You may not agree that the topic(s) of this question belong in an introductory calculus-based physics course, but I’m going to pretend they do for the duration of this post. Gradient, divergence, and curl are broached in Matter & Interactions within the context of electromagnetic fields. Actually, gradient appears in the mechanics portion of the course.
One problem with these three concepts, especially divergence and curl, is the distinction between their actual definitions and how they are calculuated. The former are rarely, if ever, seen at the introductory level and usually first appear in upper level courses. However, some authors [cite examples here] replace the physical definitions with the mathematical symbols invented by Heaviside and Gibbs to represent the calculation of these quantities. In other words, the divergence of is frequently defined as and the curl of is frequently defined as . These should be treated as nothing more than symbols representing their respective physical quantities and should not be taken as equations for calculation. If one insists on keeping this notation, then the dot and cross should at least be kept with the nabla symbol so that represents divergence and represents curl. Either way, these are operators that operate on vectors and their symbols should reflect that concept and should be interpreted as such and not as a recipe for calculation. This book by Tai was extremely helpful in getting this point across to me.
Gradient has its own unique problem in that some sources claim that one can only take the gradient of a scalar, which is patently false. One can indeed take the gradient of, for example, a gradient but the object one gets back is not a vector. If we adopt a unified approach to vector algebra and vector calculus we find that there are patterns associating the operand and the result when using these vector opators. For example, operating on a vector with doesn’t produce a vector; it produces a second rank tensor. This is one reason I would love to find a way to bring this approach into the introductory course. So many things would be unified.
But now, on to the questions I want to ask here
(a) Write a conceptual definition of gradient in words.
(b) Write a mathematical definition of gradient that does not depend on any particular coordinate system. You must not use the nabla symbol.
(c) Write a conceptual definition of divergence in words.
(d) Write a mathematical definition of divergence that does not depend on any particular coordinate system. You must not use the nabla symbol.
(e) Write a conceptual definition of curl in words.
(f) Write a mathematical definition of curl that does not depend on any particular coordinate system. You must not use the nabla symbol.
(Note: I need to revisit this post in the future to make sure the notion of applying gradient to a vector quantity can be handled in the coordinate free way I have in mind. My intuition is that it can be, but I need to work out some details. )
In this post, I will address the first question on the list in the previous post. What exactly does it mean for something to be a vector?
In almost every introductory physics course, vectors are introduced as “quantities having magnitude and direction” and are eventually equated to graphical arrows. A vector is neither of these, but is something far more sophisticated. Remember that I’m coming at this as a physicist, not a pure mathematician. I will probably get more than a few things incorrect. Let me know if/when that happens. Let me see if I can present this at a level suitable for an introductory calculus-based physics course. Imagine you walk into class on the first day and start talking. Here goes.
We live in a Universe with has measureable properties, and containing physical entities that also have measureable properties. A lot of physics consists of attempting to measure, and thus quantify, these properties (experiment). More important to some physicists is describing these properties mathematically and making predictions about them (theory) rather than attempting to measure them. We can invent mathematical objects to represent these measureable properties. The word represent is important here, because the mathematical object representing an entity is not the same thing as the entity itself. These mathematical objects themselves have properties, and these properties allows us to manipulate these objects so as to use them to make predictions about Nature.
The properties possesed by the mathematical objects we use to describe Nature collectively form something with a very strange name: a vector space. That sounds very technical and complicated. It is indeed a very technical term because it means something profound. However, as I will try to convince you now, it is not necessarily complicated at all. Let me attempt to show you.
I will use bold symbols (e.g. etc.) to represent mathematical objects with the properties that collectively form a vector space. These mathematical objects have a generic name: vectors. Yes, that’s their name. Note that there is nothing at all here to do with arrows or anything else really. Vectors are nothing more than mathematical objects with properties that let us model and make predictions about the properties of the Universe we observe and try to understand in Nature. Be careful to understand that there are two sets of properties here, those of the Universe and its inhabitant entities, and those of the mathematical objects we use to represent those things. I’m not saying this is the best way to describe this, but it’s a start.
I will use italic symbols (e.g. etc.) to represent ordinary numbers you are already familiar with. Technically, the are real numbers and every math course you have ever taken has used them whether or not you knew they had a name.
In a vector space, there are two and only two mathematical operations defined: addition and scalar multiplication. That’s all there is. You’ve known how to add and multiply for a long time, and there is nothing new to see here. Consider addition. So, in any vector space, you can take any two vectors, and add them and get a third vector. It’s just that simple, however, I must warn you that there is indeed something deeper going on here but there’s no need to bring it up yet because it’s a geometry issue. We will get to it soon enough. So for now, addition is the same addition you’ve already become familiar with. Oh, here’s a new technical gem for you. The simple fact that adding two vectors gives you a third vector is a property that we use to say that the vector space is closed under addition. All that means is that when you add two vectors, you get a vector. All three inhabit the vector space. That’s simple to understand. You cant add two vectors and get, say, a real number. You must always get a vector. That’s very simple. Let’s say it more mathematically.
