The set of rational numbers Q is countable, and be therefore be expressed as a countable union Q = Un>=1{rn}. Let R be a metric space with the usual distance. Now define a function f:R->R by setting
f(x) = 1/n if x = rn and 0 if x is irrational
(a) Show that f is continuous at each...
Show the progression (6k +1) (k is an integer) is closed under multiplication:
Firstly I should check that I remember what this means... If it is closed when you multiply any 2 elements together you get an element that is in the set?
So for this I thought just show (6k+1)(6n+1), where k...
Consider a 2f:2f filtering setup with f = 1000mm. The system is illuminated with a uniform plane wave of uni amplitude and wavelength \lambda = 1.0\mum. The input transparency (object) has amplitude transmittance g(x,y) and the spatial filter has amplitude transmittance s(x,y).
Write an...
Thanks for the help. Finally got to the bottom of it with help from a mate.
Mark44 you where right I didn't need to do an epsilon-delta proof, simple a LHS = RHS using limit properties and you hinted at! Just so use to seeing epsilon-delta proofs everywhere in real analysis.
But in short took...
Mark44
I had a look at the properties and found nothing really useful. And I do think that I had to use an epsilon-delta proof >.< (they get very cumbersome very fast!)
hamster143
I don't know where that property comes from and I don't really have any idea how to apply it to the proof.
I...
I'm trying to prove that:
lim(x->a) e^f(x) = e^lim(x->a)f(x) (Assume lim(x->a)f(x) exists)
However I am having great difficulty! My only real approach I have taken is epsilon-delta proof.
if \epsilon > 0 then there exists \delta > 0 such that if |x - a| < \delta then | e^f(x) -...
Show that the circle that is in the intersection of the plane x+y+z=0 and the sphere x2+y2+z2=1 can be expressed as:
x(\vartheta) = (cos(\vartheta)-(3)1/2sin(\vartheta)) / (61/2)y(\vartheta) = (cos(\vartheta)+(3)1/2sin(\vartheta)) / (61/2)z(\vartheta) = -(2cos(\vartheta)) / (61/2)
I'm really...
I would use cylindrical coordinates usually but the question says explicitly to use Cartesian. And also doing the integration you suggested I get -16pi. How can the volume be negative? (I always feel so bad questioning you!)
I end up with 64 times the integral from -(pi/2) to (pi/2) of (cos(t))^4 dt.
Is this right?
*I found the solution to this using trig identities (24pi), but have I ended up with the right integral?*
Find the volume of the region between the two paraboloids z1=2x2+2y2-2 and z2=10-x2-y2 using Cartesian coordinates.
I let z1 = z2 and solved this to get the intersection of the two paraboloids which gave y2+x2=4 (Which I can also use as my domain for integration?)
So the volume of the area...
Using P2(x,y), find a quadratic approximation to ln(1.25) to 4 decimal places.
The original function is f(x,y)=ln(x2 + y2) and is about the point (1,0).
I calculated P2 to be y2-x2+4x-3
however I don't know how to find a quadratic approximation. Do I just set say x=1 and y=.5?
Any...
Find a pair of fields having equal and divergences in some region, having the same values on the boundary of that region, and yet having different curls.
I really have no idea on where to start for this.
Would making up 2 arbitrary fields in spherical co-ordinates work?
a(theta) + b\phi +...
It's been a long time since I've had to do this stuff so bare with me!
Compute and graph the following:
(a) -15+i/4+2i (multiply by the conjugate and I got -2.9+1.7i
(b) (271/3)4 Now for some reason I have a feeling this has 3 solutions using complex numbers but can not figure them out...