# What is a good topic for an essay on applications of Calculus 3?

In a class I have, the professor has offered extra credit for 1 page paper on a topic in Calculus 3 that has an application in the real world. I know calculus is used a lot in physics but I do not know physics very well.

What is a good topic that is understandable to a layman and that I might write an essay on?

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Since "calculus 3" might be different from place to place, could you say what topics you are covering? –  Thomas Apr 18 '14 at 22:27
double intergrals, the gradient, partials, absolute max and min in 3 dimensions, quadratic surfaces projections, let me look at my class syllabus –  Fernando Martinez Apr 18 '14 at 22:28
real valued function several variables,limits, continuity, differential, directional derivatives, parital derivaties chain rules, multiple integrals are some topics covered –  Fernando Martinez Apr 18 '14 at 22:30
A great deal of electromagnetic theory is due to vector calculus, divergence, curl of vector fields etc, along with the vector wave equations: $$\frac{1}{c^{2}}\frac{\mathrm{d}^{2}\vec{E}}{\mathrm{d}t^{2}}-\nabla^{2}\vec{E}‌​=0,\quad \frac{1}{c^{2}}\frac{\mathrm{d}^{2}\vec{B}}{\mathrm{d}t^{2}}-\nabla^{2}\vec{B}=0‌​$$ Also leading to the fact that $c_{0}=\frac{1}{\sqrt{\mu_{0}\epsilon_{0}}}$. –  Shaktal Apr 18 '14 at 22:34
You could calculate the energy that could be gathered by a Dyson Sphere. That's kinda real! –  Alexander Gruber Apr 18 '14 at 22:37

You could look at Lagrange Multipliers, which is a method of optimisation for functions of several variables subject to some constraints.

http://en.wikipedia.org/wiki/Lagrange_multiplier

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yes the lagrange looks promising –  Fernando Martinez Apr 18 '14 at 22:41

Did you cover line integrals and Green's Theorem in your class? The planimeter is one of the coolest applications of Green's Theorem I've ever seen. It allows you to measure the area bounded by any closed curve, just by tracing along its boundary! Here's another link about planimeters.

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You can use cute, clever, pretty easy double integration to derive the Gaussian integral $$\displaystyle\int_{-\infty}^{\infty} e^{-x^2} dx=\sqrt{\pi}.$$

I would this calculation's most important use is in statistics, because it is intimately related to the normal distribution ("Bell curve") which is used all over the place. I cannot overhype how common the normal distribution is, but for one particular example see quantum field theory.

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I'll bet the teacher knows about the Bell curve. –  TrialAndError Apr 19 '14 at 2:20

I have two ideas for you. Since you are only allowed to write one page, you are not going to be able to do much. But may

1. Look at electromagnetism and Maxwell's equations. Some completely random notes: http://www.phys.ufl.edu/~thorn/homepage/emlectures1.pdf. Take a look at chapter 2.
2. Again, since you just have one part to write, you could also just consider how calculus is used in business. In business calculus people are interested in optimizing functions of several variables. You could for example discuss the terms consumers surplus, producer's surplus, market equilibrium. If you study the non-linear functions, that already takes you outside of what a lot of economics majors study these days.
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