- In a vector space, addition is a closed operation. If and are vectors then is also a vector.
Now consider scalar multiplication. You’ve known how to multiply real numbers for a long time, and again, there isn’t much new to see here. Multiplying a scalar and a vector gives another vector. We will explore the goemetric implication of this later. Like vector addition, scalar multiplication is a closed operation.
- In a vector space, scalar multiplication is a closed operation. If is a vector and is a scalar, then is also a vector.
Here is a list of remaining properties that define a vector space.
- In a vector space, addition is commutative, meaning that the order of the vectors being added doesn’t matter.
- In a vector space, addition is associative, meaning vectors can be grouped in any way as long as the order isn’t changed.
- In a vector space, scalar multiplication is associative. The things you’re multiplying can be grouped differently as long as their order isn’t changed.You get the same vector either way. Cool!
- In a vector space, when you have the sum of two scalars multiplying a vector, the thing you get back is the sum of each scalar multiplying that vector.
- In a vector space, scalar multiplication is distributive over vector addition. Some authors equivalently say that vector addition is linear. Both of these mean the same thing, but I think the second way of saying it is more important, and I will try to show why later. When you have a scalar multiplying the sum of two vectors, the vector you get back is the sum of that scalar multiplying each vector separately.
- In a vector space, there is a multiplicative identity element such that multiplying it by any vector you get the same vector back. This effectively defines a unity element, commonly called 1 (one). This is important because sometimes we can exploit what I like to call a “sneaky 1” to help manipulate a mathematical expression. More on that when we need it.
- In a vector space, there is an additive identity element such that adding it to any vector gives that same vector back as the sum. This is effectively a definition of a zero vector.Seeing zero written this way (as a vector) may seem strange, but you will get used to it.
- In a vector space, there is a member of the vector space called an inverse element such that adding it to any vector gives the identity element (zero element). For any vector we have a vector such that the two sum to zero. Do not think of the sign as subtraction. Think of it as merely a symbol that turns the vector in to its additive inverse.
We’re done. That’s it. These properties collectively and operationally define a vector space that is inhabited by mathematical objects called vectors. These properties also define the things we can do to manipulate vectors. Note there is no mention of subtraction, and there is no mention of division. There is vector addition and scalar multiplication. That’s all there is. This is really simple! Also note there is no mention of magnitude, direction, arrows, components, dot products, or cross products. If you don’t know what those three terms mean don’t worry. We will define them later.
Let me now convince you that you have dealt with vector spaces and vectors for many years and didn’t realize it. Consider the real numbers (that’s all positive numbers, negative numbers, and zero regardless of whether they’re rational or not, and regardless of whether they’re integers or not). Do they meet each and every one of the properties above? To convince yourself that they do, go through them one by one. Does adding two real numbers give a real number? Yes (3.2 + 5.9 = 9.1). Does adding 0 to 5 give 5? Yes. Does adding 6 to -6 give 0? Yes. You can do the rest. Therefore, I claim that without knowing it, you have been using vector spaces and vectors all along!
Now, let me ask you a new question. Consider only the natural numbers. Recall that these numbers are the ones you use for counting and you’ve probably been using them longer than you’ve been using real numbers! Do the natural numbers (counting numbers) form a vector space with each number being a vector? I will tell you that the answer is no, they do not, but I don’t want you to take my word for it. Go through each of the above properties one by one using counting numbers and see if you can convince yourself that these number do not inhabit a vector space.
This is a physics class, so let’s get more physicsy. In physics, as in all science, we use a system of units called the SI System. All scientists know about this system of units, but some subdisciplines (e.g. astrophysics) don’t use them yet. I hope this changes because it will make many things simpler, but I digress. The SI System consists of seven independent fundamental units that represent seven fundamental quantities: mass, length (I prefer spatial displacement), time (I prefer temporal displacement), thermodynamic temperature, amount, luminous intensity, and electric current. All physically measureable properties in our Universe can be expressed in various combinations of these seven fundamental quantities and their units. Your question is: Do these seven fundamental form a vector space? What a weird question! Still, it’s one you can address by, again, working your way through the defining properties of a vector space given above. See what you can come up with.
This may seem a very strange way to begin introductory physics, and it is! It’s strange, but I hope it will help get you to a place where your understanding is deeper than it would be had we begun in a traditional way. Accept the strangeness and uncomfortableness you feel right now, and then let it go. There’s much learning to be done, and it starts here.
