Almond Biscotti (Low Carb, Gluten Free)

I love almond biscotti. I learned to make it when I was young (like, middle school?) and somehow never got cut on gratter boxes. I also haven't had one in about 2 years, because traditionally they aren't very low carb.

But I have recently discovered the power of XANTHAN GUM. And yes it sounds like a space alien but it's 'all natural' and more importantly it acts like the protein/binder gluten, which DH can't have and isn't found naturally in any of my low carb flours, especially not my prefered one, almond flour.

The first step is to make almond paste that isn't full of sugar. Fortunately, if you have almond flour, egg whites, and your prefered granulated sweetener of choice (I like Splenda, because I don't have to convert the volume measurement, and erythritol makes my tongue break out, so goodbye Truvia) you can make almond paste! Just throw equal quantities of almond flour and sweetener into a food processor, pulse to combine, then add a couple of egg whites. Start with one and keep adding until it is roughly the consistency of play dough. Voila! Almond paste. If you want marzipan, add more sugar. A nice extra touch is to add about 1/2 tsp of almond extract per cup of almond flour.

After that, it follows standard biscotti procedure. How many it makes depends on how you cut them--if you like your biscotti thick there will be fewer. at about a 3/4 inch slice, I got about 18 large biscotti. They stay slightly moist on the inside. You could leave them in a warm oven, or slice them thinner

YUM

Recipe
 1 batch almond paste (1 cup almond flour, '1 cup' sugar sub, 2 egg whites, 1/2 tsp almond extract)
 4 tablespoons cold butter, cut into small pieces

 1 3/4 cup almond flour + 1 1/2 tsp xanthan gum
 1/2 cup sweetener (3 tbsp honey also works, but raises carb count)
 1/2 tsp baking powder
 heavy pinch salt (to taste)
 2 eggs
 1/2 tsp vanilla

Pulse almond paste and butter together in food processor (or stand mixer). Add almond flour, xanthan gum, sweetener, baking powder and salt. Pulse/mix until well blended. Add eggs and vanilla and pulse/mix until mixture is uniform. Form mixture into a large loaf measuring about 6 inches by 18 inches and 1 inch high for large biscotti,or two loaves about 4 inches by 12 inches by an inch for small biscotti, on a baking sheet lined with a silpat or parchment paper. Bake at 350 F about 30 minutes. It should be lightly golden and springy to the touch. Gently transfer to a cutting board and let cool 5 minutes. Use large, sharp knife to cut into even slices, 1/2 inch to 1 inch thick. Lay them cut side down on the baking sheet, bake another 10 minutes. Flip slices gently, and bake another 10 minutes. Cool on tray, then store in a dry place. May soften over time; if that happens, place in a warm oven(even a toaster oven works!) for 10 minutes, and the crunch should be restored.

Basic Physics: Part 0, Section 2: Vectors and Coordinate systems

In the previous sections, I cover some basic algebra topics and necessary trig functions.

In this section, I'm going to lay out some basics of coordinate systems and vectors. I'm pairing these concepts because vectors without coordinate systems are a little esoteric for this series, and coordinate systems are necessary, but easily dealt with, at least compared to things like trig functions.

These two concepts are needed because when we talk about a problem and set out to solve it, we need a way to describe where things are, where they are going and how they are getting there. The combination of vectors and coordinate systems allows us to know exactly what we are referring to  and what it's relation is to anything else that might be relevant to the problem. Without this tool, problems in more than one dimension can easily become a hopeless jumble.

Let's start with coordinate systems. There are many possible coordinate systems of which  11 of which are commonly used and  only 1 of which we need for right now. That coordinate system would be the Cartesian coordinate system, apocryphally realized by M. Rene Descartes  (he of "I think therefore I am" fame) as he lay in bed watching a fly buzzing above him. It was also realized by M. Fermat, though he failed to publish it. If you ever had to plot things by hand in a math class, you have used the two-dimensional cartesian coordinate system! The cartesian coordinate system can be thought of as a grid system in 3 dimensions, that lets you specify a  location based 3 numbers, one for each dimension. You can think of it as giving someone a latitude, longitude and altitude. You have given them all the information they need to locate a particular spot on planet earth. (I will officially note that the earth is NOT a cartesian system, since the lines of longitude are not parallel but intersect. But for small distances, say NYC, it is a decent approximation).

Formally, a location in any coordinate system  is the intersection of three orthogonal planes. Orthogonal, for our purposes, means that the lines/planes intersect at \( 90^{\circ}\) to each other. Think of the walls in your house. Your walls (hopefully!) intersect your floors at right angles most of your walls will intersect each other at right angles, unless you have a very interesting house. So your walls are orthogonal to each other and they all orthogonal to the floor.

