Calculus Theorems

Every once in awhile, Green's Theorem pops up at least conceptually for conservative fields like gravity. In that case, the right-hand side of the equation goes to zero. Kind of explains the path-independent nature of work and of potential energy. Never actually used it to solve problems.

Green's theorem is related at least tangentially to Navier-Stokes, which is important for continuum mechanics.

>>8422052
OP here
I do remember Green's theorem being related to Stokes but I also remember Stokes being way easier and straightforward to use than Green's theorem.

So Green's can pop up here and there in the terms of gravitational theory, huh, thanks for letting me know.

I also heard from my professor that if one was to pour some fluid over a surface and you knew the equation of the surface and the initial conditions, then greens theorem could theoretically tell you the direction and magnitude(velocity?) of every particle in the fluid.

Not sure if green functions are the same thing as green theorem but if it is, then its used in many-body quantum physics(perturbation theory). In general GF shows probability amplitude for particle to move from point x1 at time t1 to point x2 at time t2. Theres much more about it but im too lazy to explain.

>>8422285
Green's theorem is just stokes theorem for 2 dimensions.

As [math]\partial_x Q - \partial_y P[/math] is just [math]\nabla \times F[/math] in 2 dimensions.

While [math]Pdx + Qdy[/math] is also just [math]F \cdot dr[/math]

Of course, for physics, we have stokes theorems directly on the maxwell equations. So green's theorem is going to pop up whenever you're dealing with 2 dimensional electric fields or gravitational fields.
So we have green's theorem when dealing with 2 dimensional flux, so also useful in fluid mechanics.

In complex analysis, the proof for cauchy's integral theorem is a direct application of green's theorem.
And cauchy's integral theorem is used everywhere, like in QM for example.

>>8422285
>greens theorem could theoretically tell you the direction and magnitude(velocity?) of every particle in the fluid.

> No. You'd drive yourself batshit crazy trying to do that, anon.

It will give you generalized behavior. Avg velocity and direction for particles in close vicinity to the point of interest. Good enough for modeling fluid shear.

>>8422285
> greens theorem could theoretically tell you the direction and magnitude(velocity?) of every particle in the fluid.

> You'd drive yourself batshit crazy trying to do that, anon.

What it will do is give you generalized behavior. Avg velocity and avg direction for particles in close proximity to the point of interest.

>>8421589
I know that feeling. For the life of me I did not understand why we needed Laplace transforms until I started to getting into control theory. Turns out complex differential equations in the time domain turn into really nice algebraic equations in the frequency domain.

I'm still not quite sure what the difference is between the Laplace transform and the Fourier transform though and when I should use one over the other. Wikipedia says the Fourier transform of a function is a complex function of a real variable and Laplace is a complex function of a complex variable but that doesn't really clarify anything for me.

>>8421589
Green's theorem gives you a really easy way to calculate areas of surfaces if you know a parametrization of their border that makes the difference of the partial derivatives equal 1

>>8424216
Sorry, meant to say a vector field. Just use a trivial vector field that satisfies the condition

>>8424209
I'm also learning LaPlace Transforms in my math course right now and I was sort of confused on how or why we apply them, but it is slowly becoming clearer as we get more in depth with it. They are very usefully in solving differential equations but whether or not that certain differential equation is useful is a different question

>>8424228
You will use LaPlace in system dynamics. It's a giant LaPlace orgy.

File: pi.png (7KB, 200x144px) Image search: [Google]

7KB, 200x144px

>all these engineering and science majors

Man, it sure feels good to actually learn math rather than a pile of equations.

>>8424228
>>8424269
In Process Control class right now, can confirm that the course is 90% Laplace transforms.

>>8421589
Green's theorem allows you to relate integrals over a 2D surface to the 1D boundary.

The regular integral from first year calculus relates a 1D line to its 0D boundary, which are 2 points.

The generalized Stokes' theorem allows you to relate N dimensional integrals to their N-1 dimensional boundaries.

It's sometimes easier to go one way or the other, especially if your 2D surface is a closed sphere for instance, then your boundary doesn't exist, so it's zero.

But what if your 2D surface is say a torus or something else? Well here enters some topology to play around with.

It's not too crazy to think you'd want to know about the inside of some 3D object given only knowing the 2D surface area, use your imagination and all these theorems start to boil down simply and have applications. Electricity and magnetism is a popular one.

>>8421589
Green's Theorem is tremendously useful in physics.
Suppose you have a volume filled with some quantity that won't just vanish (a conserved quantity like energy for example). You can relate that variation of your quantity inside your volume (which is an integral on V) with the amount that crosses your boundary (an integral on the boundary). Using Green's Theorem you can put them in the same form, and deduce from the fact that your relation is true on any volume V that your integral is unneccessary : a conservation relation for your whole space takes the form of a local relation (generally a partial derivative equation for local fields).
Also, that's how you find the local form of Maxwell's equations from their integral counterparts :
Lorentz Force => Gauss's Theorem => Maxwell-Gauss equation through Green's theorem.