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  1. C

    Matrices with complex entries

    Homework Statement Let B be an m×n matrix with complex entries. Then by B* we denote the n×m matrix that is obtained by forming the transpose of B followed by taking the complex conjugate of each entry. For an n × n matrix A with complex entries, prove that if u*Au = 0 for all n × 1 column...
  2. C

    Fourier sine transform for Wave Equation

    Homework Statement Find the solution u, via the Fourier sine/cosine transform, given: u_{tt}-c^{2}u_{xx}=0 IC: u(x,0) = u_{t}(x,0)=0 BC: u(x,t) bounded as x\rightarrow \infty , u_{x}(0,t) = g(t) 2. The attempt at a solution Taking the Fourier transform of the PDE, IC and BC...
  3. C

    Two springs connected by a spring

    Homework Statement Two masses, m1 and m2, are connected to each other by a spring with a spring constant k. The system moves freely on a horizontal frictionless plane. Find the natural frequency of oscillation. Homework Equations F = -kx F = ma The Attempt at a Solution Let m1 be the mass...
  4. C

    Finite approximation of PDEs

    Homework Statement Given u_tt = F(x,t,u,u_x, u_xx), give the finite difference approximation of the pde (ie using u_x = (u(x + dx; t) - u(x - dx; t))/(2dx) etc.) Homework Equations Well, clearly, u_x = (u(x + dx; t) - u(x - dx; t))/(2dx) The Attempt at a Solution I really have no idea how...
  5. C

    Introductory PDE (diffusion equation)

    Homework Statement u_t = -{{u_{x}}_{x}} u(x,0) = e^{-x^2} Homework Equations The Attempt at a Solution The initial state is a bell curve centred at x=0. The second partial derivative of u at t=0 is {4x^2}{e^{-x^2}}, which is a Gaussian function, which means nothing to me other than its...
  6. C

    Double integrals in polar coordinates

    Homework Statement Find \int{\int_{D}x dA} where D is the region in Q1 between the circles x2+y2=4 and x2+y2=2x using only polar coordinates. The Attempt at a Solution Well, the two circles give me r=2 and r=2 cos \theta, and the integrand is going to be r2cos \theta, but I have no...
  7. C

    Max/min with partial derivatives

    Homework Statement Show that f(x,y) = -(x^2 - 1)^2 - (yx^2-x-1)^2 has only two critical points, and both are maxima. The Attempt at a Solution Set partial derivatives (wrt x and y) to zero to find critical pts. f_x = -2(x^2 - 1)(2x) - 2(yx^2 - x - 1)(2xy - 1) = 0 f_y = -2(yx^2 - x -...
  8. C

    Gradient vector for polar coordinates

    Homework Statement Find the gradient vector of: g(r, \theta) = e^{-r} sin \theta Homework Equations The Attempt at a Solution I know how to get gradients for Cartesian - partially derive the equation of the surface wrt each variable. But I have no idea how to do it for...
  9. C

    Partial derivatives of implicitly defined functions

    Homework Statement If the equations x^2 - 2(y^2)(s^2)t - 2st^2 = 1 x^2 + 2(y^2)(s^2)t + 5st^2 = 1 define s and t as functions of x and y, find \partial^2 t / \partial y^2 The Attempt at a Solution Equating the two, we get 4y^2*s^2*t = -7s*t^2. My main problem is, as simple as this...
  10. C

    Work along a line integral

    Homework Statement A particle is attracted towards the origin by a force proportional to the cube of its distance from the origin. How much work is done in moving the particle from the origin to the point (2,4) along the path y = x^2 assuming a coefficient of friction \mu between the particle...
  11. C

    Surface area of a curve around the x-axis

    Homework Statement Find the surface area traced out when the curve 8y^2 = x^2(1-x^2) is revolved around the x-axis. The Attempt at a Solution x-axis means y = 0 When y = 0, x = 0, -1 or 1. Since this curve is "the infinity symbol", the curve has symmetry at x = 0. Isolating y, 8y^2 =...
  12. C

    Surface integral in spherical coordinates question

    Homework Statement Find the surface area of the portion of the sphere x^2 + y^2 + z^2 = 3c^2 within the paraboloid 2cz = x^2 + y^2 using spherical coordinates. (c is a constant) Homework Equations The Attempt at a Solution I converted all the x's to \rho sin\phi cos\theta, y's to \rho...
  13. C

    Surface integral/point charge question

    Homework Statement A uniform surface charge lies in the region z = 0 for x^2 + y^2 > a^2, and z = \sqrt{a^2-x^2-y^2} for x^2 + y^2 \leq a^2. Find the force on a unit charge placed at the point (0,0,b) Homework Equations dF = G (\delta)(vector of point to surface) / (magnitude of surface)^3...
  14. C

    Centroid of a triangle using Green's theorem

    Homework Statement Given a curve C that starts from the origin, goes to (1,0) then goes to (0,1), then back to the origin, find the centroid of the enclosed area D. Homework Equations \bar{x} = {1/(2A)}*\int_C {x^2 dy} \bar{y} = -{1/(2A)}*\int_C {y^2 dx} The Attempt at a Solution Well...
  15. C

    Expected value problem (probability)

    Homework Statement Given Y1 and Y2 are integer values, where 0\leqY1\leq3, 0\leqY2\leq3, 1\leqY1+Y2\leq3 p(Y1, Y2) = \frac{{4 \choose y_1}{3 \choose y_2}{2 \choose {3-y_1-y_2}}}{{9 \choose 3}} Find E(Y1+Y2) and V(Y1+Y2) Homework Equations E(Y1+Y2) = E1(Y1)+E2(Y2) E1(Y1) =...
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