Homework Statement
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Given the dyad formed by two arbitrary position vector fields, u and v, use indicial notation in Cartesian coordinates to prove:
$$\nabla^2 ({\vec u \vec v}) = \vec v \nabla^2 {\vec u} + \vec u \nabla^2 {\vec v} + 2\nabla {\vec u} \cdot {(\nabla \vec v)}^T
$$
Homework...
Homework Statement
The motion described by the Eulerian velocity, v, is given as v = c(x1e1  x2e2) is
called stagnation point flow. How long will it take a particle traveling on a streamline to reach the stagnation point?
Homework Equations
The Attempt at a Solution
I really...
Homework Statement
Let R, L, C, K be suitable dimensional constants. In a certain motion of a fluid,
the mass density function ρ is given in the Eulerian description by ρ = R*(x1^2+L^2).
Is the velocity field v1 = C/(R*(x1^2+L^2)) e1 a possible velocity field?
Is v2 = K*exp(x1/L)...
Homework Statement
Hello, I have a question on how to calculate the equivalent/hydraulic diameter for a problem that I'm doing.
The problem states that I should use the crosssectional flow area shown with an equivalent diameter based upon a closed rectangle (entire perimeter)...
Homework Statement
A bare slab fuel element is 0.2 in thick. It has a kf = 10 Btu/(hr*ft*F) and a q''' = 5e7 Btu/(hr*ft^3). At x=0 (left hand face), h_lhs = 400 Btu/(hr*ft^2*F) with a fluid temp of 700F. At x=0.2 in (right hand face) the heat transfer coefficient is h_rhs = 300 Btu/(hr*ft*F)...
Homework Statement
I'm having a problem calculating the I value for this beam. I'm not sure on how to account for the missing circle in the front square and how to account for the beam changing from a 1" x 1" square to a 5" x 0.5" beam
Homework Equations
Ixx = bh^12/12
The...
Homework Statement
I'm supposed to design and analyze a shaft including all features to retain the shaft in the bearings and attach the coupling and sheave and the shaft shouldn't fail in static loading or through fatigue.
Homework Equations3
The flexible coupling attaches your shaft to the...
Homework Statement
A pressurizer is designed to accommodate an outsurge of 30,800 lbm and an insurge of 20900 lbm. The mass of the water necessary to cover the electric heaters is 4020 lbm. The system's operating pressure is 2250 psia. The hot leg is 600F and and the cold leg is 550F (For the...
Homework Statement
f(t) is given as:
from 0 to 0.2s, f(t) = 5
from 0.2s to 0.6s, f(t) = 0
from 0.6s to 0.8s, f(t) = 5,
etc
Homework Equations
for an odd function
a0 = 2/p * integral(from p/2 to p/2) of f(t) dt
bn = 4/p * integral(from 0 to p/2) of f(t)*sin(2*pi*n*t/p) dt
The Attempt at...
Homework Statement
x'' + 2\zeta \omega_{n} x' + \omega_{n}^2 x = u_{s}(t)
zeta is underdamped and u_{s}(t) is the unit step function and \omega_n is the natural frequency and there are zero initial conditions. Find the total response via the convolution integral.
Homework...
For A5I
x1 = x3
x2 = x3
so [1;1;1] is the eigenvector for \lambda = 5
[2,1,2;2,1,2;2,1,2] * [1;1;1] = [5;5;5]
But I'm still not sure how to find the eigenvectors for A0I
Homework Statement
A = [2,1,2;2,1,2;2,1,2]
Find the Eigenvectors of A
Homework Equations
The Attempt at a Solution
First I found the eigenvalues of A
det(A  \lambda I) = 0
\lambda = 0,5
__________________________________________
A5I...
I'm currently doing a project in my materials class and I wanted to check if these two material indices were right. The objectives are:
1. The material must not fracture
2. The material must be strong (no deformations)
Material Index 1

Considering a yieldbeforebreak...
Homework Statement
Find the radius of convergence and the interval of convergence for
\sum_{n=0}^\infty \frac{x^n}{n3^n}
Homework Equations
The Attempt at a Solution
Ok, so I first applied the ratio test.
\lim_{n\rightarrow\infty}

\frac{x^{n+1}}{(n+1)3^(n+1)} /
\frac{x^n}{n3^n}
...
Homework Statement
\int sin^3(2x) dx
Homework Equations
sin^2(x) + cos^2(x) = 1
The Attempt at a Solution
First I try to get the integral in the form of \intsin^3(u) and I do this by usubstitution.
\int sin^3(2x) dx
u = 2x
du = 2dx
dx = du/2
So the new integral looks as such...
Homework Statement
Determine the area of the parallelogram spanned by the vectors
< 0, 9, 6 > and < −10, −6, −4 >
Homework Equations
Area = A X B
The cross product of < 0, 9, 6 > and < −10, −6, −4 > = 0i  60j + 90k
The Attempt at a Solution
I know the area is the cross...
Homework Statement
A heliumneon laser of the type often found in physics laboratories has a beam power of 3.50 mW at a wavelength of 633 nm. The beam is focused by a lens to a circular spot whose effective diameter may be taken to be equal to 2.00 wavelengths. Calculate the intensity of the...
Homework Statement
Determine the Taylor Series for f(x) = ln(13x) about x = 0
Homework Equations
ln(1+x) = \sum\fract(1)^n^+^1 x^n /{n}
The Attempt at a Solution
ln(13x) = ln(1+(3x))
ln(1+(3x)) = \sum\fract(1)^n^+^2 x^3^n /{n}
Is that right?
One more thing. To find the interval on convergence, I know I have to take the ratio test as n>infinity. Is this how I'm supposed to set it up?
lim [x^(n+1) * 5^(n+1) / (n+1)*(n+1)] * [(n*n/x^n*5^x)]
n>inf
After canceling out some factors I got:
lim 1/(2n+1) =...
Homework Statement
Determine the series of the given function:
f(x) = 10 / (15*x)
Homework Equations
Power series of 1/(1x) = Σ from n=0 to n=infinity of (x^n)
The Attempt at a Solution
f(x) = 10/(15x)
= 10*(1/15x)
= 10 * Σ(5x)^n
= 10 * Σ(5^n)*(x^n)
= Σ (50^n)*(x^n) < Not...
Homework Statement
A particle of charge q = e moves in a circle of radius = 0.44 m with speed v = 2.350×107 m/s. Treating the circular path as a current loop with constant current equal to its average current, find the maximum torque exerted on the loop by a uniform magnetic field of magnitude...
Homework Statement
10 y′+ 10 y = 7 exp(6*x)
y ( 0 ) = 7
Homework Equations
The Attempt at a Solution
First I divided everything by 10 because the coefficient of the y' should be 1
Then I found the integrating factor, which was exp(integral of 1*dx) = exp(x) = Integrating factor...
Homework Statement
http://nplq1.phyast.pitt.edu/res/westwolf/bookauthor/Capacitors/circle.jpg
We use two circular metal plates of radius 0.55 m and thickness 8.7 mm to build a capacitor. We leave a gap of 2.5 mm between the plates and fill half of the space between the plates with a medium...
Homework Statement
Solve the inital value first order linear differential equation
y'=y+x y(0) = 2
Homework Equations
The Attempt at a Solution
y'y=x
That's as far as I got. I'm not sure how to approach this. I've looked through my notes and book, and I don't have any examples that...