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1d
awarded  Enlightened
1d
awarded  Nice Answer
Oct
16
comment Surface area of a sphere limits
If you're treating $dA$ as a form, you must notice that it doesn't pull back to an area form on the rectangle, due to the singularity at $\theta = \pi$ where it vanishes. To fix this you must reverse the orientation of the "back" before integrating, i.e. multiply by -1.
Oct
16
comment Surface area of a sphere limits
Yes, I guess "doesn't make sense" is too strong. It depends on whether you're treating the area element $dA$ as a measure or as a 2-form; in the former case there is no issue with orientation, the factor you get from change-of-variables is always positive, and intuitively you are evaluating the function 1 on a bunch of tiny parallelograms and measuring how much these get scaled when you map them from the rectangle to the sphere -- and this amount of scaling is always positive.
Oct
16
comment Surface area of a sphere limits
I think the OP is asking why you can't integrate using the parameterization where $\phi$ goes from $0$ to $\pi$ and $\theta$ from $0$ to $2\pi$, instead of the other way around, which does indeed still cover the sphere.
Oct
16
answered Surface area of a sphere limits
Oct
15
answered Volume of overlap between two convex polyhedra
Oct
10
awarded  Nice Answer
Oct
3
accepted derivative of composition of rotations
Oct
2
answered What would the Big oh be of (1/2)^n
Sep
30
comment Matrix exponential: Formal notation for power series? Or, more?
In fact since imaginary numbers are isomorphic to $2\times 2$ skew-symmetric matrices, it should not be surprising that exponentiation is defined in an analogous way in both settings.
Sep
30
answered derivative of composition of rotations
Sep
30
awarded  Explainer
Sep
30
revised Derivative of exponential map
deleted 88 characters in body
Sep
30
asked Derivative of exponential map
Sep
29
asked derivative of composition of rotations
Sep
29
comment Lagrange multiplier for more than one constraints.
Also WLOG you can drop the positivity constraint on $x$ since $-x$ satisfies the equality constraints iff $x$ does, and both give the same value of the objective.
Sep
29
comment Lagrange multiplier for more than one constraints.
Something goes wrong if you just add $\lambda^T(I-A^\dagger A)x$ to the objective, where $\lambda$ is a vector of Lagrange multipliers of the same dimension as $x$?
Sep
28
comment Is this true about the open intervals on the real line?
Look at the gap $L(k+1)-L(k).$
Sep
28
answered Is this true about the open intervals on the real line?