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My answer is probably not very useful when teaching in high school. I'll just mention here a few reasons why this definition is in fact a good one, and why it's a good idea to teach this formula at a university level mathematics course. $\newcommand{\u}{\mathbf{u}}$ $\newcommand{\v}{\mathbf{v}}$ $\newcommand{\w}{\mathbf{w}}$ $\newcommand{\R}{\mathbb{R}}$ Let ...

16

What's wrong with this?: $$\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial t} \frac{dt}{dt}$$ with $\frac{dt}{dt}=1$. It's what you would get if you had $f(x,y,z)$, except that $z=t$. You do have to be a bit careful, though. You want to be clear that $f(x,t,t)=xt^2$ ...

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I don't know if this is what you are looking for, but for Green's theorem, when you discuss how you can change the integral $\int 1 dA$ into the line integral $\int x dy - y dx$, you can discuss several real life applications. You can approximate the area of a lake by walking around it and recording your GPS coordinates $(x_i,y_i)$, then adding up $\Sigma (... 14 A place that Lagrange Multipliers comes up is in the proof of the real spectral theorem. Namely, let$A$be a symmetric matrix. Define$f: \mathbb{R}^n \to \mathbb{R}$by$f(v) = v^\top A v$. If you maximize$f$on the unit sphere in$\mathbb{R}^n$, the Lagrange Multiplier condition will show that the maximum is achieved at an eigenvector of$A$. This is ... 14 I think you're correct, strictly speaking, but overreacting. I just checked two calculus books, Rogawski and Strang. Rogawski defines the cross product as the "symbolic" determinant (his words) in question but then defines exactly what he means by this determinant in the same line, so I don't see a problem there. Strang mentions the determinant ... 13 Below are instructions for drawing various quadric surfaces. They're not sketches but drawings designed to look (sort of) pretty and be easy to draw. You can see all of them in one YouTube playlist if you want: https://www.youtube.com/playlist?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo Hyperbolic paraboloid https://youtu.be/ZofoRuDJXhM?list=... 13 By coincidence, I'm teaching multi-variable Calculus for the first time this semester, and have given some thought to how to handle this precise issue. This seems to me to be closely related to an example I first encountered at http://math.oregonstate.edu/bridge/ideas/functions/, which I strongly encourage you to visit before you read the rest of this ... 13 Sure, give up on the mixed-object determinant formula. Instead use tensor arithmetic. Let$\epsilon_{ijk}$be the completely antisymmetric symbol. Also, while we're at it, let's give up on quaternionic notation for unit-vectors and instead use$\hat{x},\hat{y},\hat{z}$or$\hat{x_i}$for$i=1,2,3$then you have an elegant formula: $$\vec{A} \times \vec{B} = ... 12 This is somewhat of a hypothesis rather than a definitive answer, but one reason why vector calculus may no longer be the first proof based course at many colleges is because vector calculus involves teaching a lot of new material*. Real analysis, is mostly (at least in the first quarter/semester), material they have already seen before in calculus - minus ... 11 I always think of \int_D \nabla \times v = \int_{\partial D} v in terms of water flow. You have a bunch of water flowing around: It's velocity at a given position (x,y) is given by the vector field v(x,y). Stick a tiny paddle wheel into the water at position (x,y) and let it start spinning: Its speed is \nabla \times v. (Probably times the length ... 11 Do you ever introduce the Jacobian matrix, or the derivative of a function at a point as a linear map? This clarifies everything. If not, and you must restrict yourself to "traditional multivariable calculus" notation, you could write the chain rule as (f \circ \gamma)'(t) = \nabla f|_{\gamma(t)} \cdot \gamma'(t) In your example f(x,y,t) = xyt ... 10 I might seem picky, but I would first refrain from saying that \nabla f is a vector. It is a vector field. This might be considered a common abuse of vocabulary, but using it amounts to assuming that student can fix it up routinely. The problem you are faced with shows very convincingly they don't. I bet we all have been confronted to someone who, asked ... 10 I guess you are aware of it, but since you don't say so explicitly: the chain rule on the top of your post does not apply to your example, since that f is not a function of x,y but of x,y,t. Also, as written, you are confounding the modern concept of "function" f:\mathbb{R}^3\to\mathbb{R} with the original notion of "function of". ... 9 A problem I used to assign on take-home tests back when I was teaching gifted high school students (late 1990s) might be of interest. Show that f(x,y) =y^{x} has the same limit along any 1st quadrant polynomial approach towards (0,0) (i.e. along any path having the form y=ax^{n}, where x>0 and n is a positive integer), but the limit varies if ... 8 Here's a crafty but perhaps crazy way to convey some ideas to a class. Have all the students gather on the football field (or another field) in a grid on a mildly windy day. Each student carries a little stick with a strand of paper to measure the direction of the wind. Perhaps with some physics they can also estimate the speed of the wind. Have them make ... 