I have to present an application of Multi-variable Calculus in front of the class tomorrow. What should I present?

Show that the polar coordinate theta has the differential [math] \operatorname{d} \theta = - \frac{y}{{{x^2} + {y^2}}}\operatorname{dx} + \frac{x}{{{x^2} + {y^2}}}\operatorname{dy} [/math] on some subset of [math] {\mathbb{R}^2} [/math].

This part is fairly easy, it is pretty much just taking the derivative of arctan.

But then do the following:

Show that [math] - \frac{y}{{{x^2} + {y^2}}}\operatorname{dx} + \frac{x}{{{x^2} + {y^2}}}\operatorname{dy} [/math] is well defined on all of [math] {\mathbb{R}^2}{\text{\backslash }}\left\{ {\left( {0,0} \right)} \right\} [/math] but can not be expressed as [math] \operatorname{d} \theta [/math] on this domain.

The point of this is too show that not all 1-forms on [math] {\mathbb{R}^2}{\text{\backslash }}\left\{ {\left( {0,0} \right)} \right\} [/math] can be written as the differential of a function.

Why this is important is because it is an example of how to establish a deep connection between calculus and algebraic topology. The existence of a closed (i.e. dw=0) differential form on [math] {\mathbb{R}^2}{\text{\backslash }}\left\{ {\left( {0,0} \right)} \right\} [/math] which is not exact (i.e. w=df) corresponds to the first De Rham Cohomology group of this space being nontrivial.

This is inline with topology because the kth Homology group of a space essentially measures the amount of k-dimensional holes in the space.

And we are looking at [math] {\mathbb{R}^2}{\text{\backslash }}\left\{ {\left( {0,0} \right)} \right\} [/math] , the punctured plane which by definition has a hole at the origin, so it only makes sense that it should have nontrivial 1st homology group.
And through poincare duality, we have the 1st Homology group of this space being isomorphic to this purely calculus defined 1st De Rham Cohomology group.

So if you can manage to understand this, you will probably impress the hell out of your teacher.

>Freshman wants help thinking of ideas for Calc 3 project
>Veeky Forums recommends cohomology

...

an optimization problem would be the most realistic tbphwy

particle in the box in 3 dimensions
or some weather modeling shit, aren't temperature gradients multivariable calc reliant?

What, how is it mutli-variable?
F=ma
-kx=ma
-kx+mx''=0
r=+/- i*sqrt(k/m)
...
...
I don't see any multivariable? Just an ODE
(I used spring constant as restoring force but SHM is defined with force with direct negative proportion to position, so 'k' is applicable to any scenario of SHM)

If you're thinking about uniform circular, the DE is still the same, there's just 2 of the same in the x and y dimension

But ofc I'm still in Calc 2 and just finished a freshman course on Mechanics so perhaps someone can enlighten me

...

Fuck realized both were negative on second line of my work and meant to make both positive, but left -kx by accident, either way result is the same but I'm not an idiot, that was an accident

bump

>memes

This is actually pretty cool. You can prove R^2\0 is homologically nontrivial simply by studying the differential of a polar angle coordinate.

Yeah, they can be.
Do this OP, heat transference is easy.

Use RK to approximate y' = cosh-1(y) + ln(y); given y0 = 1; step size of .01; at y = 1000.

Are differential equations ok? If so, I recommend this book. You'll find lots of real world applications in it.

simple linear regression

Explain how that's related to Calc 3.

....It involves minimizing the mean distance squared with respect to both slope and y-intercept? It's not hard, but it does require basic multivariable calculus.

It's weird seeing my professor posted here.

Look up harmonic motion. I'll give you a hint, it's not some physics equation.

>it's not some physics equation
Harmonic motion is definitely part of classical mechanics. Gauge theory has plenty of applications in topology and geometry but you'd be wrong in saying that the Yang Mills Lagrangian isn't a physics equation, and the same goes for harmonic motion.

>And through poincare duality
wouldn't this be de Rham theorem rather than poincare duality?

The universe.

I'm not who you replied to, but I have no idea what you're referring to, wikipedia page on SHM doesn't have any multivariable either

who is this guy

>not knowing the Wilberger meme

le upboat humble brag

what the fuck

why did you post an image of my prof

am I going to see you tomorrow?

based board erasing man has a video series for multivar. i don't even go to MIT and i know about him

I passed multivariable calculus some time ago and even a topology course and two in algebra but I don't understand this

I was thinking both, like: [math] H_{DR}^k\left( M \right)\mathop \cong \limits_{DeRham} {H^k}\left( {M;\mathbb{R}} \right)\mathop \cong \limits_{Poincare} {H_{n - k}}\left( {M;\mathbb{R}} \right) [/math]

But now that I think about it, the punctured plane isn't compact so poincare wouldn't hold.

You wouldn't learn it until like a Differential Topology, Analysis on Manifolds, or Differential Geometry class.

catapult immigrants over the border as an example

Bump, how exactly is multivariable involved with harmonic motion?

This guy really helped me in Calc 3 when my professor couldn't fill the gaps.

bumasses

Economics