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1

The oscillation of species' populations due to predation. The convergence of machine learning models using gradient descent - this is a special vector field that's tuned by uniformly multiplying the field with a scalar (bonus points for involving a segue into machine learning theory and having them tune the vector field for most rapid descent) A couple ...


1

The Hairy Ball Theorem:        wikiwand link: "there is no nonvanishing continuous tangent vector field on even-dimensional $n$-spheres."


4

The two examples you give both have zero curl, which limits their usefulness. Examples that do have a curl would be: an electromagnetic wave the magnetic field of a wire, inside the wire the magnetic field of a slab of current, inside the slab the field of a point charge that is moving inertially. The external magnetic field of a wire is also an ...


7

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. ...


4

If you are following a textbook, I would strongly recommend adopting their convention. If you are writing your own lecture notes, there are several ways I see to proceed. My personal favorite approach (if your students are sophisticated) would be to have them suggest conditions, challenge them with situations where it seems reasonable for the "...


1

It is not formally described as an AP Calculus course, but it is supposed to map roughly onto Calculus AB. It sounds like there isn't a clear definition for the level of the course or the content that needs to be covered. It may be beneficial to poll the students to find out how many are planning on skipping Calculus II in college and moving straight to ...


2

Reasons to study Newton's method: -It's an application of derivatives -It is a good example of numerical methods -It can help strengthen understanding of relationship between derivative and tangent line -It's likely going to be on the AP test I've put these roughly in order of how important they are to your situation; as nice as it is to get greater ...


7

AP classes have become much more common these days, and at many schools the result has been that very few students actually pass the AP exam with a grade that would allow them to skip the course in college. The trouble is that even if 90% of your students fall in this category, you also have a duty to serve the other 10%. Those students are going to be ...


9

If time constraints are so dire that you risk not getting to cover integrals and the fundamental theorem of calculus, then I'd cut Newton's method (and probably much more) since I don't see how you could pass the test without knowing those former topics. But, if you cut Newton's method from the teaching, wouldn't it still come up in practice AP tests that ...


7

I don't know if this is what you're looking for exactly, but I've run forms of this activity a few times when introducing partial/directional derivatives. Supply the class with materials: square grids printed on paper, scissors, compass (with attached pencil), ruler and protractor [Sometimes I forego the compass and just provide appropriately-sized circles ...


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