Tricky calculus 3 questions

>But it makea them completely useless for anything that isn't pure abstract formalism which reduces their possibility to expand in otherb areas

But we don't need other areas. The only reason we do calculus is so that when we are doing our thesis and we have to prove a certain limit or series converges or diverges we know where to start looking.

> plus it takes a dent on physical intuition which is crucial

Yeah, the professor has mentioned how physicists think the first derivative is velocity and the second is acceleration.

That sounds like a bunch of vsauce garbage to me. The only intuition you need for calculus is geometric intuition.

>I believe most hate analysis because they think is symbol manipulation and can't visualize shit.

Having intuition means nothing if you don't know how to manipulate symbols and when the theory is rich a whole set of ideas can be found only by manipulating symbols.

That said, I still need problems :(

>Yeah, the professor has mentioned how physicists think the first derivative is velocity and the second is acceleration.
>That sounds like a bunch of vsauce garbage to me. The only intuition you need for calculus is geometric intuition.

Hmmm, so what is your geometric intuition behind Green's theorem?

From the topics I am being tested on tomorrow you should know I am still not at Green's theorem.

But you clearly already passed Calc 3. Got some problems for me?

>But we don't need other areas
The amount of math majors who end in academia is a minority. There's a reason companies are hiring more physicists for modeling in areas that were dominated by mathematician (yes, mathematicians not statitians or actuarialscience majors). Also, if you want to be on academia researching a niche field, either you have to be autistic on that particular subject, or prove your worth with your intellect. Which you will not if you have no interest in ideas beyond your comfort field of study. Interdisciplinary groups are more than a meme, it's the only way we can investigate shit acknowledging how specialized the fields are becoming.
>Yeah, the professor has mentioned how physicists think the first derivative as velocity and second derivative as acceleration.
If you had physical intuition you woukd see how those interpretations add to the geometric intuition and would let you model things more clearly. The first derivative as velocity literally is the justification for the word "rate of change" as the "slope" the curve has at that point. Obviously in unidimensional functions there isn't really much place to expand on this shit, but in multivariate calculus and vector calculus there is much more shit going on such as diferential geomtry, "density" functions and other shit that really ia best explained with physical situations. Most mathematicians cry when things as curl and divergence is motivated in class because the best way to understand it is through physicak vector functions such as velocity fields.
>Having intuition means nothing if you don't know how to manipulate symbols and when a theory is rich a whole set of ideas can be found from only by manipulating symbols.
I never implied you shouldn't learn how to prove shit, but the world doesn't has an exercise list, and if you want to do research you need motivation. Generalizations come from extrapolation of physical situations.

>still in calc 3
Lol, find the volume inside of a cone, inside of a sphere, but outside of a cylinder who's axis is on the cone's.

This really makes me think, but a problem would make me think even more.

I would be glad. In class we are just finishing partial derivatives so integrals come next my man. For now I need problems about vector functions.

Get a book moron.

use calc 3 methds to find [math]\int\limits_{-\infty}^\infty e^{-x^2}dx[/math].

Okay. Any suggestions?

I recognize that Gaussian integral and it'd be fun to poke at it eventually but vector functions my main man.

So show that the curvature of a function [math]f:\mathbb{R}\rightarrow\mathbb{R^2}[/math] at point t is equal to the inverse of the radius of the tangent circle tangent to the curve at the point [math](x(t),y(t)){R^2}[/math] .

But there are many such circles. What could you possibly mean ? ;^)

Ok, I was not precise enough, show that the curvature is equal to 1/r, where r is the supremum of radii of tangent circiels.

Here's a nice proof-based question:

Let p and q be polynomials in two variables, and suppose that

[math] p_xq_y-p_yq_x=c[/math]

where c is a non-zero constant. Let F(x,y)=(p(x,y), q(x,y)).

Show that the components of the inverse function F^{-1}(x,y) are polynomials.