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
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...
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...
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
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 ??
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...
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?
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
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...
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.
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...
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 -...
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
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...
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...
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...
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...
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...
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
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
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 +...
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
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)
=...
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.
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???
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.
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)...
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??