An example is probably the easiest way to show this. Let's start with a basic three-dimensional (3D) cartesian coordinate system:

In the picture above, I have drawn the same coordinate system from two slightly different perspectives. The top one you are staring down the barrel, so to speak, of the z-axis, looking at the x-y plane straight on. In the bottom one the picture has been rotated \(45^{\circ}\) about the y-axis so you can see along the z-axis as well. This becomes very useful if you are talking about things in 3D, while the top one is fine if you are only worried about two dimensions.

Another word about terminology and notation. What is an axis, and why are those letters wearing hats? As with a lot of math-stuff, it comes down to the dual needs for conciseness and precision. Let's start with the hats. When you draw a coordinate system for a problem and you label the axis, you are defining your directions. It's as if you are creating a mini universe and saying "this is East/West, this is North/South and this is Up/Down". But rather than label things in poetic victorian manner as "easterly direction" mathematicians and their ilk like to label things with letters. So "easterly direction" becomes "x-direction", but that's still too wordy. So 'direction' becomes 'axis', and that can get further shortened with vector notation as \( \hat{x} \) said "x-hat".  So an axis defines the direction of your coordinate system, but it also serves as a point of reference, much like the equator, the Greenwich meridian, and sea level  serve as reference points for finding places on the earth. So if you are on the x-axis, you are not moving in a y-direction or a z-direction. If you want to give a location in the coordinate system, you can notate it either as \( \left\langle a, b, c \right\rangle \) or you can use vector notation, to get a little ahead of ourselves: \( a \hat{x} + b \hat{y} + c \hat{z} \). The latter notation is preferably simply because it is more flexible, as we shall see.

Getting back to our example. Let's say we want to find a point \( \left\langle 3, 2, 2 \right\rangle\) (\( 3 \hat{x} + 2 \hat{y} + 2 \hat{z} \) ). For the moment we don't care what the units are. We start by locating the \( x = 3\) plane, that is, the plane that contains every point of the form \( \left\langle 3, b, c \right\rangle \) where \( b \), \(c\) are every real number. Then our diagram looks like this

with the red dot noting the point where the plane intersect the x-axis  in the bottom view, since it's a little hard to see. Next, we locate the \( y = 2\) plane.

The blue dot notes it's intersection in the y-axis in the bottom diagram because again, it's hard to tell. It's much easier to see in the top image, but there' a reason why the bottom diagram is actual preferable in some ways. This can be most easily seen when we try to add in the last point, the \( z = 2\) plane to give us our three-plane intersection.

Amazing what you can do with a basic drawing program and a little insanity.

The top image doesn't really allow us to visualize that last necessary dimension. You can mentally add it, but you can't draw it into the top one. The bottom one you can see the last plane and pinpoint their intersection (marked with a black dot here). 

That's pretty much all there is to coordinate systems. They let you pick a frame of reference so you can locate things in a mini-universe for the purpose of problem solving. What I find particularly neat is that you can place your coordinate system anywhere you like and the problem will still be solvable. It may be easier to solve from a computational standpoint if you center it nicely, but you don't have to. Why this is the case is something that I'll get into more when we start doing physics properly. 

Now, on to vectors. If a coordinate system gives you a frame of reference, vectors are what let you move around in that frame of reference and deal with more than just static problems. Now, what they are precisely requires linear algebra and is way outside the scope of this series, so we are going to stick to just definitions and not get into the nitty gritty. So, here goes.
 

A vector pairs a quantity with the direction that quantity is in, going or pointing to. A vector has both "magnitude" (quantity) and "direction". So long as you can describe a quantity as having these two qualities, you can express it as a vector*. We've already shown how we can describe position as a vector. You can also describe velocity as as a vector. "He's going 80 miles per hour" gives you a speed (a magnitude). "He's going 80 miles per hour due north" gives you a magnitude and a direction. We'll get more deeply into what physical quantities can be described using vectors in the first section of Part 1. 

For now, let's just work with two arbitrary vectors and see what we can do with them. As discussed in the algebra post, we'll use letters to stand in for numbers that we can plug in later. 

$$\vec{v} = a \hat{x} + b \hat{y} + c \hat{z}$$

$$\vec{w} = d \hat{x} + e \hat{y} + f \hat{z}$$

The little arrow above \(v\) and \(w\) indicates that they are vectors. In math everything has its own shorthand because you never know when you will want to deal with something in its entirety, or just don't want to write out the whole thing for the umpteenth time. 

