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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 ...

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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=...

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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 $\Sigma (... 12 By coincidence, I'm teaching multi-variable Calculus for the first time this semester, and have given some thought to how to handle this precise issue. This seems to me to be closely related to an example I first encountered at http://math.oregonstate.edu/bridge/ideas/functions/, which I strongly encourage you to visit before you read the rest of this ... 11 This is somewhat of a hypothesis rather than a definitive answer, but one reason why vector calculus may no longer be the first proof based course at many colleges is because vector calculus involves teaching a lot of new material*. Real analysis, is mostly (at least in the first quarter/semester), material they have already seen before in calculus - minus ... 9 I might seem picky, but I would first refrain from saying that$\nabla f$is a vector. It is a vector field. This might be considered a common abuse of vocabulary, but using it amounts to assuming that student can fix it up routinely. The problem you are faced with shows very convincingly they don't. I bet we all have been confronted to someone who, asked ... 8 A problem I used to assign on take-home tests back when I was teaching gifted high school students (late 1990s) might be of interest. Show that$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$is a positive integer), but the limit varies if ... 8 I always think of$\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$. (Probably times the length ... 8 Here's a crafty but perhaps crazy way to convey some ideas to a class. Have all the students gather on the football field (or another field) in a grid on a mildly windy day. Each student carries a little stick with a strand of paper to measure the direction of the wind. Perhaps with some physics they can also estimate the speed of the wind. Have them make ... 8 I don't have much to add to mweiss' nice answer in terms of teaching suggestions. But I would like to add my own point of view on what the abuse of notation$\mathbf{r}=\mathbf{r}(s)=\mathbf{r}(t)$means and where it comes from historically. The first person to do this abuse of notation (implicitly) seems to have been Jacobi around 1830 (I suggest you read ... 7 I think the situation is clearer in$\mathbb{R}^2$than in$\mathbb{R}^3$. Draw an open curve and ask students how much area it enclosed. Presumably they will recognize that the question has no real answer -- there is no way to tell what is "inside" and "outside" an open curve. (Perhaps one way to dramatize this is using a Paint-type program... They ... 7 (This is not bounty-worthy; just consider these illustrated comments.) Smith & Minton's textbook Calculus emphasizes drawing curves in each coordinate plane, and a few cross-sections (echoing Gerhard Paseman). To get an idea of what the graph looks like, first draw its traces in the three coordinate planes. Although they still rely on software, they ... 7 Search arxiv.org for "unit vectors i, j, k" and you will find examples of research papers where the notation is used. Many are from physics communities investigating phenomena in three-dimensional space. Clearly we should not expect to find this notation in research relating to general dimensionality. In the image see "unit vectors i, j, k" at the bottom. 7 One idea that comes to mind is the fact that a connected graph has a Eulerian circuit if and only if every vertex has even degree. Another idea that comes to mind is just the Fundamental Theorem of Calculus. If you have a rather boring video of a car's speedometer as the car travels from A to B, then you can figure out the total distance the car has ... 6 A question with two constraints might make the method seem preferable to finding a parameterization (which I assume is the "easier" technique you refer too in the OP). For example, maximizing$f(x,y,z)=x−y$subject to the two constraints that$x^2+y^2+z^2=1$and$x+y+z=1$might be difficult for students to do without Lagrange multipliers. Is this form of ... 6 I have some experience teaching students to use Maxima for evaluating integrals. I did it in a classroom with almost as many computers as students. Maxima was installed on all the computers locally, so they didn't have to install Maxima themselves, which would have been somewhat of a barrier. It worked fine, and they seemed to enjoy it, but Maxima has ... 6 I'm not sure this a research setting, but the cross product relations$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$with Lie bracket given by cross product. 5 I am an undergraduate student at Portland State University, having finished my major requirements I have decided to add a math minor. I did quite well in the lower division calculus, differential equations, and intro to linear algebra courses; yet I have had no experience with proofs. I am planning on taking a "Intro to Math Reasoning" course first for ... 5 I think the whole problem is connected to the problem of quantifiers, like "for all" and "there is". See also this question which I feel is relevant here. The important point is to stress that the definition of the limit ($\varepsilon$-$\delta$or sequences, does not really matter) means that you have to get the same limit when approaching the point ... 5 The divergence theorem or Gauß’s theorem can be nicely linked to reality: In terms of hydrodynamic flows you could start with the following statement: The total signed water flow through the border of an area or surface of a volume (without sinks and sources) is zero – what goes in, must come out. Then you can add sinks and sources, i.e., points with a ... 5 As far as I know, vector calculus is applied by financial analysts in exotic derivatives pricing. The Black-Scholes Model is actually a special form of Schrödinger equation. Thus, if you want to establish high precision models to price exotic derivatives, you will have the chance to apply vector calculus. However, these kinds of applications presumes ... 5 I wouldn't say it's common (none of the calculus books on my shelf use it). It's in some math books (e.g., this one, and this one and this one) but it's mostly in physics books. 5 You can tell them the values of some of the partial derivatives without giving them the entire function. This prevents the students from creating the composite function, since they don't even know one of the pieces of it. For example: You know from your chemistry class that for a fixed amount of gas, the pressure, volume, and temperature are related by$...

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Here are two similar ideas. (1) The shortest geodesic between two points on the surface of a polyhedron (generally) bends as it crosses edges, when viewed globally in $\mathbb{R}^3$, but from the local point of view of an ant walking along the path, it is straight, i.e., straight when each crossed edge is unfolded flat. This is commonly illustrated on a ...

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I'd suggest having a look at Geometric Mechanics by Darryl Holm. The theme of developing mechanics in the framework of differential geometry has a considerable record at the graduate level, as indicated in Dave Renfro's comment. At a level suitable for undergraduates, though, the pickings may be slim. Somewhat more accessible than Abraham and Marsden ...

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Welcome to SE. As you are asking on a side for math educators, I assume that you want to explain this to students. My advice: don't. Start with much, much easier examples to get the students accustomed to such problems, then come back to it later; if at all. If you, as the teacher/professor, are not able to find a solution for this problem, it is too hard ...

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When I was a student learning the difference, I found the unit-speed interpretation to be more helpful. Try using an animation in Matlab, Python, or Mathematica to illustrate this for simple curves by simultaneously tracing out $r(t) = (\cos{2t}, \sin{2t})$ and $r(s)$ or $\tilde{r}(t) = (t, t^2)$ and $\tilde{r}(s)$. Seeing the curve being traced out in its "...

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Here is an argument that has convinced (or at least silenced :-)) students in office hours. Making it into a rigorous proof would require more time talking about what volume MEANS than I do. First, suppose I have $k$ vectors $\vec{v}_1$, $\vec{v}_2$, ..., $\vec{v}_k$ in $\mathbb{R}^k$. Let $A$ be the $k \times k$ matrix with columns $\vec{v}_1$, ..., \$\vec{...

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Use level curves and contour plots (click link for video examples of their use). With contour plots, use color to represent third dimension. Here's a familiar example: :

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