; they will be discussed in turn for each of the **equations,** and it is more important now for you to focus on *what comes into your mind* when you see these **equations.** You should—by the end of this document—know what all the symbols mean; they will not be defined for you on the final exam.)

week one:
`U`_{interaction} = (^{3}/_{2})`k`_{B}T

week two:
1 Calorie = 1000 calories = 4.18 kJ

week three:
`E = `^{ kBT}/_{l}^{3} (Elasticity)

` ν = l `_{×} c (Molecular viscosity)

week four:
`T`(`t`) = (`T`_{initial} - T_{external})`e`^{(-t/τ)} + `T`_{external}
` τ = `^{L}^{2}/_{πD}

week five:
`U`_{hydrophobicity} = k_{B}NT + U_{electrostatics}

week six:
`U`_{surface} = σ _{×} 4`πR`^{2}

Δ`P =` ^{2σ}`/`_{R}

week seven:
`L`_{shell} = √πDt *here `D`_{Ca} was changed to `D` for HTML typesetting reasons

week nine:
`N`(`t`)` = N`_{0}exp(`kt`)

(Remember, in week eight we used the same equation as we did in week four.)

Understanding how to apply these equations is difficult, mainly because when speaking the the **Math language**—a *language* written in **equations** and spoken in **jargon**—there is a lot of information embedded in the *context* of the **equations**.
That is to say, much of the information needed to understand a statement in the **Math language**—a mathematical expression; an equation—is not explicitly written; it is assumed that the reader will understand the *concepts* based on their previous knowledge. Unfortunately nearly every specialized field is rife with this imprudent practice, and this typically prevents extremely capable individuals from approaching ostensibly arcane subjects. Our goal in this course was to break through these barriers and teach you soft matter science using the *language* of cooking; we did our best to teach you the **Math language** along the way. Listed below are the **equations of the week** written in a form that explicitly states information typically embedded in the *context* of the **equations**. These slight modifications made to the original symbolic representation have not changed their meaning; only the symbols we use to express this meaning have changed. “That's great, but why didn't you write the **equations of the week** *this* way from the start?” Unfortunately, much of our understanding of how you learn and the way that we teach was gleaned from the myriad individual conversations and questions that arose throughout the semester. We have all learned a lot, and in future courses, the teaching staff will use these new forms of the **equations of the week** to present the *concepts*. For now, the best I can offer is my apologies for not preparing this document sooner, my hopes that your frustrations will fade as your understanding solidifies, and my immense gratitude for the lessons I have learned with you all. Hopefully these newer, more explicit forms will clarify some of the misunderstandings from the semester:

week one:
`U`_{Interaction} = (^{3}/_{2})`N`_{Bonds}k_{B}T_{Bonds breaking}

week two:
1 Cal = 1000 cal = 4.18 kJ = 4180 J (in symbols)

1 Calorie = 1000 calories = 4.18 kilojoules = 4180 joules (in words)

week three:
`E = `^{ kBT}/_{l}^{3} (Molecular elasticity; here *l* = length between crosslinks)

` ν = l c ` (Molecular viscosity; here *l* = length between molecules)

week four:
`T`(`t,L`) = (`T`_{Initial} - T_{External})`e`^{(-tπD/L2)} + `T`_{External}

week five:
`U`_{Hydrophobic} = k_{B}N_{Molecules}T_{Denaturation} + U_{Electrostatic}

week six:
`U`_{surface} = σ _{} 4`πR`^{2}

Δ`P =` ^{2σ}`/`_{R}

week seven:
`L`_{shell} = √πDt *here `D`_{Ca} was changed to `D` for HTML typesetting reasons

week nine:
`N`(`t`)` = N`_{0}`e`^{kt}

Herein I aim to elucidate that *context* for our **equations of the week** and present a full view of the course content along the way. Before taking a close look at the *concepts* presented each week, I begin by discussing a few ideas about the way scientists and engineers use **equations.** In other words, I start with a discussion about a few of the finer points of talking in the **Math language**.