So, what can we do to these things? Well, we can add them. The trick is that you can only combine things attached to like 'hats'. So you combine all the x-hat components, all the y-hat components and all the z-hat components, but you can't combine x-hat components with non-x-hat components. So 

$$\vec{v} + \vec{w} = a \hat{x} + b \hat{y} + c \hat{z}+d \hat{x} + e \hat{y} + f \hat{z}$$

$$\hspace{10 pt} = (a+d) \hat{x} + (b+e) \hat{y} + (c+f) \hat{z}$$

Subtraction works the same way:

$$\vec{v} - \vec{w} = a \hat{x} + b \hat{y} + c \hat{z}-d \hat{x} - e \hat{y} - f \hat{z}$$

$$\hspace{10 pt} = (a-d) \hat{x} + (b-e) \hat{y} + (c-f) \hat{z}$$

At this point you are probably wondering about multiplication and division, since addition and subtraction have been relatively straight forward. The answer is that there are two types of multiplication for vectors, and no types of valid division. Why this is starts getting into linear algebra and "outside the scope of this course". So I'm going to ask you to trust me on this one, because it's absolutely true even if I can't show you right now why it's true. They are also rather more involved than vector addition/subtraction, so I am going to move them to their own post so we can really take our time with them. 

I hope that this all was clear. If it wasn't, please let me know in the comments!

*There are also a few things that we'll get to over the course of this series that you wouldn't think you could describe as vectors, but they behave identically to the ones we deal with here. 

Shifting Cooking Gears

Cooking has always been about experimenting for me. Going by the recipe is fine for somethings, like candy making or if I know I enjoy this person's particular formulation of the dish, but by and large I like to tweak recipes. Non-baking recipes I'll usually just make up as I go along. Baking recipes are more chemistry dependant and therefore harder to do on the fly.

Lately though, I feel like I am having to reinvent the wheel, knowing what a wheel looks like but having to make it with very limited and somewhat unsuitable materials. It's exhausting trying to do that for every dinner.

Lately I've been having PCOS flare ups, partly just because and partly because there has been some stress in my life, so I've had to switch back to a stricter low-carb/low-GI diet. This would be annoying but par for the course, if we hadn't started a gluten-free diet for DH. He's had 'stomach problems' all his life, which I have been trying to solve for the 3 years we've been married and I've been in charge of procuring his food. Gluten-free was the last on a long list of things we've tried, and so far seems to be the most successful. We'll look into having proper testing done at some point, but since he just took a new job in a new city, the timing is not right for finding a specialist in our current area.

So in short, I am facing the challenge of cooking both low-carb/low-GI (LCLGI) and gluten-free. Lots of LCLGI food is gluten-free because if you aren't using any grain-flours you aren't going to be including gluten. It's also rarely recognizable as analogous to its carb-loaded counterparts, and to a certain extent just requires recognizing that there is no substitute for pasta or bread. Gluten-free foods, of which there are TONS on the market right now all nicely labeled, are rarely LCLGI because they are made with rice starch, tapioca starch, potato starch, etc. Pretty much everything I can't eat. Thus I am faced with the choice to make two different dinners, or to try and find food that lies in the overlap that we both find palatable.

Of course, some things don't really change. Meat is gluten-free and low-carb. Vegetables, pace potatoes, ditto. But there is something so fundamental to having some kind of starchy thing, and that's mostly where the problem lies. DH can have rice, but I can't. There are both low-GI and gluten-free pastas on the market, but of course they occupy opposite ends of the spectrum. I can have rye or spelt bread in small amounts, but he can't. I can make risotto with rice for him and risotto with barley for me, but that seems absurd. The best LCLGI and gluten-free recipes feature coconut flour, which has a noticeable taste for me that I don't always want. Most of the recipes I've come up with use almond flour which is unavoidably gritty, or oat flour which is gritty and whole-wheat tasting unless you really work to hide it.

 Cookbooks are typically one or the other, and if they are both they are typically one of the crazier diet fads, like paleo. While that is the closest to what we are eating, I just can't say we are going paleo. The whole diet is based on bad or non-existent science, cheese is something I rely on, and I can't get over the absurdity whenever I see a paleo recipe call for things bananas or brussel sprouts. Those yellow bananas you get in the grocery store have existed for less than 200 years, and look nothing like a paleolithic banana. Brussel sprouts only popped into existence in the 1300s. Coconut flour also did not exist in paleolithic times.

In short, food posts are probably going to be a little less "here's a recipe I made up last week" and more musing on what works and doesn't work as I try to reformulate, replace and otherwise revamp my repertoire of foods in the coming months. A journaling of success and failures so I hopefully don't have to repeat the latter too often. Also most likely they will be shorter interludes as I work on my Basic Physics series. And if you happen to follow me on Twitter (@PhysicsGal1701), now you know what all the food posting is about.

