Search results

  1. A

    Pauli-Lubanski pseudovector

    Hi can anyone help me prove the result of W2 of the Pauli-Lubanski pseudovector : This is very new to me and I've read I must use terms such as J13 and P3 Where totally antisymmetric symbol is defined by: \epsilon1234=1 and \epsilon1243=-1
  2. A

    Differential coefficients & metric tensor (Urgent)

    Consider a flat 2-dimensional plane. This can be described by standard Cartesian coordinates (x,y). We establish a oblique set of axes labelled p and q. p coincides with x but q is at an angle θ to the x-axis. At any point A has unambiguous co-ordinates (x,y) in the Cartesian system. In...
  3. A

    Transformation from Cartesian to spherical polar coordinates

    Transformation from Cartesian to spherical polar coordinates In dimensions: x=r sinθ cos \varphi and y= r sin θ sin \varphi z=r cos θ Show one example of: ∂z\alpha/ ∂xμ . ∂xμ/ ∂z\alpha = δ\alpha\beta Now here is my answer: δyx=(∂y/∂r . ∂r/∂x) + (∂y/∂θ . ∂θ/∂x) + (∂y/∂\varphi...
  4. A

    Consider the vector Vμ(3,1)

    Consider the vector Vμ(3,1) Find VμVμ Now here is my attempt Using the following: Vμ=gμvVv I could calculate: Vv=(3,1) But how can I now manipulate this to obtain Vμ
  5. A

    Basic Probability

    In a shop 1/3 of the TV's are brand A, the probability that a brand A TV is faulty is 0.2. What is the probability person 1 buys a faulty brand A TV? Here's my attempt Im not sure if it seems almost too easy but should I use the rule: P(A) x P(B) = P(AnB) Giving: 0.33333x0.2=0.066666 ??
  6. A

    Model for single lane of traffic

    A model for a single lane of traffic is given by the following pde p dv/dx + v dp/dx + dp/dt = 0 Where: v = kx/p Show that dp/dt = -k Here is my attempt v = kx/p dv/dx = k/p p= kx/v dp/dx = k/v Substituting into original pde: p (k/p) + v(k/v) + dp/dt=0...
  7. A

    Solve the first order hyperbolic equation

    Solve the first order hyperbolic equation 3 du/dx + 2x du/dt =2u With initial condition: u(x,0) = 2x+4 My attempt at a solution I usually adopt the method of characteristics: dx/a = dt/b = du/c So from the above: a=3, b=2x and c=2u am I on the right track here?
  8. A

    Differentiating exponential

    Differentiate the following: u = 2 (x-t2 / 3)et/3 Here is my attempt I have to find du/dx and du/dt du/dx = 2 (x-t2 / 3)et/3 du/dt = -4t/9 (x-t2 / 3)et/3
  9. A

    Simplify the follwoing Equation

    Sounds waves in a pipe of varying cross-section are described by the equation V2 d/dx (1/A dAu/dx) = d2u/dt2 Where A = 0.2+0.3x So first I substituted A into the equation: V2 d/dx (1/(0.2+0.3x) d(0.2+0.3x)u/dx) = d2u/dt2 V2 d/dx (0.3u/(0.2+0.3x) du/dx) = d2u/dt2 This is as far as I can...
  10. A

    Simultaneous Congruence

    Solve the following simultaneous congruence 3x= 14 mod 17 7x= 13 mod 31 My attempt I have no problems solving when in the form: x= 2 mod 13 x= 4 mod 27 for example but am confused by the above question, any help would be great.
  11. A

    Need Matrix help

    Find the determinant and trace of matrix, and use these and that 6 is an eigenvalue to find all eigenvalues of matrix ( 4,-4,-8 -2, 2, 6 0, 0, -1) Here's my attempt determinant = 4((2x-1)-(6x0) - (-4)((-2x-1)-(6x0)) +(-8)((-2x0-(2x0)) =-8+8+0...
  12. A

    Parametric Form

    Calculate in parametric form and describe how the planes intersect Where: P1 = x-3y+5z=6 P2 = 2x-7y+9z=2 My attempt Put planes in matrix form: 1, -3, 5, 6 2, -7, 9, 2 Find Echelon Form 1, -3, 5, 6 0, -1, -1, -10 Z = free variable = a So: -y-z=-10 y = 10 -...
  13. A

    Vector Subspace

    Is the following a vector subspace W1 {(x1,x2,x3): x1=x2=x3=0} I usually begin my attempt by finding two members of the set then check which axioms are valid.However I can only think of 1: (0,0,0) Any help would be great thank you
  14. A

    Galois Field (3)

    Describe how the field GF(3) may be extended by postulating the existence of a root a of q(x) and list all the elements of this larger field. Show that a is not a generator of the extended field. q(x) = x2+1 My attempt First replace x with a we obtain: 0 = a2+1 a2=-1 a2=2...
  15. A

