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    Wave Equation for a Vibrating String

    Hm, I was able to work through the problem, correcting for the period, and it looks like I'm now only off by a factor of 2. A friend of mine is also having the same problem. Not sure if we can chalk it up textbook error, or just something we're not seeing... Ah nevermind, feeling rather silly...
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    Wave Equation for a Vibrating String

    Hm, ok, fair enough. Thank you so much for all the help. I knew it was going to come down to something silly.
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    Wave Equation for a Vibrating String

    Yep, I just got that. Makes sense now. Thank you so much! Although, can I ask, how would one know (without knowing the answer) that the period is in fact 2l and not l, like I wrongly assumed it was, just from reading the problem/looking at the image. Ah man, feeling rather stupid.
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    Wave Equation for a Vibrating String

    http://en.wikipedia.org/wiki/Fourier_series Everything I've looked up suggests the 2 should be there. I'm also pretty sure I've been using the 2 in all my fourier coefficients calculations up until now. :( I thought the two arises from the fact that ω=2pi/T. I'm sorry. I guess I'll run...
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    Wave Equation for a Vibrating String

    I'm almost certain the 2pi factor belongs there. At least it certainly does in the equations for the coefficients of a fourier series, where the 2pi emerges from ω. Unless for some reason those equations are modified in this case? I know the book tends to leave off the 2pi, but that's when...
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    Wave Equation for a Vibrating String

    So actually, my An includes the prefactor of 2h/\pi^{2}. There's on the other hand, is independent of the constants out front. That's why I was thinking that maybe the two answers are equivalent, in that they factored those pre-factors out, but then the fact that our sin terms are different is...
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    Wave Equation for a Vibrating String

    I used y= 4hx/l for 0<x<l/4 2h-4hx/l for l/4<x<l/2 0 for l/2<x<l So this basically gave me two equations for An, where the first is equal to An = 2/l (from 0 to l/4)∫(4hx/l)*sin(2\pinx/l)dx and the second, for l/4<x<l/2 An = 2/l ∫(2h-4hx/l)*sin(2\pinx/l)dx Do those look ok...
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    Wave Equation for a Vibrating String

    That's exactly what I did, but I guess I just convinced myself there was something I wasn't seeing there. Thanks for the confirmation!
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    Wave Equation for a Vibrating String

    Homework Statement A string of length l has a zero initial velocity and a displacement y_{0}(x) as shown. (This initial displacement might be caused by stopping the string at the center and plucking half of it). Find the displacement as a function of x and t. See the following link for...
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    Tensor Notation for Triple Scalar Product Squared

    Hm, ok, I'm not sure I follow here. If I computed the dot product of two cross products, isn't that wrong, in that I should first compute the cross product of (BxC) x (CxA), before taking the dot product that I did in the step before?
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    Tensor Notation for Triple Scalar Product Squared

    Thanks for the response! Ok, well, I know that (AxB)_{i} = ε_{ijk}A_{j}B_{k}. And I can say that, (BxC)_{i} = ε_{ilm}B_{l}C_{m}. If I write that all as one term, ε_{ijk}A_{j}B_{k}ε_{ilm}B_{l}C_{m} then that equals, (δ_{jl}δ_{km} - δ_{jm}δ_{kl})A_{j}B_{k}B_{l}C_{m} and I know...
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    Tensor Notation for Triple Scalar Product Squared

    Homework Statement Hi all, Here's the problem: Prove, in tensor notation, that the triple scalar product of (A x B), (B x C), and (C x A), is equal to the square of the triple scalar product of A, B, and C. Homework Equations The Attempt at a Solution I started by looking at the triple...
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    Center of Mass of a Cardioid

    Homework Statement Find the center of mass of a solid of density \delta = 1 enclosed by the spherical coordinate surface \rho = 1-cos\phi. Homework Equations The Attempt at a Solution I'm a bit confused about how to start here, mainly because the surface is defined by spherical...
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    Setting up Triple Integrals over a bounded region

    Ah ok. How's this? $$\int_{0}^{\pi/2}\int_{0}^{1}\int_{0}^{r} r(6+4rsinθ) dzdrdθ$$ Thanks again for all the help by the way. Definitely starting to feel a bit better about these conversions.
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    Setting up Triple Integrals over a bounded region

    Yeah, you're absolutely right. I realized that right after I submitted that last post. Here are my new integrals. Cartesian Coordinates: $$\int_{0}^{1}\int_{0}^{\sqrt{1-y^{2}}}\int_{0}^{\sqrt{x^{2}+y^{2}}} 6+4y dzdxdy$$ Cylindrical Coordinates: I'm not sure about this one, particularly...
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    Setting up Triple Integrals over a bounded region