Cheers!

Basic Physics: Part 0, Section 1: Trigonometry

In Part 0, Section 0 I gave an overview of two basic algebraic skills needed to learn physics effectively. In this post I will explain several trigonometric (hereafter abbreviated to 'trig') ideas that are indispensable when physics problems are in more than one dimension. Since the world we live in is three dimensional, trig is really quite useful. \(\require{cancel}\)

Now, there is a lot more to trig than I am going to discuss here, mainly because we only need this one small portion to be able to tackle physics at this level. But trig is a much deeper subject with really interesting insights, if your fancy is tickled by beautiful interconnected math.

So to begin, we are going to need a triangle. This should be clear just from the name, which comes from the Greek for 'triangle' and 'measure'. More specifically, we need a right triangle, which is a triangle that possesses a \(90\) degree angle (a 'right' angle). You may have run into them before if you ever had to do anything with the Pythagorean Theorem. Right triangles are very nice to work with because you automatically know one angle, so all you need is is either two side lengths or a side and one of the other two angles and you can deduce everything else. So let's begin with a basic right triangle:

You will be subject to the same drawings and handwritten labels as my students are/will be

This is just a run of the mill right triangle. Each of the sides is labeled with a term that both labels it and can be used to indicate or find its length. The little square in the bottom righthand corner of the triangle tells you (as it says in the picture) that that angle is \(90^{\circ}\) (degrees). One of the other angles, in this case the one in the lower left hand corner of the image, is specified using the Greek letter \(\theta \) (pronounced 'theta') to stand in for whatever the actual measurement is.

Before we go any further with triangles, I want to say a few words about the use of Greek letters in math and physics. When I was younger, I thought that weird looking symbols were the sign of TRUE MATH and required REAL SMARTS to use. An equation could look like pure gibberish to me but so long as it included some Greek letter it must contain a great truth about the universe. Then I started taking physics and math classes and realized NOPE. They are, basically, there because we ran out of useful letters in the Carolinian alphabet we use in western European languages. We set aside a bunch of Greek letters to stand for quantities that pop up a lot (e.g. \(\phi, \theta \) are generally reserved for angles) and free up \(a\), \(b\), \(c\), to be used as needed. In other words, Greek letters do not imply fancy-shmancy calculations. They say "we needed more letters and  these were easy to co-opt".

So, getting back to our representative right triangle, which I am going to post again  so you don't have to scroll back up to see what I'm talking about.

 For this triangle, we are given all the information we could possibly want or need. Yes, even that unlabeled corner. First of all, all triangles* have angles which must sum to \(180^{\circ}\). The right angle eats up half of that (\(90 ^{\circ}\) ) automatically. The remaining \(90^{\circ}\) are divided between the other two angles. So if we know \(\theta\) we know the other angle is going to be \(90^{\circ}-\theta\). But secondly,  and most importantly, we don't need that third angle. Trigonometry lets us get away with just one angle, using the power of

Repeated like a prayer at every physics 1 exam

You may have seen or heard this mnemonic before. Maybe carved into a desk in a high school math classroom, maybe muttered by some flustered-looking college student. This acronym is a way to remember the names and definitions of the three most important trig functions: sine, cosine, tangent. Each of these functions describes a ratio, or a particular relationship between two sides of a right triangle and one of its angles.  So in this picture,

our chosen angle is \(\theta\), so all of our ratios, all of our functions will be found with respect to this angle. The side of the triangle that connects our angle and the right angled corner is known as the 'adjacent' side. In our triangle, that would be side '\(a\)'. The other side (or leg) of the triangle that is connected to the right angled corner is called the 'opposite' side, which in our case would be side '\(b\)'. The last side that does not connect to the right angle at all is called the 'hypotenuse', and is the longest side of any right triangle. In this case, side '\(c\)' is our hypotenuse.  The reason that the hypotenuse has to be the longest side is because of the Pythagorean Theorem which states that for a right triangle, the square of the first leg plus the square of the side of the second leg will equal the square of the side of the hypotenuse. Mathematically, $$a^2 + b^2 = c^2,$$ and so the hypotenuse ('\(c\)') will always be bigger than either leg. You can prove this to yourself with a bit of graphing paper. Although you do not have to do so, I can tell you it is easiest to demonstrate this with a 3-4-5 triangle (a triangle whose legs measure 3 units and 4 units and whose hypotenuse is therefore 5 units).

