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comment $\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz$ where $f(z) = z^{1/2}$ with branch such that $\Re f(z) \geq 0$
I'm a little rusty with contour integration so maybe wait for (hopefully) verification from others on this, but: It seems to me that if the problem calls for integration along $|z|=2$ then, no, you don't do the keyhole contour, you stick with the $|z|=2$ circle. Since the square root introduces a branch, you have to do more than one loop around the circle to close the contour. I think two should do it, but double check that.
Aug
15
comment Understanding the notation of a book when derivating
I think senx is meant to be $\sin x$. Do you mean that you don't understand derivatives? If not, it's usually a month or so in a calculus class to get derivatives. SOrry if I am misunderstanding.
Aug
15
comment I need some help understanding the tensor algebra done this problem.
Would you be satisfied with a verification, or do you actually want to see the derivation?
Aug
15
answered chances of repeating numbers
Aug
15
accepted Is this a manifold?
Aug
14
comment Is this a manifold?
@Ted Shifrin Thanks for the answer, and the positive feedback. Now, to your question, my messed up thinking is that there wouldn't be a homeomorphism between $I$ and our "circle+whisker" because the point where the two objects connect would have to map to two different coordinates on $I$. But then I think: we're supposed to be able to cover the object with overlapping "coordinate patches", and that seems to me -- incorrectly, no doubt -- to offer a way out.
Aug
14
comment Is this a manifold?
Yep, that's the idea.
Aug
14
asked Is this a manifold?
Aug
13
comment Is this approach for testing orthogonality/parallelity of vectors wrong as I think?
" ... by normalizing both vectors the expression will be as simple as Cosθ ... " Exactly!
Aug
13
answered Determine variables that fit this criterion…
Aug
12
comment Is $y^2=x(x-1)^2$ an immersed submanifold?
If I can ask a qualitative question: Let's say you take that "loop" out of the graph and replace it with a straight line running from (0,0) to (1,0). Do you still have an immersed submanifold? Do you have a submanifold at all?
Aug
12
answered Is this approach for testing orthogonality/parallelity of vectors wrong as I think?
Aug
2
comment Outer interval of circle intersection
What exactly is given in the problem? I would expect it would be the radii of the circles and the distance between the centers. Is that it?
Aug
2
comment Looking for proof that $SO(3)$ is a submanifold of $\mathbb R^3$
@John OK, I guess I'm just going to have to admit it to myself and everyone else: I'm in over my head. In math, I've done partial diff eqs and some complex analysis (contour integrals and the like). That's about as far as I've gone. What do I need to look at to fill in the gap between where I am and where I need to be to understand your proof? Thanks.
Aug
2
answered Oblique projection for which the projection vector is at an angle of 45 degrees
Aug
2
asked Looking for proof that $SO(3)$ is a submanifold of $\mathbb R^3$
Jul
30
comment Geometric interpretation of complex eigenvalues
I think you want "A transforms" instead of "a transforms". Other than that, I really like this.
Jul
28
accepted Derivative of matrix product: is it true that $\frac{d}{dt}(A^TA) = 2A^T \frac{dA}{dt}$?
Jul
28
comment Derivative of matrix product: is it true that $\frac{d}{dt}(A^TA) = 2A^T \frac{dA}{dt}$?
I appreciate your help, but if you feel like giving more, I have spent alot of time looking at exactly the equation you finish with, but I just can't see how to do the requisite index shuffling. If the second term were known to be symmetric, it would be obvious. But I don't see why I can conclude that it is symmetric. Thanks.
Jul
28
asked Derivative of matrix product: is it true that $\frac{d}{dt}(A^TA) = 2A^T \frac{dA}{dt}$?