I might be necroing, but this question is also relevant for me. I am also reading the same example from Griffiths. If the uncharged metal sphere has a zero potential, then why does it imply that the xy plane has zero potential?
You are not given any information about the charge density right?
Is my question too cluttered and messy? I'll revise my question if it is. Because I'd really appreciate it if someone could give their insight on this.
I was reading about Cauchy-Goursat theorem and one step in the proof stumped me. It's the easier one, that is, Cauchy's proof that requires the complex valued function f be analytic in R, and f' to be continuous throughout the region R interior to and on some simple closed contour C. So that the...
Hm, well, I also thought so. But see, I was trying to figure out how to show that an operator like L = i(d^3/dx^3) is hermitian for functions f(x) in the interval -(infinity)< x < (infinity) where as f approaches infinity, it also approaches zero. I tried using the straightforward method where I...
Thanks, that's pretty straightforward huh.
Anyway, I wonder if you guys can answer a barely tangential question? Let's say I have a complex valued function f(x) of a real variable. If the limit f(x) as x-> infinity is zero, are the derivatives of f(x) as x-> infinity also zero?
I have a question about complex valued functions, say f(z) where z=x+iy is a complex variable.
Can every such complex valued function be represented by:
f(z)=u(x,y)+iv(x,y)?
Also, is the limit of the conjugate such a function equal to the conjugate of the limit of the function?
Something like...
That's cool, I get it now. Then that means the roots are either purely imaginary or complex (but not purely real) right? Then why is it required that the discriminant be less than zero? Is it because it will make sure that part of the solution will have an imaginary part?
Hello, I was looking at Riley's solution manual for this specific problem. Along the way, he ended up with a quadratic inequality:
If you looked at the image, he said it is given that λ is real, but he asserted that λ has no real roots because of the inequality. Doesn't that mean λ is...
I have studied multivariable calculus of course, along with some basic calculus of vector fields, and I am reviewing some ODEs too (I am forgetting the theory, for some reason, I don't like the idea that I am forgetting what little I know about math theory).
Well, coincidentally, I am reviewing...
^ I can understand these, at least mechanically, but well, I guess, what I am desperately trying to find is a general method that applies to all cases; something in which the usual basic methods is actually a special case of it. What you outlined above is intuitive, but it doesn't seem to fit...
Oh this, I did some searching. Apparently, what I said above about straight forward integrating over the surface S isn't exactly correct. You parametrize the surface (to vector valued function), I think, usually using spherical/cylindrical coordinates then transform it to rectangular components...
Yes, exactly. But the problem here is, he didn't actually tell that the surface integral is a double integral (at least from what I currently know) that can be solved by actually integrating over the surface. In the textbook, after constructing something that's starting to look like a Riemann...
I am saying all those given that my textbook only offers an introduction to calculus of vector fields, so I know asking for a more general approach is a bit unreasonable.
Anyway here's an example of one:
For a function of three variables G(x,y,z), the surface integral of G over the surface S...
I have been restudying vector calculus, especially on topics pertaining to line integrals, surface integrals (and the accompanying vector forms). One problem I have encountered from the book I have been using is that it seems there are some theorems and results that are only restricted to...
Hm, yes, if I understood the question correctly, then yes I am working with matrices and vectors restricted to reals. About change of basis, I'm not quite sure about that, I have learned how to make orthonormal basis vectors out of linearly independent vectors though (Gram-Schmidt method)...
For a person observing another one moving relative to him, the time interval for an event 'in there' (ie, the event in which the time is proper) is slower for a person observing that, from what I remember in basic special relativity that much is true. Your next statements though, seems a little...
Let's say I have a matrix M such that for vectors R and r in xy-coordinate system:
R=Mr
Suppose we diagonalized it so that there is another matrix D such that for vectors R' (which is also R) and r' (which is also r) in x'y'-coordinate system:
R'=Dr'
D is a matrix with zero elements except for...
Sorry for necroing an old thread, but I've gone back, and am reviewing my infinite series knowledge. I thought the previous discussions here might be useful for my query.
My question is, can we generalize an infinite series so that it can be denoted by:
\Sigma ^{\infty} _{n=m}a_n where m is any...