    Linear Transformations

    Identify the Hermite form of the following linear transformations and the basis for its kernel (x,y,z) = (x-y+2z,2x+y-z,-3x-6y+9z) So when finding basis for kernel we have to set equal to 0, giving: x-y+2z=0 (1) 2x+y-z=0 (2) -3x-6y+9z=0...
  16. A

    Wave Equation

    Wave Equation (urgent) Sounds waves in a pipe of varying cross-section are described by the wave equation v2 d/dx .(1/A.dAu/dx) = d2u /dt2 Where A = 0.2 +0.3x simplify the equation My attempt at a solution Sub in A: v2 d/dx ( 1/(0.2+0.3x) . d(0.2+0.3x)u/dx) =d2u/dt2 Not to...
  17. A

    Harmonic Function

    Show that the following is not a harmonic function: u = x+y2 This is what I know: du/dx = 1 this must be equal to dv/dy , therefore: v = y du/dy = 2y this must be equal to -dv/dx, therefore: v = -2xy Putting expressions together: v = y-2xy. Not sure where to go...
  18. A

    L'Hospital's Rule does not work

    Confirm that L'Hospital's rule does not for the following: SQRT(8x+1) / SQRT (x+1) As x tends to infinity Attempt at a solution Given that: f(x) = SQRT (8x+1) f'(x) = 1/2 SQRT(8x+1) g(x) = SQRT (x+1) g'(x) = 1/2 SQRT(x+1) Now I am unsure where to go from here...
  19. A

    Finding singularity

    Find any singularities in the following,say whether they are removable or not f(x) = (x2 + 2)/x Attempt at a solution when f(0) = ((0)2 + 2)/0 =2/0 which isn't defined So 0 is a singularity
  20. A

    Diffusion Equation

    Water pressure in a filtration bed is given by the following diffusion equation: \frac{\partial p}{\partial t} = -k Homework Statement Homework Equations The Attempt at a Solution
  21. A

    Partial differential equation

    A model for single land of traffic is given below: p.dv/dx + v. dp/dx + dp/dt = 0 Where v = kx/p Show that with the expression for p, the PDE becomes: dp/dt = -k Here is my attempt v = kx/p dv/dx = k/p Sub into pde: p (k/p) + (kx/p) .dp/dx + dp/dt = 0 k +...
  22. A

    Differentiate the following

    Differentiate the following: 3(2x2 +1 ) Heres my attempt 4x.3(2x2+1)^0 anything the the power of 0 = 1 So we have: 12x.(1) = 12x Is this correct???
  23. A

    Find the characteristic lines

    Find the characteristic lines for the equation: 2 du/dx + 8x du/dt = 16x Here's my attempt a = 2 b = 8x c = 16x Using dt/dx = b/a = 8x/2 = 4x t = 2x2 + C C = t1 -2x12 Hence the characteristic is: t = 2x2 + t1 - 2x12
  24. A

    Use L'Hospital's Rule to find limit

    Use L'Hospital's rule to find the limit of the following: (SQRT (3+x) -2)/(x-1) As x tends to 1 Here's my attempt Let f(x) = SQRT(3+x) - 2 g(x) = x-1 f'(x) = 1/2 (SQRT(3+x)) g'(x) = 1 Therefore f'(x)/g'(x) is: 1/2(SQRT(3+x)/1 =1/2(SQRT(3+1) =...
  25. A

    Find any singularities in the folloqing fucntion

    Find any singularities in the following function, say whether they are removable or non-removable. Indicate the limit of f(x) as x approaches the singularity. (x^(2) + x + 1) /( x-1) Not to sure where to start, as the numerator does not factorise easily.
  26. A

    Improper Integral

    Decide if the following improper integral converges or not Integrate 1/x2 limits are 1 0 Now by simple integration we can see that: [x-2] = -x-1 Substituting in 1 and 0: -(1/1) - 0 So series converges to -1 Is this correct???
  27. A

    Find equation of the tangent plane

    Find the equation of the tangent plane to the level surface of the scalar field \xi(x,y,z) = x2+y2+z2 at the point (1,1,2) Looking the work through this question with someone, not to sure where to start.
  28. A

    Find equation of the line

    Find the equation of the straight line which is perpendicular to the plane 4x+3y+2z=1 Which goes through the point (1,1,7) Is the point (5,7,15) on this line? By inspection we can see direction of the normal to the plane: (4,3,2) Therefore equation of straight line is: r(t)...
  29. A

    Find any fixed points for the following mapping:

    Find any fixed points for following mapping f(z) = z2 - z + 1 Map onto the same point gives: z = z2 - z + 1 0 = z2 - 2z + 1 Therefore z = 1 and z = 1
  30. A

    For scalar field find grad

    For the following scalar field: \psi(x,y,z) = (y-1)z2 Find grad \psi Here is my attempt at: Multiplying out brackets: yz2 - z2 Therefore grad \psi = 0+Z2 J -2ZK Is this correct??