Search results

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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??