http://sites.science.oregonstate.edu/BridgeBook/ 2020-01-26T16:00:28-08:00 http://sites.science.oregonstate.edu/BridgeBook/ http://sites.science.oregonstate.edu/BridgeBook/lib/images/favicon.ico text/html 2011-07-29T08:54:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/acknowledge?rev=1311954840 This work was supported by NSF grants DUE--9653250, DUE--0088901, and DUE--0231032, by the MAA Professional Enhancement Program (PREP), and by the Oregon Collaborative for Excellence in the Preparation of Teachers (OCEPT). Our home institution, Oregon State University, encouraged us in this activity; we thank in particular the Departments of Mathematics and Physics, and the University Honors College. text/html 2011-07-29T17:27:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/activities?rev=1311985620 This part of the guide discusses the individual group activities, in the order in which we would use them ourselves. In a large lecture environment with recitations, we typically cover one activity each week, including weeks with exams. In smaller classes, we introduce the activities as needed, averaging one per week. text/html 2011-07-29T12:06:55-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/anecdotes?rev=1311966415 Do students use geometric reasoning? Judge for yourself. Vectors The first group activity, discussed in more detail in Part II, asks the students to draw a particular vector, expressed with respect to 2 different orthonormal bases. We call two students to the board to present their answers, one for each basis, but on the same diagram. Frequently, the second student uses a different scaling than the first; the resulting vectors differ. On several occasions, it has taken several attempts … text/html 2011-07-29T08:54:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/authors?rev=1311954840 Tevian Dray received his BS in Mathematics from MIT in 1976, his PhD in Mathematics from Berkeley in 1981, spent several years as a physics postdoc, and is now Professor of Mathematics at Oregon State University. He considers himself a mathematician, but isn't sure. (Neither is his department.) Most of his research has involved general relativity. text/html 2011-07-29T09:46:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/beast?rev=1311957960 Students need to be able to identify the objects they work with. Is it a vector or a scalar? Finite or infinitesimal? What units does it have? This ability is essential when setting up a problem, especially word problems. It is also a useful technique for checking whether the answer makes sense. Are both sides vectors? Infinitesimals? Do the units match? text/html 2011-07-29T12:07:02-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/bridge?rev=1311966422 The Vector Calculus Bridge project~[ 1 ] originated in our own failure to communicate with each other despite years of collaboration in mathematical physics. We regularly taught vector calculus in our separate departments, with many students taking both courses. It was a shock to both of us to realize how little overlap there was between the two courses even though the syllabi were remarkably similar. text/html 2011-07-31T08:44:12-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/changes?rev=1312127052 Figure 3.1: The infinitesimal displacement vector $d\rr$ along a curve, shown in an ``infinite magnifying glass''. In this and subsequent figures, artistic license has been taken in the overall scale and the location of the origin in order to make a pedagogical point. text/html 2011-07-30T13:09:32-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/classroom?rev=1312056572 We recommend explicitly discussing the process used to decide what sort of integral needs to be done. In the above example it goes something like this: \begin{itemize}\item ``I want to add up $\lambda$, which is a scalar, so I need to multiply it by a scalar, $ds$, and do a scalar line integral.'' \item ``The linear density is $\lambda$, so the (small) amount on a small piece of the curve is $\lambda$ times the length, $ds$, of the small piece.'' \item ``The density is $\lambda$, with dimensi… text/html 2012-09-23T20:45:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/coordinates?rev=1348458300 It is important to realize that $d\rr$ and $ds$ are defined geometrically, not by~($\ref{drdef}$) and~($\ref{dsdef}$). To emphasize this coordinate-independent nature of $d\rr$, it is useful to study $d\rr$ in another coordinate system, such as polar coordinates ($r$,$\phi$) in the plane. It is also natural to introduce basis vectors $\{\rhat,\phat\}$ adapted to these coordinates, with $\rhat$ being the unit vector in the radial direction, and $\phat$ being the unit vector in the direction o… text/html 2011-07-31T10:07:52-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/cross?rev=1312132072 We strongly discourage teaching, or even reviewing, the dot and cross products at the same time --- students tend to get them mixed