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1h
comment In which cases are $(f\circ g)(x) = (g\circ f)(x)$?
Related: If $f \circ g = g \circ f$ does that mean that both functions are to and from the same set and both are bijections? Does it tell us anything else?
1h
comment Differentiable, Parametrized Curve for Trace $y = |x|$
Suggestion: Pick your favorite smooth, increasing function $f$ that satisfies $f'(0) = 0$, and put $x = f(t)$, $y = |f(t)|$. (For extra fun, see how smooth you can make the parametrization.)
3h
comment Prove that multiplication is well defined
In this situation, it probably helps to "cheat": Multiply out $(a - b)(c - d)$ as if you know about integer arithmetic, then interpret the result as an ordered pair of natural numbers. Then use the fact that $(a - b) = (a' - b')$, etc. As long as the end result refers only to pairs of natural numbers, you haven't technically done anything illogical.
3h
answered When is a manifold also a vector space?
4h
comment What simple topological properties of conic sections can be explored?
Could you please add details about which metric spaces you're considering, and how you define a "conic section" in these spaces? (I didn't vote to close, but tend to agree the current wording is difficult to address substantively.)
1d
comment Why don't infinite sums make any sense?
@Rob: Welcome to Math.SE. You might consider re-titling your question to something like "Seeming discrepancy with geometric sum formula." :)
1d
answered Is velocity a function of displacemnt?
1d
comment locus of a variable straight line
"taking the equation of a line in parametric form and substitute the given line equations. we get 6 constants, solve" looks like a sketch of a strategy. Could you please clarify where exactly you are unable to make progress in carrying out the details?
1d
comment How did the rule of addition come to be and why does it give the correct answer when compared empirically?
@bzal Here are some Wikipedia links: base-$b$ positional notation, and three examples common in computer science: binary (base $2$), octal (base $8$), and base $16$ (hexadecimal). Wikipedia also has a more extensive list of examples.
2d
answered How does a C-constant in a substitution affect the result of integration?
2d
comment When is a diffeomorphism analytic?
Or perhaps you read "A class $C^{\omega}$ diffeomorphism is said to be analytic." (N.b., superscript is \omega, not \infty.)
2d
answered How did the rule of addition come to be and why does it give the correct answer when compared empirically?
2d
comment Describing Area of Spherical Cap as Sum of Spherical Triangles
The circular boundary of a spherical cap isn't a geodesic (unless the cap is a hemisphere), so a spherical cap can't be partitioned into finitely many spherical (geodesic) triangles. Similarly, a Euclidean disk can't be partitioned into Euclidean triangles.)
Feb
5
comment How do you differentiate the integral from $ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt$
Voted to close as a duplicate, but in case the fine points of your question aren't addressed elsewhere: 1. The integral can be "differentiated with respect to $x$", but not with respect to the dummy variable $t$; 2. The derivative of the integral isn't an antiderivative of the integrand, so there's no reason it should look like one. :)
Feb
4
comment Solving $(4y+2x-5)dx+(6y+4x-1)dy=0$ using 2 methods produced 2 different answers!
After integrating (and algebraic rearrangement) you have an equation of the form $\ln|U| = C$, which can be written $|U| = e^{C}$, and then $U = \pm e^{C}$, with $A = \pm e^{C}$ representing an "arbitrary constant". Substituting the initial condition at this stage gives the "correct" value of $A$, i.e., the desired solution. (By contrast, substituting the initial condition before removing the absolute value sign gives |U| = e^{C}$, namely, both signs $U = \pm e^{C}$, i.e., "extra" solutions.)
Feb
4
answered Solving $(4y+2x-5)dx+(6y+4x-1)dy=0$ using 2 methods produced 2 different answers!
Feb
4
answered Geodesics on surfaces of revolution about z axis with negative curvature
Feb
3
comment Can the concept of orientability be applied to more general spaces?
For what it's worth, a Möbius strip with one segment collapsed to a point is homeomorphic to a cylinder with one generator collapsed to a point. (As for the question, I've never encountered "orientability for non-manifolds." A natural place to look would be the theory of stratified spaces. If $X$ is a topological space containing some manifold $M$ as a dense open subset, you might define $X$ to be orientable if and only if $M$ is orientable. Offhand I don't know if this is useful, however.)
Feb
3
comment Holomorphic Frobenius Theorem
In a word, "yes". :) You can view $JY'$ as the contraction of $J \otimes Y'$, so by the Leibniz rule for the Lie derivative, $[X', JY'] = L_{X'}(JY')$ is the contraction of$$L_{X'}(J \otimes Y') = (L_{X'} J) \otimes Y' + J \otimes L_{X'} Y'.$$This gives the advertised formula.
Feb
3
comment An alternative derivation of radius of curvature (2D functions). How valid is it?
Your idea and formula are correct, but (unfortunately) well-known, see, e.g., curvature of a graph. (The curvature is the reciprocal of the radius of curvature.)