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May
16
comment How should I solve integrals of this type?
@JackM Differentiating $e^x\cos x$ or $e^x\sin x$ gives combinations of terms similar to your integrand. It stands that some combination of these might differentiate to give the integrand precisely; we then must figure out which combination.. You equate coefficients because you want to find the particular case where the (differentiated) functions are the same.
May
13
comment How to deduce whether $\sqrt{3}i \in \mathbb Q (u)$ or not, where u is a root of $x^3-2$
@Improve yes, with no extra effort. It's not a separate identical looking argument for the other roots or anything, you already have the entire result by that very statement you gave. Algebra can't tell the difference between the roots: it has no sense of 'where' they are, only the algebraic relationships they satisfy. So if it's not possible for one of them it can't be possible for any of them, because that would imply that algebraic relationships could somehow make a distinction between two quantities that satisfy all the same algebraic relationships, which is a contradictory statement.
May
12
answered How to deduce whether $\sqrt{3}i \in \mathbb Q (u)$ or not, where u is a root of $x^3-2$
May
3
comment What is a high rank tensor?
metric tensors are always $n\times n$, since they always take 2 vectors as input. The Riemann curvature tensor, which takes 4 inputs, might be closer to what you're looking for.
Apr
22
comment Why do folded concentric circles and rectangles form a hyperbolic paraboloid?
@KyleKanos Origami would tell you how to make it, not why it folds the way it does. The solution clearly lies in analyzing what internal forces and stresses in the paper are created by the folds and bending, which is pretty squarely in physics territory.
Apr
17
answered How to tell if points in $Z^2$ belong to a half-plane?
Apr
11
awarded  Yearling
Apr
11
answered $A \oplus C = B \oplus C$ but $A\neq B$
Apr
6
comment Sketch the graph of the polynomial function $P(x)= x(x-3)(x+2)$
Your intervals have their ends at each of the roots. These are $-2, 0,$ and $3$. So yes, $x=0$ should be included as a boundary and you should test on either side of it to figure out whether the polynomial is positive or negative there.
Apr
6
comment Calculus on manifolds
It's tedious in comparison. You can do it (by 'it' I mean set up an appropriate surface integral, not direct integration in $\mathbb{R}^3$ which doesn't work as Nicholas has pointed out), but you end up doing all the intrinsic calculations anyway. Adding other calculations and things to look out for on top of that doesn't make life easier in the long run.
Apr
6
answered Sketch the graph of the polynomial function $P(x)= x(x-3)(x+2)$
Mar
22
answered Intuitive idea of axiom of choice
Mar
18
comment Application of Composition of Functions: Real world examples?
A function is just a process that turns one thing into another thing. Anytime you're describing something that chains processes together one after the other you're composing functions. You find a probability distribution and then want to find its average. Find a particle's position as a function of time, and then its distance from its start point. Almost any time you want to do multiple things to a function you're composing it with other functions.
Mar
5
awarded  calculus
Feb
12
comment What are the practical applications of the Taylor Series?
@Ruslan you can use the symmetries and periodicity of $\sin(x)$ to restrict your calculation to $[0,\pi/2]$. It's this range that has the maximum error of $8\%$.
Feb
1
comment Are there any other purposes for variables in math other than functions?
@tazheneryduck0 No, that doesn't make any sense. If you divide by $x$ you need to know that any results you get are only okay if $x\neq 0$, because otherwise the division wasn't allowed. But you can definitely divide by variables to find solutions to equations, it's used all the time. Your teacher seems to be confused. If $x$ is a root then $f(x)$ is what ends up being zero. That doesn't prevent you from dividing by $x$ at all.
Jan
31
answered Are there any other purposes for variables in math other than functions?
Jan
28
comment Why are particular combinations of algebraic properties “richer” than others?
While I don't have a full answer, I would like to point out that associativity is what allows you to move your focus around when solving equations. Commutative but non-associative operations only let you switch two things around, not several in a row, so you barely gain any freedom. Associativity without commutativity is much less restricting. Also function composition is associative, (regardless of formalism; any sensible definition would be) so any algebra that can be interpreted as a collection of transformations (which is a large number of them) has to be associative.
Dec
8
awarded  Nice Answer
Dec
3
comment Explaining probability theory versus statistics
Doesn't this assume that you can't take a Bayesian approach to statistics?