    Hm, I am thoroughly confused now, haha. I thought the region was the inside of the cone? What I'm picturing: If I drew a cylinder and placed a cone directly inside it, so that the top of the cone meets the top of the cylinder, the region I'm looking at is inside the cone.
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    Setting up Triple Integrals over a bounded region

    Should your theta limits be from 0 to $$\theta/2$$ since it's only the first octant? So this is what my integral in cartesian coordinates looks like: $$\int_{0}^{1}\int_{0}^{\sqrt{1-y^{2}}}\int_{\sqrt{x^{2}+y^{2}}}^{1} 6+4y dzdxdy$$ I guess I don't quite understand ##\rho##, in that case. I...
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    Volume enclosed by a spherical coordinate surface

    Thank you so much. I really appreciate all the help.
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    Volume enclosed by a spherical coordinate surface

    Hehe fair enough. I get 2pi^2. Looks like a nice and neat enough answer -- hopefully that's what you got! I went through the problem a second time, though, and now I'm finding myself confused about one of the trig substitutions I made. Once I've expanded (1-cos(2ϕ)/2)^2, I'm left with a few...
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    Setting up Triple Integrals over a bounded region

    Homework Statement Set up triple integrals for the integral of f(x,y,z)=6+4y over the region in the first octant that is bounded by the cone z=(x^2+y^2), the cylinder x^2+y^2=1 and the coordinate planes in rectangular, cylindrical, and spherical coordinates. Homework Equations...
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    Volume enclosed by a spherical coordinate surface

    Ok, so when I expand (1-cos(2\phi)/2)^2, I get 1/4 - (cos(2\phi))/2 + (cos^2(2\phi))/4 Replacing the last term with a half-angle formula, I have 1/4 - (cos(2\phi))/2 +1/8 + (cos(2\phi))/8 At that point, I take the integral of that with respect to \phi. I wind up with 1/4*\phi -...
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    Volume enclosed by a spherical coordinate surface

    Ok, so as far as my limits of integration go, those are ok? I also get (1/3)\rho^3, and evaluating at 2sin\phi I get 8/3∫∫sin^4(\phi) d\phid\theta. I'm not sure where I ought to be getting 16 from? As for ∫∫sin^4(\phi) d\phid\theta, I think I'm simply screwing up some basic calculus. I...
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    Volume enclosed by a spherical coordinate surface

    Homework Statement Find the volume enclosed by the spherical coordinate surface ρ = 2sin∅ Homework Equations dV = ∫∫∫(ρ^2)sin∅dρd∅dθ The Attempt at a Solution (Sorry about my notation!) Alright, here's what I've done so far... Since the region is a torus, centered...
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    Volume of a region bounded by a surface and planes

    Oh wow, thank you! I couldn't help but feel like I didn't quite have it. Thanks for checking it out!
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    Volume of a region bounded by a surface and planes

    Homework Statement Find the volume of the region bounded by the cylinder x^2 + y^2 =4 and the planes z=0, and x+z=3. Homework Equations V = ∫∫∫dzdxdy V=∫∫∫rdrdθ The Attempt at a Solution Alright, so I feel as though I'm missing a step somewhere along the way, but here's what I've gotten...
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    Proof - Express in Clyndrical Coordinates

    I think that was my hang up exactly. Thanks!
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    Proof - Express in Clyndrical Coordinates

    Proof -- Express in Clyndrical Coordinates Homework Statement Show that when you express ds^2 = dx^2 + dy^2 +dz^2 in cylindrical coordinates, you get ds^2 = dr^2 + r^2d^2 + dz^2. Homework Equations x=rcosθ y=rsinθ z=z The Attempt at a Solution EDIT// I was really over thinking...
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    Parametrization of a line formed by 3 points

    A, B, and P are co-linear, right? And I'm saying that based on the fact that AP and BP are anti-parallel vectors. And since they all lie along the same line, I wouldn't need three points, and so a parametrization using AP is just as valid as one using BP, even though they appear different? I'm...
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    Parametrization of a line formed by 3 points

    Homework Statement Find a parametrization of the equation of the line formed by the points A, B, and P. A(2,-1,3) B(4,3,1) P(3,1,2) Homework Equations x=x_0+v_1*t y=y_0+v_2*t z=z_0+v_3*t The Attempt at a Solution Alright, so, I've already determined that P is equidistant from the points...
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    Find the Sum of The Geometric Series

    Thanks, I think I got it. I just get totally thrown off when the index isn't in the correct form, or when the series is blatantly obvious, haha. Thank you, though!
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