up! The dot product is needed right away, in order to discuss line integrals, whereas the cross product isn't really needed until one discusses surface integrals. text/html 2011-07-30T21:12:52-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/curl?rev=1312085572 At any point $P$, we define the curl of a vector field $\FF$, written $\grad\times\FF$, to be the vector which gives the circulation of $\FF$ per unit area around an arbitrary small loop around $P$. That is, given any unit vector $\nn$ at $P$, $(\grad\times\FF)\cdot\nn$ should be the circulation of $\FF$ per unit area at $P$ around a small loop with axis $\nn$. In order to compute the curl, we must compute the circulation around such loops. text/html 2015-08-29T17:59:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/curvilinear?rev=1440896340 Rectangular Coordinates The arbitrary infinitesimal displacement vector in Cartesian coordinates is: $$d\rr=dx\,\xhat + dy\,\yhat +dz\,\zhat$$ Given the cube shown below, find $d\rr$ on each of the three paths, leading from $a$ to $b$. Path 1: $d\rr=$ text/html 2011-07-29T17:55:27-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/diffint?rev=1311987327 First-year calculus courses often emphasize that derivatives are slopes, and integrals are areas. These interpretations are only sometimes correct, and in fact can cause problems for students in multivariable calculus. We prefer to make differentials fundamental. text/html 2011-07-29T17:51:52-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/diffmany?rev=1311987112 What about functions of several variables? Suppose that $f=f(x,y,z)$. Then small changes in, say, $x$ will produce small changes in $f$, and similarly for small changes in $y$ or $z$. But changing one variable while holding the others fixed is precisely what partial differentiation is all about, so it should come as no surprise that the differential of $f$ is now \begin{equation} df = \Partial{f}{x}\,dx + \Partial{f}{y}\,dy + \Partial{f}{z}\,dz \end{equation} text/html 2011-07-29T17:51:15-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/diffone?rev=1311987075 Given a function $f=f(x)$, the differential of $f$ is often defined to be \begin{equation} df = {df\over dx} \,dx \end{equation} We can think of $dx$ as an infinitesimal change in the $x$ direction, and $df$ as the corresponding infinitesimal change in $f$. Then the above equation shows how these infinitesimal changes are related. That's exactly what a derivative is, a ratio of infinitesimal changes! text/html 2011-07-30T12:36:59-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/dim?rev=1312054619 The traditional approach to multivariable calculus emphasizes functions of two variables as a way to learn about functions of several variables. There is even an excellent text~[ 11 ], which works exclusively in two dimensions. Yet the real world is made up of functions of three variables. text/html 2011-07-30T20:24:38-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/div?rev=1312082678 At any point $P$, we define the divergence of a vector field $\FF$, written $\grad\cdot\FF$, to be the flux of $\FF$ per unit volume leaving a small box around $P$. In order to compute the divergence, we must compute this flux. Consider a rectangular box whose sides are parallel to the coordinate planes. What is the flux of $\FF$ out of this box? Consider first the vertical contribution, namely the flux up through the top plus the flux down through the bottom. These two sides each have ar… text/html 2011-07-30T12:08:42-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/divgradcurl?rev=1312052922 Rectangular coordinates \begin{eqnarray*} d\rr &=& dx\,\ii + dy\,\jj + dz\,\kk \\ \FF &=& F_x\,\ii + F_y\,\jj + F_z\,\kk \end{eqnarray*} \begin{eqnarray*} \grad f &=& \Pf{x}\,\ii + \Pf{y}\,\jj + \Pf{z}\,\kk \\ \grad\cdot\FF &=& \PF{x}{x} + \PF{y}{y} + \PF{z}{z} \\ \grad\times\FF &=& \CF{z}{y}\ii + \CF{x}{z}\jj + \CF{y}{x}\kk \end{eqnarray*} text/html 2011-07-31T09:59:19-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/dot?rev=1312131559 Most students first learn the algebraic formula for the dot product in rectangular coordinates, and only then are shown the geometric interpretation. We believe it should be done in the other order. Students tend to remember best the first definition they use; this should not be an algebraic formula devoid of context. The geometric definition is coordinate independent, and therefore conveys invariant properties of the dot product, not just a formula for calculating it. text/html 2011-07-29T19:38:20-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/elements?rev=1311993500 Below is a link to a worksheet students can use to construct the vector differential $d\rr$ in both cylindrical and spherical coordinates. We have used this worksheet successfully both as an in-class activity and for homework. * Worksheet: Calculating Line Elements text/html 2011-07-30T11:53:16-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/formulas?rev=1312051996 