1
vote
2answers
47 views

What is the potential function of the field $\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$

The vector field is obviously conservative on every closed domain that doesn't encompass the point $(0,0)$, so there must be a potential function. I've got $\arctan(\frac{x}{y})$ for $x$ unequal to ...
2
votes
3answers
111 views

How to prove the one-variable calculus definition of derivative extends to $\Bbb C$ *only* because $\Bbb C$ is a field?

I have been told the one-variable calculus definition of derivative extends to $\Bbb C$ only because $\Bbb C$ is a field. See : Higher dimensional analogues of the argument principle? $$ ...
0
votes
0answers
73 views

Conservative field and potential functions

How can I prove that the field $F=\frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy$ is a conservative vector field, and isn't local conservative on the domain $U=(x-5)^{2/3}+(y-7)^{2/3}<1$? I tried to ...
9
votes
1answer
187 views

Is it possible to do calculus on any field with a topology?

I'll try to make my point clear: when we consider the field of complex numbers $\mathbb{C}$ we can do calculus there because we have properties of a field and in the same time we have a topology to ...
3
votes
1answer
35 views

Chracterizing quartic polynomials F such that $F, F',F''$ have only real rational roots.

When designing friendly problems for a calculus class one comes up with such a question. (The cubic case is relatively easy.) Of course one can generalize: Characterize degree $n$ polynomials such ...
0
votes
1answer
44 views

probability inequality resolution finite field

I'm trying to find out whether my communication protocol should have redundant information padded, in order to help the receiver correct the error (error correction code, ECC) without needing a ...
1
vote
2answers
244 views

Question about the definition of a field…

Just out of curiosity - when we define a field, why bother mention multiplication, when its nothing more then repeating the same addition operation? Here's the definition we were taught in calculus ...
1
vote
3answers
262 views

What properties of numbers allow us to remove parentheses from expressions?

I've seen it asserted in several places (e.g., Spivak's Calculus, p.3) that the fact that "parentheses can be freely rearranged" in expressions involving only addition ($+$) is based solely on (P1) ...