if I shower rain drops on a hemi sphere Then , amount of area covered by the rain drops equal pi * r^2 . I am not able to understand why is it like this and not 4 * pi *$ r^2$ / 2.
@Sarabsrimt There is a great confusion in what the "amount of area covered by raindrops" means. If it means the surface area of the hemisphere, then in fact what you say is correct.
@Sarabsrimt But
If you imagine looking at the hemisphere from the top (where the raindrops are coming from) then from the top you can't see that the hemisphere has any height, so it basically looks like a circle of radius $r$ (try looking at a hemisphere from the top, it looks like a circle). Then the area is pi*r^2 "from the top".
which the raindrops fall into. But this is all so imprecise because of what I said in my first sentence.
I can see what is going on somewhat : see, the photons are hitting the hemisphere surface, and what matters is the kind of "cross-section" area which is hit by the photons, because the 3-dimensional nature of the hemisphere doesn't matter . For example, imagine you did not have a hemisphere but rather a long cone with radius $r$ and height $l$ very large. If you shine photons from the top, then they still only penetrate a region that has area pi r^2 from the top.
So there's no difference "from the top" between a long cone , and a hemisphere, they still have the same cross section which is hit by photons i.e. pi * r^2, although the surface area of both which is hit, could be very different.
Is there way to share a picture with you here. I can just write the maths there, since it requires loads of matrices. Things would be easier fir you and I will be able to explain quickly
But if I guess you said you use an iPad. The whole doubt is written in a notes file. Will it work if I upload the notes file and then you can see it or not?
@Shashaank I dont have my IPad with me right now, so it is better if you upload to Google drive. Or see how it can be done with imgur, I thought you can do it using that as well. I have never done it, unfortunately.
Basically if you can upload the image anywhere online it would be helpful.
Then we can get the link from that and post it as a hyperlink here.
Yeh why is $S T S^-1$ not diagonal when T is not written in the standard basis but Somme other basis say B and S is the matrix of eigenvectors of T written in basis B
If you would have written T in standard basis and S was the matrix of eigenvectors of T written in standard basis then the above expression would give a diagonal matrix
Why doesn’t the above expression give a diagonal matrix when T is written in basis B ( other than standard basis) and S is the transformation matrix from the basis of eigenvectors vectors to this other basis , B?