Now, let's get down to the nitty-gritty definitions. This is where most people start to have problems, because while you can express the trig functions as simple ratios, what  they *are* is a little tricky. Lets start by expressing them as simple ratios, and work backwards to what it all means.

The function \( \sin{(\theta)} \) (said 'sine of theta') gives us the ratio between the 'opposite' side ('\(b\)') and the hypotenuse ('\(c\)')**. Explicitly, $$\sin{(\theta)} = \frac{b}{c}, $$ and thus the mnemonic for the sine function being "SOH": Sine [is] Opposite [over] Hypotenuse.

The function \( \cos{(\theta)} \) (said 'cosine of theta') gives us the ratio between the 'adjacent' side    ('\(a\)') and the hypotenuse ('\(c\)'). Explicitly, $$\cos{(\theta)} = \frac{a}{c}, $$ and thus the mnemonic for the sine function being "CAH": Cosine [is] Adjacent [over] Hypotenuse.

And lastly, the function \( \tan{(\theta)} \) (said 'tangent of theta') gives us the ratio between the 'adjacent' side ('\(a\)') and the 'opposite' side ('\(b\)'). Explicitly, $$\tan{(\theta)} = \frac{b}{a}, $$ and thus the mnemonic for the sine function being "TOA": Tangent [is] Opposite [over] Adjacent.

I should note that there are functions for the opposite ratios (hypotenuse over adjacent, etc) but they are rarely, if ever, used in physics at this level and I honestly don't see them pop up in physics at any level.

That's all well and good, you might say to me through your computer screen, but what is 'the sine of theta'? What if I just have an angle? How do I find the sine of it then?

Simple answer? Calculator or an online tool like Wolfram Alpha. But let's face it, that's a cop out, not an explanation.

The way to find the sine (or cosine or tangent) of any angle is using a thing called a unit circle, illustrated so very nicely below.

Via WikiCommons, Public Domain

A unit circle is simple a circle of radius \(1\). Right triangles are created using the radius of the circle as the hypotenuse of the triangle and drawing a vertical line connecting the x-axis and where the radius meets the circle. If you move the location of the radius, 'sweeping' though the full \(360^{\circ}\) of the circle, the triangle changes accordingly.

Now I want you to imagine that at every increase of \(1^{\circ}\), we measure '\(b\)', the vertical leg or the 'opposite' side. We then plot, or map out, that leg height against the number of degrees in our angle. We do the same for  '\(a\)', the horizontal leg or 'adjacent' side. Since our hypotenuse has a value of \(1\), by plotting out the height and length of the two legs we are plotting the sine and cosine functions, as shown in this nifty graphic***.

Via WikiCommons, Public Domain

This goes on forever. One of the cool, and occasionally frustrating to scientists, thing about the trig functions is that they go on forever. They never slow down, approach any sort of limit. If you could keep tracing the unit circle forever, you would keep producing sine and cosine waves.

So, how does this help us with our triangles back there? Remember that at their hearts, trig functions are ratios. And ratios scale, as any cook can tell you (though in cooking you do reach some practical limitations on the ability to scale). If I want to make a double batch of spaghetti sauce, I don't have to reinvent my recipe. If my one-batch recipe calls for one garlic clove for a pint of pureed tomatoes, I know that for my double batch I need two garlic cloves for \(2\) pints of pureed tomatoes, and I will still get the same tomato sauce. Similarly, if I know the sine of an angle, say \( \sin{(30^{\circ})} = 0.5 \), I am saying that on the unit circle, the vertical leg is half a unit long. So for any right triangle that has a \(30^{\circ}\) angle, the side opposite that angle will be half the length of the hypotenuse!

Let's work through an example of this. Here we have right triangle, with one angle given and one side length.

It's considered good practice, and it becomes much easier to do this when pictures become more complex, to label the picture with symbols and give the value to those symbols elsewhere. It makes the image cleaner, and leads to fewer errors.

So, what can we find out about this triangle? We have chosen our angle (or it was chosen for us) and we have the 'adjacent' side length. We could start by finding the 'opposite' side length using the tangent, which gives us the relationship between the angle and the two legs. Let's call the 'opposite' side '\(y\)'. Then we can say $$\tan{(\theta)} =  \frac{y}{l}$$ and we can solve for \(y\) using just a touch of algebra $$l\times \tan{(\theta)} =  \frac{y}{\cancel{l}} \times \cancel{l} $$ $$l \times \tan{(\theta)} =  y$$ and then to keep with convention we flip the sides of the equation to the thing we are solving for is on the left hand side (frequently abbreviated LHS): $$y =l \times \tan{(\theta)}.  $$ Now we can plug in our values if we like to find the value of '\(y\)'. Since \( \theta= 30^{\circ} \) , \( l = 5\, \text{cm}  =  .05\, \text{m}  \) and  $$y=\left(0.05\,\mathrm{cm}\right)\tan\left(30^{\circ}\right)= 0.03\, \mathrm{m} =3 \,\mathrm{cm} $$ You may be wondering why I converted from centimeters (cm) to meters(m) and then back again. This is again just a 'best practice' thing. It does not make a big difference in situations, like this, were there is only one unit (length), but when we start calculating things like forces, we will be combining different units into other units and it becomes very important to make sure all your units are base units (e.g. meters, seconds) or you can easily end up with an answer a thousand times bigger or smaller than it should be.