Below is a link to a page that summarizes the formulas for the gradient, divergence, and curl in rectangular, cylindrical, and spherical coordinates, as well as giving appropriate expressions for $d\rr$. * Formula Sheet: Div, Grad, Curl text/html 2011-07-31T09:55:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/geometry?rev=1312131300 The computations in this chapter, while geometric in origin, are still somewhat tedious. How then can one emphasize the geometry to students? Technology can be used to explore the properties of different vector fields. One useful tool is the Vector Field Analyzer, a Java applet developed by Matthias Kawski~[ 9 ], which makes a great classroom demo, and which can be made available for students to use outside of class. text/html 2015-08-27T18:03:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/gradient?rev=1440723780 Take another look at the formula for the differential of a function of several variables, namely \begin{equation} df = \Partial{f}{x}\,dx + \Partial{f}{y}\,dy + \Partial{f}{z}\,dz \end{equation} Each term is a product of two factors, labeled by $x$, $y$, and $z$. This looks like a dot product! Separating out the pieces, we have \begin{equation} df = \left( \Partial{f}{x}\,\ii + \Partial{f}{y}\,\jj + \Partial{f}{z}\,\kk \right) \cdot (dx\,\ii + dy\,\jj + dz\,\kk) \end{equation} The last fact… text/html 2011-07-30T21:05:30-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/gram?rev=1312085130 As discussed in (ss)9.1 in Chapter 9, the component of a vector $\vv$ along a unit vector is simply the dot product of the two vectors, and if the vector projection is desired, it points in the direction of the unit vector. To project $\vv$ along a vector $\ww$ which isn't a unit vector, one constructs the unit vector $\ww\over|\ww|$ in the direction of $\ww$, and proceeds as before. Thus, the (scalar) component of $\vv$ in the direction of $\ww$ is \begin{equation} \hbox{component of $\vv$ … text/html 2011-07-30T12:36:02-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/graphs?rev=1312054562 Many students view a graph as the fundamental representation of a function. But you can only graph functions of one or two variables! While graphing techniques have their place, we believe it is essential to place at least as much emphasis on techniques which do generalize to functions of three variables, such as the use of contour diagrams and color. text/html 2011-07-29T09:52:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/green?rev=1311958320 Why teach Green's Theorem? The best answer I have heard to this question is that one might as well teach Green's Theorem since there's not going to be time to cover Stokes' Theorem! Yes, Green's Theorem is important in its own right, it has for instance applications to the theory of complex variables. But Green's Theorem is Stokes' Theorem; it takes very little effort to get from one to the other. text/html 2011-07-30T20:30:06-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/high?rev=1312083006 Two of the most fundamental examples in electromagnetism are the magnetic field around a wire and the electric field of a point charge. We consider each in turn. The magnetic field along an infinitely long straight wire along the $z$ axis, carrying uniform current $I$, is given by \begin{equation} \BB = \frac{\mu_0I}{2\pi} \frac{\phat}{r} = \frac{\mu_0I}{2\pi} \frac{x\,\jj-y\,\ii}{x^2+y^2} \end{equation} where $\mu_0$ and $I$ are constant. Note that the first expression clearly indicates … text/html 2011-07-30T14:41:57-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/history?rev=1312062117 This instructor's guide was originally drafted in 2002 as part of the Vector Calculus Bridge Project. From 2003--2005, it was used in Bridge Project faculty workshops, during which time it was slightly revised. Most of the contents were later included in the Paradigms in Physics website, and/or in the Bridge Project textbook. text/html 2011-07-28T17:39:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/identities?rev=1311899940 We first review some well-known properties of conservative vector fields. Recall the master formula \begin{equation} df = \grad f \cdot d\rr \end{equation} leading to the fundamental theorem for gradient \begin{equation} \Lint \grad{f} \cdot d\rr = f \Big|_A^B \end{equation} This says that the work done by a gradient vector field is independent of the path (with given endpoints), that is \begin{equation} \Int_{C_1} \grad{f} \cdot d\rr = \Int_{C_2} \grad{f} \cdot d\rr \end{equation} so long as $… text/html 2011-07-30T12:32:05-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/implicit?rev=1312054325 Implicit differentiation is easy using differentials --- simply take the differential of each term. For instance, if \begin{equation} x^2+y^2=4 \end{equation} then adding the differentials of each term results in \begin{equation} 2x\,dx + 2y\,dy = 0 \end{equation} This