What if the limit of S(n) as n approaches infinity is known to be S, I've read that the function S(n-1) as n approaches infinity is also S. How do we verify this? My idea is that, after finding an N such that: |S(n)-S|<ε whenever n>N; and we adjust ε so that we find a 1<N.
So that...
I think defining the sum of an infinite series using the associated sequence of partial sums is pretty ingenious. The limit of a sequence is already defined, which is the limit of the sequence function, I think this makes everything pretty neat.
You can think of the sequence of partial sums as...
You might want to give more details, what is the orientation of the wire? This might be a circular wire, but we are not sure. Where are those angles measured from?
Ah, yes, yes, now that I have completely rewritten the theorem, and with your clarification I now understand better.
I sometimes feel like I'm hunting Easter eggs when I'm looking at theorems.
Sorry, I might not remember it very well, I'll just do a direct quote from a textbook then.
Woah, I now realize I'm completely wrong, my 'theorem' up there was a product of my cluttered head, so please excuse it. Here goes the correct theorem:
If \Sigma _{n=1} ^{\infty} a_n and \Sigma _{n=1}...
Isn't the occurrence of the golden ratio to 'beautiful' things subject to confirmation bias? We might have an 'ugly' thing that still has golden ratio in it, but we might not be counting these instances.
For the series such that: \Sigma _{n=1} ^{\infty} a_n =\Sigma _{n=1} ^{\infty} b_n A certain theorem says that these series are equal even if a_n = b_n only for n>m. That is, even if two infinite series differ for a finite number of terms, it will still converge for the same sum. I am thinking...
Hm, well I could think that obviously if n≥N then this implies n+1≥N (if N>0 and n is restricted to positive integers). Does that mean I can say: if for any ε>0, and N>0 such that |S(n)-S|<ε whenever n≥N is true by hypothesis, then |S(n+1)-S|<ε whenever n+1≥N is also true?
The catch is I've...
I have a question about limits at infinity, particularly, about a limit I have seen in the context of infinite series convergence.
Let's say we have an infinite series where the the sequence of partial sums is given by {S(n)} and also, it is convergent and the sum is equal to S. Then we know...
No problem, I was at least relieved that I got the simple stuff first. As for the frequency dependence, I guess I'll take it for granted for now. Thanks.
I get the notion that it's better to refer to frequency when we are talking about specific EM wave in the spectrum since it doesn't change when it propagates through mediums.
Thanks anyway, I think I got it, the expression is valid only for a specific frequency of a wave. Now, I guess my real...
Looking back, I think the texts are actually referring to the wavelength in vacuum. The expression above seems to be valid at least if we are concerned with the wavelength of the light as it passes through the medium. I'm now half convinced that we cannot conclude anything about the dependence...
From the elementary texts, dispersion is the phenomenon where the refractive index of a medium depends on the wavelength of electromagnetic radiation through it. From what I've read, it is the wavelength of the radiation in vacuum. Also, it is said that the refractive index increases with...
Well, I get that. Just so I'm making myself clear, I actually thought about the circularity for some time and I tried to resolve it by adhering strictly on what the principle says, and that (i) and (ii) should be true; (ii) which states that the implication itself must be true and not just...
Every time a very neat theorem is presented in front of me, I'm having similar thoughts. It's just a bit of a shame though, a lot of my peers, or even my instructors seem to see theorems only as a very useful tool, and wouldn't really care how they are proved or expound on them.
About...
Actually, I've found out that to 'parametrize' the variables into x=t, y=t is a more comforting method to do it. At least intuitively, I see it as tracing the path of integration when we set the x and y variables into that parametric equation.
Edit: Yes, I didn't see it, but I was using x as...
Suppose I have a vector V and I want to compute for the line integral from point (1,1,0) to point (2,2,0) and I take the path of the least distance (one that traces the identity function).
The line integral is of the form:
\int _a ^b \vec{V} \cdot d\vec{l}
Where:
x=y, \ d\vec{l}...
You might want to look up Kirchoff's Rule, also. I think you are trying to ask what the ammeter readings will be in that circuit diagram, it can be answered by the said computational method. Intuitively, at the split, the current reading should be lower, this is a direct consequence of the law...
So, at the negative part of the wave where the slope is negative, is that where the current is increasing and but goes the opposite direction (intuitively, becoming more and more negative)?
That means there are different ways to interpret a di/dt or dq/dt that is negative.