So now we have the 'opposite' side of the triangle. We now have several options on how to find the hypotenuse. We could use the sine or the cosine to find the hypotenuse (call it '\(h\)') in much the same way that we used the tangent to find '\(y\)'. Or we can use the Pythagorean Theorem to find it, which is my preferred method. Plugging in our particular terms to the formula we get $$l^2 + y^2 = h^2.  $$ To extract the '\(h\)' value we need to take the square root of both sides: $$ \sqrt{l^2 + y^2} = \sqrt{h^2}$$This is the same as raising both sides to the power of 1/2, so \( (h^2)^{1/2} \) becomes just \( (h) \). So  we find that $$ h = \sqrt{l^2 + y^2} = \sqrt{(.05\, \mathrm{m})^2 + (.03\, \mathrm{m})^2 }  = .05773627...  $$ $$ h \approx 6\, \text{cm} $$ And now we know everything about this triangle!

So, why do we need this? What arcane bit of knowledge am I trying to put in your head? What good does it do to know anything about this *one* triangle? How will this help me in physics?

Good questions.  First of all, I want to emphasize that, while math is beautiful and worth knowing for it's own sake, for a student of physics I find it useful to think of math concepts less as "yet another thing I have to learn" and more like a tool in your tool kit. Just like you need multiple, different tools to attack house repair (a hammer to drive nails, a screwdriver for screws) you need different tools to break down a problem and then build up your answer. Having these basic trig functions in your back pocket gives is like having a drill and driver set for a power drill. You can use it to both break down and rebuild the problem.

Secondly, by knowing about this one triangle, you are now able to attack any right triangle, no matter its shape or orientation in space.

All equally susceptible to attack by trig

And by knowing about one triangle, you allow yourself to know about triangles which can be drawn as extensions of that main triangle. Why exactly you would want to do that will be made more clear in a few weeks when we get into physics, so you will have to take me for my word on that for now.

Lastly, by being able to translate between angles and the sides of a right triangle we have set ourselves up to deal aptly with another very important concept for physics--vectors!

As always, please leave any questions in the comments section!

*In Euclidean, or flat, space. If you drew a triangle on a sphere, it would not have angles totaling \(180^{\circ}\)

**You may notice that I kept 'opposite' in quotation marks, but not hypotenuse. This is because while the hypotenuse is fixed, which side is called the 'opposite' depends on which angle you chose for your reference.

***Here's another graphic that shows a little better the cosine function, but it's not as smooth and it's labels are in German

Basic Physics: Editorial Consortium

The next promised post on trigonometry is in the final polishing stages, but in the meantime I would like a post to mention several people who have graciously agreed to help me in this endeavor to bring the first year of a physics majors schooling in physics to a non-math, non-science types audience.

 I know full well that as a grad student in physics I am a very bad judge of what is and isn't understood or common knowledge. Teaching has helped rein me in enormously, but my students are assumed to have at least basic calculus knowledge. So I anticipated myself having a problem recognizing what needed more explanation, what was over-explained or even patronizing. I don't want to be the detective novel criminal who spells all the easy words wrong and all the hard words right (in reverse, kinda). Dear Husband, my usual editor, is too well versed in math to much use in this particular arena, so I reached out to some other family members, specifically my mother, my sister, and my brother, to help make sure I do this right. I asked them to do this because each of them brings something that I felt I really needed on what I am dubbing my Editorial Consortium.

My mother is in the demographic group, you might say, that always gives me deer-in-the-headlights or horrified looks when I say I do physics and protest it was too hard for them. Though very talented, she has not directed her talents in a STEM field direction. She is, however, the only reason that I can do long multiplication or division and light years ahead of me in mental arithmetic (also cooking, social skills, language, and checkbook balancing). She is also a natural copyeditor of high standards who is not shy of letting me know when I have fallen short of the mark.

