As part of one of my university courses, Multi-variable Calculus, I need to write a 5000 word essay on an a research question that is on the course. Before telling you what I have as an idea here are the key things that the essay should cover.

  1. The topic must be specific, yet have enough to it to have a research question, not a technique, and be in the 5000 word limit
  2. The topic of research must be connected to the Multi-Variable Calculus

Here are my thoughts so far:

  1. Researching surface areas of a 3+ dimentional function while estimating the function with taylor series. The problem is that this doesn't really fit with the first bit where it says that it should be a question rather then a technique
  2. Researching applications of multivariable calculus in differential geometry. For this bit I would need a topic in differential geometry but I have no idea which. (If someone has any knowledge of differential geometry please leave a suggestion)
  3. Furtherone investigating vector fields but I really don't have an idea of what a research question on this topic would look like that will be specific but also lenghty.

PS: I forgot to say that I will be expected to go over deffinitions, derivations on more complex topics and explaining example graphs.

Also just note that I am not asking for someone to hand me an essay, I am just looking for ideas/suggestions for a possible Research question

Any help will be greatly appriciated.


  • $\begingroup$ Are you expected to come up with new novel results that hasn't been done before? Or are you just looking for a problem that has been solved before which you can replicate the results and you can share a bit of the history of? One of my favorite such problems would be the napkin ring problem (take a sphere and drill a cylindrical hole through the center so that the height of what remains is constant. Find the volume and recognize it doesn't depend on the size of the sphere) $\endgroup$ – JMoravitz Jun 13 '18 at 20:16
  • $\begingroup$ @JMoravitz Thanks for the quick response. We are not expected to come up with a novel or a new mathematical invention. The way that I view it is researching a topic (or even a problem, as suggested) and then applying these skills to other areas of mathematics or showing the significance of the problem for pure or applied math/science based theories. Also thanks for the suggestion but it seems to be little too specific. But thanks for the suggestion anyways! $\endgroup$ – JonhSmith Jun 13 '18 at 20:22
  • 3
    $\begingroup$ The generalized Stokes theorem unifies Green't Theorem, and Gauss' Theorem and the Fundamental Theorem of calculus in a tidy package. You can fill quite a few pages on the implications of this little gem. $\endgroup$ – Doug M Jun 13 '18 at 20:24
  • $\begingroup$ @DougM Thanks for the suggestion! Do you have any applicants of the theorem on top of your head that are 5000 word lenghty? $\endgroup$ – JonhSmith Jun 13 '18 at 21:14
  • $\begingroup$ Is this for an undergraduate math course? What level, introductory or advanced? Are you doing analysis with differential forms, or is multivariable calc new to you? $\endgroup$ – Larry B. Jun 13 '18 at 21:46

DeRham cohomology could be an interesting topic as it underlies the relationship between divergence, gradient and curl.

  • $\begingroup$ Hey I know that it is really unprofessional but is it possible that you link me a source on the DeRham chomology where it underlies the releationship between divergence, gradient and curl because from the sources that I have looked at I can't find anything on it. Thanks $\endgroup$ – JonhSmith Jun 13 '18 at 21:50
  • $\begingroup$ I learned it from "From Caclulus to Cohomology". $\endgroup$ – JohnKnoxV Jun 13 '18 at 22:09

One of the most difficult parts of writing an undergraduate essay is the feeling that you're just copy-pasting whatever someone else derived. Here's a couple ideas to make it your own:

  • Make a review of literature. Once you've gotten your topic, stick with it, and find the relevant literature for your topic. Don't just cite and forget about it, but make your citations relate with each other. Just like relationships between people, it's the interactions between the citations that matter the most, not the material in-and-of itself (otherwise, they could just read that paper!)

  • If you're handy with computers, try applying algorithms and libraries to a multivariable calculus problem. Make it incremental, so that even if you don't complete your project you'll have something to show for it.

  • Find problems in your textbook that are difficult and interesting. You can write your essay on that problem, and extensions or generalizations to that problem. What different variations can you give? Be careful to research these variations, because some of them could actually be open!

  • How about a history essay? You can explain when math topics came about, why they were developed in such-and-such a way. This can also give you perspective on current popular topics in math.

To answer your question, here's some ideas for a topic.

Physics is a great source of multivariable essays, but it helps to limit it to a small subfield.

You could write about Aerodynamics. This is especially cool if you like sports, because you could write about the way a ball curves and how it affects its path. Might be kind of difficult, but there's a lot of material for you to cite.

You could write about electricity and magnetism, or how am AM/FM radio works. Someone in the comments mentioned Maxwell's equations, and those might be interesting to explore.

Going more into the differential equations territory, you could write about ecology and the Lotka-Volterra or Birth-death processes.

On a more linear-algebra point of view, you could write about image processing. For example, edge detection is a differential process which depends on the change between neighboring pixels, i.e. the gradient in the color space.


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