On the Math language

The **Math language** is a dragon: Replete with power and peril. As with any *language*, its power resides in the practically infinite number of *concepts* it can communicate. There are also varying levels of fluency, dialects, written versus spoken word, homonyms, et cetera which present the typical *language*-learning barriers. On top of this, the *language* of mathematics comes with its own unique difficulties (which we'll get back to at the end of this aside). As a receiver of information communicated in and a student of the **Math language**, it is your job to understand that there are *language* barriers to overcome. (This, and the next statements are true of *every* *language* you encounter; e.g. the Science *language*, the Art *language*, the Economics *language*.) Without consciously focusing on overcoming these barriers, it becomes *extremely* difficult—even impossible—to understand the *concepts* being communicated. Saying it differently, it is paramount to recognize that you are learning the **Math language** *and* the math *concepts* at the same time; isolating the two will allow you to focus on the main ideas of a mathematical lesson without worrying about the subtleties of the vocabulary, syntax, and grammar—these will come with time and practice. (As a test, always try to explain and question new math *concepts* in your native *language*. If you speak English and you can ask a math question in the English *language*, you've successfully isolated the content from the form—the *concepts* from the *language*.) To illustrate how you already do this: If someone started speaking to you in Spanish—assuming you can speak a little bit of, but are not fluent in the Spanish *language*—you would stop them, or at very least ask them to slow down so you had time to comprehend what they were saying. This is because you are trying to do two things at once: Translate the statement from the Spanish *language* into your native *language* *AND* understand the *concepts* embedded in the statement. Communication cannot occur until you do both of these successfully. This task becomes difficult in mathematics because, for one, the *concepts* themselves are often quite difficult to comprehend. Conflate this with the common practice of shortening every **Math language** statement into the smallest possible form, and mathematics quickly becomes the quintessential subject people are “bad” at. This last point is critical: The main peril of the **Math language** lies in its added complexity of compactness. That is, **equations** are typically written in the shortest “sentences”possible. The job of the receiver of mathematics information is to bring to mind all of the unarticulated *concepts* embedded in the elliptic statements characteristic of mathematics. (By elliptic, I do not mean the definition: “of, or relating to, or having the form of an ellipse” Rather, I chose the word elliptic for two reasons: First, it has definition which is precisely what I mean: Mathematical statements typically “lack a ‘word’ or ‘words’, especially when the sense can be understood from *contextual* clues.” Second, it illustrates the fact that at times the English *language* can be equally as confusing as the **Math language**!) For the mathematical novice, this is impossible to do without critically asking questions and digging deeper, until you are certain you understand the *concepts* associated with the situation; then you can begin to read the mathematical expression, and eventually you can put it to work for you—you can use it to understand new *concepts* about the world.

Equations, more or less.

**Equations** are simply a *relationships between parameters.* Every part of an **equation** is either a number (e.g. 1, 5, π, *e*, Avogadro's number, 10^{-5}), a measurable quantity (something with units—5 kJ, 10^{2} mol, 15 centimeters, 0.004 volts—and typically written symbolically with American or Greek letters; e.g. *k*_{B}, σ, T, L, ρ), or a relational operator (e.g. +, -, =, (,), ×, Σ, ∇). In this course we used **equations** in two ways. For better or worse, during this course we relied in part on the *context* of the **equations** to communicate fundamental *concepts* related to the lenses scientists and engineers use to view the physical world (i.e. the way that scientists and engineers talk about what happens). Here I draw *explicit* attention to the two most common ways **equations** can be used to learn something interesting about the world. The first way is to view an **equation** as a *relationship between some known and unknown parameters;* when you use an **equation** to determine a particular value of a parameter (or set of parameters) based on some information that you either know or have observed, we say you are *solving the equation.* The second way that we use **equations** is to view them as a *balance*; by varying the quantities on one side or the other, we can see which way the *balance* will “tip” for a given set of parameters. That is, we can determine how a physical system will react to a specific state of the world. Knowing when to employ each of these views is paramount; though they are related, using the wrong method will lead to unfounded conclusions and misunderstandings. For example, if you *solve* for the *hydrophobic energy* (`U`_{Hydrophobic}) in the *energy balance* from **week five** (`U`_{Hydrophobic} = k_{B}NT + U_{electrostatic}), you would incorrectly conclude that higher *temperatures* (`T`) would cause `U`_{Hydrophobic} to increase. The correct way to view the **equation** here is to consider a specific value of `U`_{Hydrophobic} and see how you can vary the two right-hand terms (`k`_{B}NT and ` U`_{electrostatics}) such that they sum to value greater than `U`_{Hydrophobic}. In the Science *language* we would say something like: For a given value of *U*_{Hydrophobic}, how do changes in *temperature* and *electrostatic energy* affect the system. To be complete, you could—and did—*solve* the **equation of the week** from **week five,** but here you did not consider just any *temperature*: You considered the specific *temperature* at which the proteins would denature (`T`_{Denaturization}). This example illustrates the subtle differences in how we can and cannot use **equations** and also how the same **equation** can be used as both a *balance* *AND* as a means to *solve* for a parameter. This is why *context* is paramount!

Equations of the Week

Included here are the slides presented by Emily Russell at the Monday night Review Session. They do such a beautiful job, distilling the content of the course through the Math language, I had to include them as the conclusion to this document. Their strength is in their simplicity, and I honestly do not think the important messages of the course could have been presented more clearly. Enjoy!