equation could be used to find the slope of the tangent line to this circle at any point, by solving for $dy\over dx$. But there is no need to think about which variables are dependent, and which are independent! We find it eas… text/html 2011-07-29T23:29:31-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/integrals?rev=1312007371 There are two common notations for multiple integrals, one being to use one integral sign for each iterated integral which will ultimately be performed, the other is to use a single integral sign for each integral, since an integral just means ``add things up''. We use the latter notation for surface and volume integrals, but iterated integrals are written out in full. For instance, when finding the flux of $\kk$ upwards through the unit disk, we write \begin{equation} \Int_D \kk\cdot d\AA = \… text/html 2015-08-12T13:57:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/labs?rev=1439413020 The links below will take you to the Instructor's Guide for each activity, located on the Paradigms in Physics project website. A link on that page will take you to the activity itself, but you will need to use the Back button on your browser to return here (unless you open these links in another window. text/html 2011-07-30T20:29:55-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/line?rev=1312082995 The ``use what you know'' philosophy is especially powerful in the context of vector line integrals. Suppose you want to find the work done by the force $\FF=x\,\ii+y\,\jj$ when moving along a given curve $C$, using the formula \begin{equation} W = \Lint \FF\cdot d\rr \end{equation} Curves can be specified in several different ways; let us consider some examples, all of which start at $(1,0)$ and end at $(0,1)$. text/html 2011-07-30T20:45:37-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/low?rev=1312083937 The reader may have the feeling that two quite different languages are being spoken here. The tilted plane was treated in essentially the traditional manner found in calculus textbooks, using rectangular coordinates. While the ``use what you know'' strategy may be somewhat unfamiliar, the basic idea should not be. On the other hand, the examples in the previous section will be quite unfamiliar to most mathematicians, due to their use of adapted basis vectors such as $\rhat$. Mathematicians s… text/html 2011-07-31T09:06:26-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/mmm?rev=1312128386 Figure 12.1a:Potential functions made easy (2-d). Figure 12.1b:Potential functions made easy (3-d). This chapter is adapted from~[ 12 ]. We describe here a variation of the usual procedure for determining whether a vector field is conservative and, if it is, for finding a potential function. We have used this method, which we call the murder mystery method, in our own classes for many years; students love it. text/html 2011-07-30T12:34:24-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/naming?rev=1312054464 Real problems rarely come with labels. An important skill is to give names to things you don't know. Working in groups is a good way to develop this skill --- it's hard to talk about something without giving it a name! A problem illustrating this technique is included in Group Activity~2 (Acceleration), and discussed in Part III. text/html 2011-07-29T16:32:24-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/organizing?rev=1311982344 We find that groups of 3 work the best, with groups of 4 an acceptable compromise; 5 students is too many. We also find it best to assign the groups ourselves, and we do so in such a way that those sitting next to each other wind up in different groups. We also try to avoid groups with 2 men and 1 woman when we can, as this combination often fails to work well. text/html 2011-07-30T20:49:16-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/other?rev=1312084156 Applications in physics and engineering often involve high symmetry, and are therefore expressed in terms of an adapted orthonormal basis. We considered the gradient in (ss)4.2 in Chapter 4; it is also useful to know the expressions for the divergence and curl in such bases. These formulas are usually looked up when needed; they can be found for instance on the inside front cover of some texts, such as Griffiths~[ 6 ], and are reprinted at the end of this chapter. text/html 2011-07-29T09:30:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/parameter?rev=1311957000 What is a curve? What is a surface? We find it productive to ask students to answer these questions. There are of course many possible answers, but the one we like best is that a curve is simply a 1-dimensional set of points, while a surface is a 2-dimensional set of points. In particular, we view parameterizations as a subsidiary concept, which is not part of the definition. text/html 2011-07-30T20:28:38-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/parametric?rev=1312082918 The traditional approach to