Homework Statement
f(z) = \frac{z*exp(+i*z)}{z^2+a^2}
Homework Equations
Res(f,z_0) = lim_z->z_0 (1/(m-1)!) d^{m-1}/dz^{m-1} {(z-z_o)^m f(z)}
The Attempt at a Solution
I have no clue how to do this because I don't know how to determine the order of the pole for a function of...
There is a part in the proof of sard's theorem where we restrict our discussion to a point x such that Df(x)=0, and then declare that f ' (x) is a proper (n-1) subspace (f is n-dim). What I don't understand is, the argument then goes by considering any two points in a sub-rectangle around this...
Okay... I have since figured out the solution ... the real question then becomes why is the second equals sign true (below):
×_L=1^k (a_i)_L ((e_i)_σ(1),... (e_i)_σ(k)) = ∏_L=1^k ((a_i)_L)(e_i)_σ(L) = 1 if and only if is identity and 0 otherwise.
where × denotes multiple(indexed) tensor...
{(a_i)_j} is the dual basis to the basis {(e_i)_j}
I want to show that
((a_i)_1) \wedge (a_i)_2 \wedge... \wedge (a_i)_n ((e_i)_1,(e_i)_2,...,(e_i)_n) = 1
this is exercise 4.1(a) from Spivak. So my approach was:
\BigWedge_ L=1^k (a_i)_L ((e_i)_1,...,(e_i)_n) = k! Alt(\BigCross_L=1^k...
Why take the absolute value squared, if the function is real valued? Just take the square of the function, right? Or is that the supremum norm of the function? i.e. |f| = sup(f)
OKAY! We get it, there are a lot of different terms for the same thing, and sometimes the same term means different things. Let's focus on the post's question, not some terms; all we need to know is that the function "f is continuous means f is infinitely differentiable", and that's all we...
Actually, I think he DOES mean smooth, a function is smooth (by definition) if it is infinitely differentiable. So your "counter-example" is not valid; in other words, ##\displaystyle\dfrac{x\cdot\left|x\right|}{2}## is not smooth.
Yeah! That's more what I was looking for, that really cleared things up for me; I was having difficulty because I was stuck on thinking that it only made sense to use trig functions like this if it is to describe something like the way chiro did. But your analogy with the norm of p shows that...
If we have any two orthonormal vectors A and B in R^2 and we wish to describe the circle they create under rigid rotation (i.e. they rotate at a fixed point and their length is preserved), how can we describe any point along this (unit) circle using a linear combination of A and B? I was...
No it doesn't prove that Df(a)=id_n because it assumes that Dg(a)=id_n.
Then s/he goes on to prove IF A is invertible (assumed) and g is invertible then f^-1 = g^-1 A^-1. This DOES NOT prove that A = Id. It merely gives a value for the inverse of f given the composition of invertible...
You assumed exactly what I'm questioning about! WHY can we assume that Dg(a) = I. But that wasn't even quite my question, it was WHY can we assume that Df_a = id_n
Please try to answer the question that was asked,and not just restate the assumption without explanation.
I don't understand why all authors of this proof assume that Df_a = id_n, how doesn't this destroy generality?
For example, see https://www.physicsforums.com/showthread.php?t=476508.
The λ in his post (and the post he quotes) is always Df_a (its not stated in that post, but in the book and the...
I need the electron volt in atomic units (mainly because I need "atomic unit" temperature).
I believe that one atomic unit of temperature is eV (electron volts) / kB (Boltzmann's constant). So I know that the elementary charge e = 1 = h-bar.
How do i get eV, temp, Boltmann's constant...
What sort of answer are you looking for, and what do you think that the solution should be? In other words, are you certain that Mathematica is right but that you might be looking at it in a different light. I find this happens a lot with Mathematica; often, I will use it (or Wolfram Alpha)...