8 I don't have much to add to mweiss' nice answer in terms of teaching suggestions. But I would like to add my own point of view on what the abuse of notation \mathbf{r}=\mathbf{r}(s)=\mathbf{r}(t) means and where it comes from historically. The first person to do this abuse of notation (implicitly) seems to have been Jacobi around 1830 (I suggest you read ... 8 Air speed/direction on a weather map) is a very intuitive one. There's also other fluid velocity (and flux) vector fields in various chemE, mechE, and nukeE applications. I personally think the air speed is most intuitive as something where you really need speed and direction (i.e. a vector, not a scalar) and it's something people encounter in daily life. ... 7 I think the situation is clearer in \mathbb{R}^2 than in \mathbb{R}^3. Draw an open curve and ask students how much area it enclosed. Presumably they will recognize that the question has no real answer -- there is no way to tell what is "inside" and "outside" an open curve. (Perhaps one way to dramatize this is using a Paint-type program... They ... 7 The divergence theorem or Gauß’s theorem can be nicely linked to reality: In terms of hydrodynamic flows you could start with the following statement: The total signed water flow through the border of an area or surface of a volume (without sinks and sources) is zero – what goes in, must come out. Then you can add sinks and sources, i.e., points with a ... 7 (This is not bounty-worthy; just consider these illustrated comments.) Smith & Minton's textbook Calculus emphasizes drawing curves in each coordinate plane, and a few cross-sections (echoing Gerhard Paseman). To get an idea of what the graph looks like, first draw its traces in the three coordinate planes. Although they still rely on software, they ... 7 Search arxiv.org for "unit vectors i, j, k" and you will find examples of research papers where the notation is used. Many are from physics communities investigating phenomena in three-dimensional space. Clearly we should not expect to find this notation in research relating to general dimensionality. In the image see "unit vectors i, j, k" at the bottom. 7 I'm not sure this a research setting, but the cross product relations i \times j = k, j \times k = i, etc are nice for illustrating the connection between quaternions (where ij=k, jk=i, etc) and the cross product. Specifically, the Lie group consisting of unit length quaternions has Lie algebra \mathbb{R}^3 with Lie bracket given by cross product. 7 One idea that comes to mind is the fact that a connected graph has a Eulerian circuit if and only if every vertex has even degree. Another idea that comes to mind is just the Fundamental Theorem of Calculus. If you have a rather boring video of a car's speedometer as the car travels from A to B, then you can figure out the total distance the car has ... 7 This is a service course for students who are mostly engineering majors. Therefore any drastic change in notation like this is likely to be a bad idea. Leaving out d\textbf{S} and dV would be particularly unfortunate, since leaving out the dx is such a common student mistake anyway in freshman calculus. Also, any notation that has the wrong units is a ... 7 I had a lecturer who thought the same as you, so suggested we just learnt the definition of the cross product (which he derived along the same lines as James S Cook's answer). Nobody liked his lectures because he pitched them way to high for our first-year brains. The point is, it's a very useful mnemonic device, and when you teach it make that clear, but ... 7 this makes no sense But it does. A determinant exists for any n\times n matrix for which the Leibniz formula$$\det A:=\sum_{\sigma\in S_n}\epsilon_\sigma\prod_{i=1}^nA_{i\sigma_i}$$is fully antisymmetric. This is why, for example, quaternionic matrices lack determinants: their elements don't commute. But if we apply it to the case at hand, we don't have ... 6 A question with two constraints might make the method seem preferable to finding a parameterization (which I assume is the "easier" technique you refer too in the OP). For example, maximizing$f(x,y,z)=x−y$subject to the two constraints that$x^2+y^2+z^2=1$and$x+y+z=1$might be difficult for students to do without Lagrange multipliers. Is this form of ... 6 I have some experience teaching students to use Maxima for evaluating integrals. I did it in a classroom with almost as many computers as students. Maxima was installed on all the computers locally, so they didn't have to install Maxima themselves, which would have been somewhat of a barrier. It worked fine, and they seemed to enjoy it, but Maxima has ... 6 You can tell them the values of some of the partial derivatives without giving them the entire function. This prevents the students from creating the composite function, since they don't even know one of the pieces of it. For example: You know from your chemistry class that for a fixed amount of gas, the pressure, volume, and temperature are related by$...

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