My sister, hereafter to be referred to as Sylvia, Historian Extraordinaire, just graduated college with an absurd amount of honors with a major in History and a minor in French, her thesis work (yes, thesis for undergrad) being on Dorothy L. Sayers. She has a good math background, but hasn't used it much, having no call to do calculus as a literary historian. Her one and only basic physics class was the same one in high school that inspired me to physics. She is also representing a group that I want to reach--younger adults--and she would know if a reference is too obscure. She is also incredible at calling me out for being obtuse and/or patronising.

Last, but not least, is my brother, who will start high school in the fall. I included him for three reasons. First of all, he has had all of the math that I claim is required to understand the blog, but has never taken a physics class in his life. He's interested in the sciences, but he is yet untainted by misconception and bad teaching (other than my own). Second of all, it turns out he inherited Mother's copy editing skills and is very good at noting my inconsistent use of single and double quotation marks. Thirdly, I'm curious if the explanations are clear enough for younger persons who might be interested, but don't have much of a background. The flip side of my mother, so to speak.

They have all agreed to read, edit and comment every post that I write in this series. Between them all I think there is a fair shot that I will do what I set out to do. But I won't know if I am actually succeeding unless you, the reader, let's me know. You are the other part of this Editorial Consortium. If something is not clear, if I mess something up or forget something or just plain gloss over with the hated "the reader can obviously see", let me know! There is a comments link below each post. I'd love your feedback.

Basic Physics Part 0, Section 0: Algebra

[This post is the first in a series intending to teach basic physics concepts in a blog format.]

As I mentioned in my introductory post, math is the language of physics. Physics cannot realistically be understood or done without math. While advanced physics requires some advanced math, basic first-year type physics requires some relatively basic math and math concepts. The first math topic that I want to cover is one that everyone who graduated high school should have covered at some point: algebra. 

Algebra has a kind of strange reputation. Among STEM people, it's the boring math that you needed to do to do the REAL math, or at least the non-boring stuff. It carries the same emotional connotations as diagramming sentences. Among non-STEM people, it's the boring math that they forced you to do and you never ever used again.

Until I really got into teaching and my research, I was mostly of the opinion that algebra was best left to machines. It was tedious and beneath my dignity to spend hours and pages rearranging symbols. When I started teaching, I began to understand the subtle power of algebra to make or break a solution. When I finally started to understand my research, I saw not only its power, but its beauty. Algebra is a tool that allows order to arise out of chaos.

To do the kind of physics this series is going to look at, you really only need 2 major algebra skills: the FOIL method, and some equation manipulation skills. The quadratic equation can come in handy, but that is one time that I am ok using a math program for because it doesn't pop up as frequently.

But before we get to that, I think some terminology definition is in order. When I speak of a "variable" I am referring to a symbol that can take on any value on the real number line (i.e., any where between negative infinity and infinity) within the confines of the equation and/or is the quantity we are solving for. A coefficient is a symbol that has a fixed value for that particular problem. Most physics texts I've seen and used have the convention that any letter from p-z can be used as a variable, while letters a-m are used as coefficients. The letter 'n' is a special case because it is typically used for integer numbers only. The letter 'd' is sometimes used as a variable because it's just so convenient to use it to stand for 'distance'. The letter 'o' is never used, because in handwritten notes it can all too easily look like a zero. A constant, for our purposes, is a symbol that has a fixed value that does not change from problem to problem. For example, \( \pi = 3.14159...\) no matter what problem we are doing. A 'term' is a catchall, just denoting that a symbol stands for something, without specifying type.

Now, on to algebra!

FOIL Method

The FOIL method (First Outside Inside Last) is one of the first things I was taught in algebra class, way back in 7th grade. It's basically a method for multiplying mathematical expressions together in a way that doesn't let you double multiply or leave something out.  If you are multiplying just two terms together, say \(a\) and \(b\), its easy to know when you got it all.
$$ (a)(b)  = ab$$
But what if you don't have just two items, but two expressions, \( (a+b) \) and \( (c+d) \) ? FOILing the two expressions makes sure you do all available multiplications without double counting.  You multiply the first terms from each expression, here \( a \) and \( c \), then the outside ones, here \( a\) and \( d\). Then you do the inner ones, \( b\) and \( c\), and the last ones from each expression, \( b\) and \( d\). Thus
$$ (a+b)(c+d) = ac + ad + bc + bd $$
This method can be logically extended to cover expressions with more than two terms, with the corresponding result being proportionately longer.

When I first learned this, it seemed incredibly useless. Why on earth would I need such a simple method? The answer is 'everywhere in physics'. From the simplest two-body problems to the most complex problems I've worked on for research, FOILing turns up again and again and again. Becoming not just  proficient, but a master at this technique has been crucial to my work. It is something that my students consistently underestimate, to their detriment, every semester.

