I have to present an application of Multi-variable Calculus in front of the class tomorrow. What should I present?
>>8035527
derivatives
>>8035534
Yeah, what this guy said.
I recommend going the distance --> velocity ---> acceleration route
How the formulas for marmonic motion are derived. Its multivariable and its a differential equation.
>>8035546
Harmonic*
>>8035527
gaussian integral
Take an application problem from the book thats kind of tough, solve it step by step and present it to the class.
Right now I have this:
Given a velocity vector field F(x,y) that represents the flow of water, find the optimum location over an area D to place a paddle wheel so that it spins the fastest.
I then say that it is known that:
w = (1/2) |curl F|
where w is angular velocity.
I compute this and get a function w(x,y). Then I find the maximum of this function on D.
>>8035557
Are you going to give em the D?
>>8035565
I'll give em the F too.
>>8035527
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.
>>8035596
>Freshman wants help thinking of ideas for Calc 3 project
>/sci/ recommends cohomology
...
>>8035527
an optimization problem would be the most realistic tbphwy
>>8035527
particle in the box in 3 dimensions
or some weather modeling shit, aren't temperature gradients multivariable calc reliant?
>>8035546
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
>>8035596
>>8035743
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
>>8035747
>memes
>>8035596
This is actually pretty cool. You can prove R^2\0 is homologically nontrivial simply by studying the differential of a polar angle coordinate.
>>8035723
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
>>8036361
Explain how that's related to Calc 3.
>>8036447
....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.
>>8035527
It's weird seeing my professor posted here.
>>8035743
Look up harmonic motion. I'll give you a hint, it's not some physics equation.
>>8036499
>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.
>>8035596
>And through poincare duality
wouldn't this be de Rham theorem rather than poincare duality?
>>8035527
The universe.
>>8036499
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
>>8035747
who is this guy
>>8036852
>not knowing the Wilberger meme
>>8035527
what the fuck
why did you post an image of my prof
am I going to see you tomorrow?
>>8037804
based board erasing man has a video series for multivar. i don't even go to MIT and i know about him
>>8035596
I passed multivariable calculus some time ago and even a topology course and two in algebra but I don't understand this
>>8036587
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.
>>8037819
You wouldn't learn it until like a Differential Topology, Analysis on Manifolds, or Differential Geometry class.
>>8036834
Bump, how exactly is multivariable involved with harmonic motion?
>>8035527
This guy really helped me in Calc 3 when my professor couldn't fill the gaps.
>>8038083
bumasses
Economics