This is the first in a series of posts in which I want to share some hopefully interesting things about mathematical descriptions of rotational motion. This series was inspired by a talk given at the 2015 winter AAPT meeting in San Diego. The author claimed to have found a way to represent angular displacement as a vector (true, such an expression exists and is not widely used) and that angular displacements commute (false, in general they do not except when infinitesimal). The same author presented an updated poster on this topic at the recent winter meeting in Atlanta. In researching the arguments presented in these two talks, following up on the references therein, and in searching the undergraduate and graduate physics and mathematics teaching literature on descriptions of angular quantities, I stumbled onto some of the most interesting topics I’ve ever encountered. As you may have already guessed, I want to find ways of bringing these gems of understanding into the introductory courses so students won’t be so mystified when then encounter the in upper level courses. By the way, the papers from these talks aren’t availble online; I only have paper copies and I do not have the author’s permission to distribute them.
I am sure most of this will be trivial for many readers, so apoligies in advance. Even though I too studied out of Goldstein in grad school, it was not the case that all my existing conceptual mysteries were solved. As always, I tend to frame things from the point of view of that introductory physics student for whom we want to provide an unparalleled physics experience. I don’t want that student to ever say, “Well that was never pointed out to me in intro physics.” I want that student’s conceptual foundation to be better than mine was when I was that student.
In this initial post, I will list as many of the questions I can think of that arose as I researched this topic. I will not answer any of them in this post, but will attempt to do that in subsequent posts. I will put the questions into some preliminary order, but I can’t guarantee that order won’t change later. Some questions may change to more accurately reflect what I’m trying to explain.
- What does it mean to be a vector?
- What do vector dot products and vector cross products mean geometrically?
- What is the physical significance of the double cross product (aka triple cross product)?
- Is there a coordinate free expression for the total time derivative of a vector?
- Is there a coordinate free expression for the time derivative of a unit vector (a direction)?
- Can angular velocity be described as a vector?
- Can angular displacement be described as a vector?
- If work is calculated as the dot product of two vectors, then when calculating rotational work how can angular displacement not be a vector?
- If angular velocity is a vector, shouldn’t its integral also be a vector and not a scalar?
- Why does translational displacement commute?
- How, if at all, are translation and rotation (revolution?) related?
- Why do infinitesimal angular displacements commute?
- Why do finite angular displacements not commute?
- What is the distinction between rotation and orientation?
- Is angular velocity the derivative of a rotation?
- So then what is angular velocity the derivative of anyway?
- Can angular velocity be integrated to get angular displacement?
- Can these ideas be brought into the introductory calculus-based or algebra-based physics courses?
I think that’s all, at least for now. I don’t claim this list to be comprehensive. The number of questions isn’t significant either. Let’s see where this goes.
This week was a very short week consisting of only two days. We met as usual on Monday, but Tuesday was a “flip day” and ran as a Friday. This class doesn’t meet on Fridays so we only had one day this week, and we devoted it to tying up loose ends from chapter 17.
Next week, barring losing days to winter weather as I sit here and watch the forecast deteriorate, we will hit circuits the M&I way!
In an interesting development, I was informed by my coworker (a PhD physicist) that our department chair had approached him Tuesday morning to ask if he would like to take my calculus-based physics courses from me next year on the grounds that I’m not rigorous enough. Needless to say, I was shocked becuase the chair had not mentioned this to me and indeed has not spoken to me about it at all. Had my coworker not told me I would not have known.
My chair, a PhD chemist, seems to think that because M&I emphasizes computation over traditional labs, and that what labs we do are not as rigorous as chemistry labs, either M&I or I or perhaps both are not appropriate for our students and indeed may be causing them to be ill prepared for transfer to universities. Of course this is all nonsense, but my chair actually said to my face that she knows more about computation, theory, and experiment than I do and that labs must be done the “chemistry way” or they’re not valid. If this weren’t so disgustingly true, it would be mildly funny. It’s not funny. It’s true.
I don’t know what I’m going to do, but it’s clear both M&I and I are probably on our way out at my current instituion. My colleague (who by the way has no interest in teaching calculus-based physics) and I are both exploring numerous options, including leaving for another instiution.
This week was devoted entirely to programming for chapter 17 on magnetic fields. At least two students had difficulties with lists, which was surprising they’d used them in a previous chapter. It was like they’d never seen them before. Must be something in the water.
Next week is basically a waste of time becuase thee are only two class days, and only one for which this class will meet. Monday will be a normal day but Tuesday will run as a Friday. Strange? It’s an artifact of the fantasy world I inhabit during the week.
Yesterday in my first semester astronomy class, I did something I’d previously threatened to do. I walked in, tossed my personal checkbook onto the floor in the middle of the room, and told students to write checks for any amount they felt appropriate in exchange for them becoming more engaged both in and out of the classroom. I promised I would sign all the checks and they could cash them. That’s right, I openly bribed them.
