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  1. M

    Induced charge density on uncharged metal sphere placed in uniform Electric field

    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?
  2. M

    Cauchy-Goursat Theorem question

    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.
  3. M

    Cauchy-Goursat Theorem question

    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...
  4. M

    Complex-Valued Functions, limits, and conjugates

    Thanks, I think that kind of proof is something I would think of if I was working with matrices.
  5. M

    Complex-Valued Functions, limits, and conjugates

    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...
  6. M

    Complex-Valued Functions, limits, and conjugates

    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?
  7. M

    Complex-Valued Functions, limits, and conjugates

    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...
  8. M

    Quadratic inequalities for complex variables?

    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?
  9. M

    Quadratic inequalities for complex variables?

    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...
  10. M

    Any good books in Vector Calculus?

    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...
  11. M

    Any good books in Vector Calculus?

    ^ 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...
  12. M

    Any good books in Vector Calculus?

    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...
  13. M

    Any good books in Vector Calculus?

    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...
  14. M

    Any good books in Vector Calculus?

    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...
  15. M

    Any good books in Vector Calculus?

    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...
  16. M

    Diagonalization question

    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)...
  17. M

    Is it natural to be scared?

    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...
  18. M

    Diagonalization question

    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...
  19. M

    Infinite Series question

    Hm, does that mean, the sequence of partial sums of the said series will also have 'm' as its starting index?
  20. M

    Infinite Series question

    I'd really appreciate it if someone would reply. Is my question too trivial?
  21. M

    Is this statement valid?

    About 0.999...=1, is representing it as a geometric infinite series a valid proof?
  22. M

    Infinite Series question

    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...
  23. M

    Funny maths and physics answers

    That circular reasoning almost gave me cancer.
  24. M

    Limit on infinity

    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...
  25. M

    Series, what are they for?

    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...
  26. M

    I don't understand this equation

    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?
  27. M

    How Michio Kaku, Alex Filippenko, Laura Danly, et al. earn their pay

    Whoa, watch out we've got some badass over there.
  28. M

    Infinite Series question

    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.
  29. M

    Infinite Series question

    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}...
  30. M

    Can Beauty be described in terms of mathematics?

    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.
  31. M

    Differentiating f(-x)?

    Chain rule: \frac{d[f(g(x))]}{dx} = \frac{d[f(g(x))]}{d[g(x)]} \frac{d[g(x)]}{dx} . In your case, g(x) = -x.
  32. M

    Infinite Series question

    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...
  33. M

    Limit on infinity

    I think that's same things as what I've said. Haha, sometimes I have a hard time wrapping my head around stuff like this. Thanks anyway.
  34. M

    Limit on infinity

    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...
  35. M

    Limit on infinity

    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...
  36. M

    Dispersion question

    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.
  37. M

    Dispersion question

    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...
  38. M

    Dispersion question

    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...
  39. M

    Dispersion question

    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...
  40. M

    Problem with the principle of induction

    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...
  41. M

    Problem with the principle of induction

    The principle of induction states that: Suppose\ that\ P(n)\ is\ a\ statement\ involving\ a\ general\ positive\ integer\ n.\ Then\ P(n)\ is\ true\ for\ all\ positive\ integers\ 1,2,3,...\ if\ (i)\ P(1)\ is\ true,\ and (ii)\ P(k) \rightarrow P(k+1)\ for\ all\ positive\ integers\ k. I found...
  42. M

    Memorising formulas or deriving them?

    Are you, in any way, trying to reconstruct/rediscover mathematics/science from scratch yourself?
  43. M

    Memorising formulas or deriving them?

    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...
  44. M

    Line Integral question

    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...
  45. M

    Line Integral question

    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}...
  46. M

    Your last month on earth

    Hush, my disciple.
  47. M

    Current electricity

    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...
  48. M

    Your last month on earth

    I'd try starting a new religion on the spot. If there would be people who'd survive, I might just become the new prophet haha.
  49. M

    Ok I slapped someone and I feel bad

    An atheist meeting sounds pretty hippie actually, were there any purpose for that? Do people hold those like it's some hobbyists convention?
  50. M

    What does it mean to have a negative change in current?

    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.
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