Manipulating Equations

This isn't so much a single method as the Rules of Engagement for math. Equations are pretty flexible, but there are some rules. The underlying principle to these rules is that you have to do the same thing to each side of the equation, and you have to do it to everything on each side. For example, lets say we have this equation $$ 5 x + 2 y = 6, $$ and we want to solve for \( y\). We can start by subtracting \( 5x\) from each side like this $$ 5x + 2y - 5x = 6 - 5x$$ where you can see we have explicitly taken \( 5x\) from each side and thus have not changed the equation. By adding the same thing to both sides, we have effectively added zero, just like if you add a one pound weight to either side of a balance scale, it won't change position.  So now we have the equation $$ 2y = 6 - 5x,$$ but we still have not completely isolated \( y \). So now we have to divide both sides by 2, which is the coefficient of the variable \( y\). $$ \frac{2y}{2} = \frac{6 - 5x}{2}$$ Again, it is important to note that we have done exactly the same thing to both sides of the equation and in the case of division or multiplication we have applied that change to every term. $$y = \frac{6}{2} - \frac{5 x}{2}$$ $$ y = 3- \frac{5}{2}x$$ is the correct solution in this case. Do not, I repeat, DO NOT make the mistake I see so often, which is to only apply the division to one (usually convenient) term. The following 'solution' is wrong for this problem: \( y = 3- 5x\)

In certain cases, this also involves remembering the Order of Operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. The Multiplication/Division and Addition/Subtraction orders are less critical, since they are just two sides of the same coin. Division is the same as multiplying by a fraction, subtraction is the same as adding a negative number. But the parentheses-> exponents->multiplication/division->addition/subtraction ordering is inviolate. It is impossible, outside of sheer fluke, to get a correct answer if you do not abide by this rule.

And that's the basics of algebra that you may have forgotten  (accidentally or on purpose) that you need for physics, other than the kind that you, honestly, do intuitively. Next week, we'll cover some basic trig[onometry] that everyone should know.

Basic Physics: Introduction

I have done a few introductory physics posts in the past, but I have never been happy enough with them to continue them as a series. After writing my post yesterday on the problems with science communication, I have thought more deeply about why I wasn't happy with them. I've decided that I didn't like them because they weren't able to adequately explain what I wanted to convey. This is mostly my limitations, but also because I hadn't set in my mind who my audience was, and because I had never done posts that explained what I understood to be background material to the topics.

So, I am going to try again in a more cohesive manner. By which I mean that I am going to do a year-long series of blog posts that roughly coincide with what physics majors (and engineers, and other interested parties) learn in their first year, covering basic classical mechanics, electrostatics, magnetostatics and circuitry.

Why? Why would I do this? First of all, I think it will be good practice for when I (fingers crossed) become a professor. I have mostly been working off of the curricula and methods of other professors--I would like to find my own. Secondly, I want to show that physics isn't "hard". Nearly every time I mention that I do physics for a living, I hear the same story--the person I'm talking to either took it in high school and did miserably, thus putting them off the whole thing, or they never took it because they were never any good at that brainy stuff. I want to write a series that, even if it doesn't make physics converts, gives people the confidence that they understand key physics concepts, and maybe understand why physics geeks geek out over physics.

To that end, I am going to be writing for people who have never taken a physics class, but who have some basic math background. I am going to assume a high school level of education, though even that seems to be a somewhat variable standard anymore. I am not going to hold back on "vocabulary words" as my students put it, because it's a blog and you have instant access to a dictionary, but I will explain any technical terms or words that are used in a manner different from their colloquial usage.

To start out with, I am going to do a series of crash-course algebra, trig, vectors, and calculus, so that we have a common math starting point, and a kind of reference guide. Math is the language of physics and it is very hard to really understand what physics is without being able to speak about it using math. Otherwise you kind of end up with something more like Aristotle's physics than Newton's, simply because it is very difficult to describe it using just words.

Then there will be a longish series on basic classical mechanics, which is the one physics topic most people can grasp with at least a bit of intuition. We have all thrown balls, used a seesaw, and spun in an office chair. It will cover more or less the same material you would see in first semester physics class.

The last part will be on what I have taught for 5 semesters now--introductory electromagnetism, or Intro E&M. This will cover basic electrostatic forces/fields, currents, simple circuitry and basic magnetism.

I'm going to try to stick to a schedule of posting one a week, again, roughly like it would be in a classroom setting. This will also give me enough time (hopefully) to properly proof read them and weed out errors.

So, without further ado, on to Part 0, Section 0: Algebra!