curves and surfaces involves parameterization, which we have deliberately saved for last. Recall from~(2) of Chapter 3 that for a parametric curve we have \begin{eqnarray*} d\rr = {d\rr\over du} \,du \end{eqnarray*} On a parametric surface described by \begin{equation} \rr=\rr(u,v) \end{equation} it is natural to construct $d\SS$ using curves of constant parameter. But along a curve with $v=\hbox{constant}$, we have \begin{equation} d\rr_1 = \Partial{\rr}{u} \,du … text/html 2011-07-30T17:09:03-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/physmath?rev=1312070943 Mathematicians think differently from other scientists. There is nothing wrong with this, but the differences deserve explicit acknowledgment. We illustrate this by outlining some of the differences between mathematicians and physicists, as this is what we know best. But our experience is that most other scientists are much closer to the physicists' view of mathematics than that of the mathematicians. text/html 2011-07-30T20:55:12-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/polargrad?rev=1312084512 The master formula can be used to derive expressions for the gradient in other coordinate systems. We illustrate the method for polar coordinates. In polar coordinates, we have \begin{equation} df = \Partial{f}{r}\,dr + \Partial{f}{\phi}\,d\phi \end{equation} and of course \begin{eqnarray*} d\rr = dr\,\rhat + r\,d\phi\,\phat \end{eqnarray*} which is~(5) of Chapter 3. Comparing these expressions with~($\ref{Master}$), we see immediately that we must have \begin{equation} \grad f = \Partial{f… text/html 2011-07-30T16:51:13-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/potential?rev=1312069873 Many calculus texts treat conservative vector fields in two and three dimensions as separate problems. In two dimensions, one checks the mixed partial derivatives of the components; in three dimensions, one checks the curl. Some authors attempt to connect these two cases by introducing the ``scalar curl'' in the 2-dimensional case; we feel this only makes things worse. text/html 2011-07-29T08:54:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/preface?rev=1311954840 Each of us has taught vector calculus for many years: one of us (TD) as part of the second year calculus sequence in the mathematics department, and the other (CAM) as part of the third year physics course on mathematics methods. The latter course officially has the former as a prerequisite, although there is considerable overlap in the material covered. Part of the reason the physics course exists is to teach physics majors the ``right'' way to use this material, given that the mathematics dep… text/html 2011-07-29T09:01:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/reasoning?rev=1311955260 A key problem solving skill used by physicists can be described mathematically as geometric reasoning. A picture is worth a thousand equations! Geometric reasoning empowers students to construct correct solutions from simple ideas. Let's begin. text/html 2011-07-29T16:15:18-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/references?rev=1311981318 \begin{enumerate}\item The Vector Calculus Bridge Project, <http://www.math.oregonstate.edu/bridge>. \item The Paradigms in Physics Project, <http://www.physics.orst.edu/paradigms>. \item William McCallum, Deborah Hughes-Hallett, Andrew Gleason, et al., Multivariable Calculus, 3rd edition Wiley, 2001. text/html 2011-07-29T17:32:09-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/roles?rev=1311985929 Small group activities often work best if each person in the group has a particular responsibility or task. One of the most important outcomes of small group activities is that one's own ideas are clarified by discussing concepts. Teachers tend to learn more than students! In addition, during group work one learns to communicate these ideas to others --- a vital skill for the workplace. Small group activities are not a competition to see which group can get done first. If the group moves on… text/html 2011-07-29T09:45:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/size?rev=1311957900 One of the most common errors students make when dealing with differentials is to confuse them with derivatives, writing things like $d(x^2)=2x$, rather than $d(x^2)=2x\,dx$. The easiest way to avoid such errors is to emphasize that both $d(x^2)$ and $dx$ are really, really small, in fact, infinitesimal, whereas derivatives are the ratios of differentials, which can be large. For instance, ${d(x^2)\over dx}=2x$, which can be as large as you like, depending on the value of $x$. An easy check i… text/html 2011-07-31T08:52:29-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/sphere?rev=1312127549 The Problem Nearly everybody uses $r$ and $\theta$ to denote polar coordinates. Most American calculus texts also utilize $\theta$ in spherical coordinates for the angle in the equatorial plane (the azimuth or longitude), $\phi$ for the angle from