I did this for two reasons. The first is I’m quite literally at my wit’s end on how to motivate students. Honestly, I don’t feel that I should have to go out of my way to motivate college students because they’re supposedly already sufficiently motivated to make it this far. However, most of my students (not all, but certainly most) arrive underprepared for college work and I have to pretend that I can fix that when I can’t. Still, I feel personally obligated to at least try to figure out why most of my students aren’t motivated to learn and only motivated to get a grade, and even then not a high grade. One student in another class yesterday said “Cs make degrees.” Yeah…see I can’t fix that. Fail. I’ve tried everything else I can think of to install motivation. I’ve been patient with underprepared students and tried to bring them up to speed. I’ve given them published papers describing findings that show the approach we take has distinct advantages over traditional approaches. At some point, students have to consciously decide to engage and learn and I just don’t see how I can force them to do that.
The second reason is I hope to jump start a discussion about motivation in general. To my surprise, this worked! One student, one of the very few who actually engages in the class and sees the big picture, said her motivation is to get an A. Perfectly acceptable. Now I know. She also said that those who don’t engage are cheating themselves in the long run, and she’s right. Another particularly quiet student disclosed that her lack of engagement was becuase the course was not what she expected it to be in terms of structure and organization. She expected a very tradional lecture based course and is still unsure of how to “exist” in a course where there are no lectures and no traditional grading. At last I got something I can understand and potentially work with, and this because the starting point of a thirty minute discussion of the differences in the two instructional models. Oh, and this student also referenced the course’s catalog description and suggested that it be changed to match how the class is run. I told her that our course descriptions are mandated to us by the NCCCS office and the UNC System, which effectively dictated them to us when we changed to a semester system in 1997/1998 and has veto power over not just our course descriptions, but also over course existence (we’re not permitted to create and offer transfer courses that any one campus of the UNC System will not accept despite being under a legislated mandate to do so).
My successful students are successful, I think, becuase they see that I’m trying to get across more than just “astronomy” in my course and I can provide tangible evidence of this in the form of emails from former students who, as undergraduates at large state universities, see the value in that approach now. I’m trying to instill a big picture mindset, one of becoming an independent learner and a critical thinker. Those are much harder to “see” than the basic astronomy and science content in the course. Students too often see the latter as an obstacle to the former despite the former being directly addressed in classroom activities. Science/astronomy is simple at this level, and students create so many excuses to not see it, and we let them get away with it. I don’t know that there are any good answers. If too many students fail the course, administrators threaten my job and entertain frivilous student “complaints” (like the student who failed the course and lodged a complaint that having to write in complete sentences constituted unfairness on my part becuase he hadn’t been required to use complete sentences in high school…that one got an audience with the dean). If too many students earn an A, that too raises suspicion of grade inflation. In a small class with a cap of twenty-four, statistics don’t mean much, especially with student performance varying widely from semester to semester but generally always low with a few exceptions.
In the end, no student actually wrote a check so maybe there’s hope. Some days, hope is all that keeps me going.
This week, I was away at the winter AAPT meeting in Atlanta. Students began working on the experiments from chapter 17, which serve to introduce magnetic fields.
I want to emphasize some really cool things about the mathematical expression for a particle’s magnetic field:
This is really a single particle form of the Biot-Savart law. I’m going to morph it into something really interesting. I’m going to make use of the fact that , which I assert to students will be derived in a later chapter.
THIS IS AMAZING! This demonstrates that this new thing called magnetic field is kind of like a velocity dependent electric field. That’s an oversimplification, but it hints that something deep is revealing itself here. Velocity connotes reference frame, and we see a big hint here that magnetic field depends on one’s reference frame. This is foreshadowing special relativity! We can show something else with one more slight rearrangement.
This means that if we express velocity in fractions of , then the quantity has the same dimensions as and can thus be expressed in the same unit as electric field! This conceptualization allows for some beautiful symmetry to show itself later on when we get to the Maxwell equations. In some ways, electric fields and magnetic fields are interchangeable. Again, this is a hint of some underlying unification of the two, the electromagnetic field tensor, which I’m working hard to find a way to introduce into the introductory course. If students can understand simple Lorentz transformations, then they should be able to understand how the electromagnetic field tensor transforms from one frame to another within the framework of special relativity and we can show some beautiful physics. I realize I’m in the minority when it comes to something like this becuase we tend to think of our students as not being mathematically prepared. I’ve come to realize that perhaps…just perhaps…that is our perception only becuase we aren’t giving them the best mathematical foundations upon which to prepare for physics. Maybe it’s our fault. Maybe.
Anyway, these ruminations are things I want, and hope, students see on their own but all too often I find they, at least in my case, have difficulty even engaging at a minimal level. I struggle with this, and like to think and hope that maybe it’s because they don’t see the beauty. That’s why I nudge them in these new and different directions. Like I said above, it may very well be our fault.
Feedback is welcome as always.