the positive $z$-axis (the zenith or colatitude), and $\rho$ for the radial coordinate. Virtually all other scientists and engineers --- as well as mathematicians in many other countries --- reverse the roles of $\theta$ and $\phi$ (and use some ot… text/html 2011-07-30T11:06:20-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/spheres?rev=1312049180 Surprisingly, it often turns out to be simpler to solve problems involving spheres by working in cylindrical coordinates. We indicate here one of the reasons for this. The equation of a sphere of radius $a$ in cylindrical coordinates is \begin{equation} r^2 + z^2 = a^2 \end{equation} so that \begin{equation} 2r\,dr + 2z\,dz = 0 \label{spherecyl} \end{equation} Proceeding as for the paraboloid, we take \begin{eqnarray} d\rr_1 &=& r\,d\phi\,\phat \\ d\rr_2 &=& dr\,\rhat + dz\,\zhat ~=~ \left… text/html 2011-07-29T17:41:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/supervising?rev=1311986460 When the instructor asks questions during lecture, somebody is usually able to answer, leaving the impression that everybody knows everything. When the students initiate the questions, almost every conceivable question gets asked, leaving the impression that nobody knows anything! text/html 2011-07-30T10:57:09-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/surface?rev=1312048629 The approach we have set up for line integrals can easily be extended to surface integrals. The key ingredient is to note that, for a given surface $S$, the vector surface element $d\SS$ is just the (appropriately ordered) cross product of the $d\rr$'s computed for (any!)~two non-collinear families of curves lying in the surface. That is \begin{equation} d\SS = \nn\,\dS = d\rr_1 \times d\rr_2 \end{equation} where $\dS$ is the (scalar) surface element of $S$, and $\nn$ is the unit normal vector… text/html 2011-07-30T11:11:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/thm?rev=1312049460 The divergence and curl are traditionally defined by their component representations in rectangular coordinates. The Divergence Theorem and Stokes' Theorem are then statements about properties of the divergence and curl, respectively, which in turn lead to the geometric interpretation of divergence in terms of flux, and curl in terms of circulation. text/html 2011-07-29T17:41:57-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/tricks?rev=1311986517 The best group activities are open-ended, requiring thought even to decide what the problem is. Students should be expected to interpret the words for themselves --- and instructors must realize that the ``right'' answer may depend on how they do so. text/html 2011-07-29T09:45:00-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/units?rev=1311957900 What does the equation $y=x^2$ mean? This depends on whom you ask. To a mathematician, this is simply the equation of a parabola. Yet $x$ and $y$ typically have dimensions of length, in which case the above equation is nonsense; the dimensions don't match! Similar problems arise with expressions like $\sin(x)$, which only makes sense if $x$ is dimensionless. text/html 2011-07-30T20:28:25-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/use?rev=1312082905 Perhaps the key problem-solving strategy we teach our students is to use what they know, rather than trying to apply a set strategy to all problems of a given type. In particular, we discourage students from explicitly parameterizing curves unless they have to. text/html 2011-07-31T08:31:06-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/vecdiff?rev=1312126266 Vector derivatives are now easy. The vector differential $d\rr$ along a curve is the vector between points (along the curve) which are infinitesimally close. But what does this mean? ``Infinitesimal'' simply means to zoom in on the curve so much that it becomes straight. text/html 2011-07-30T13:03:47-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/vecscal?rev=1312056227 The fundamental objects in vector calculus are vector fields, such as the velocity field of a fluid, and scalar fields, such as the density of chocolate on a pretzel. Students must be able to tell which is which. This starts with notation. We never write vectors as pairs or triples of numbers; this notation is reserved for the coordinates of points, a quite different concept. The symbols we use for vectors have arrows on them (to match what we write by hand) as well as being bold-faced (t… text/html 2011-07-29T23:24:24-08:00 http://sites.science.oregonstate.edu/BridgeBook/book/guidecontent/vectors?rev=1312007064 As already noted, we never write vectors as pairs or triples of numbers; this notation is reserved for coordinates, a quite different concept. The symbols we use for vectors have arrows on them (to match what we write by hand) as well as being bold-faced (to match the notation usually used in textbooks). The one exception to this rule is that we put hats on unit vectors, rather than arrows.