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Homework Statement [/B] 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...
2. ### Stagnation Point flow

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...
3. ### Fluids Homework Problem

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)...
4. ### Question on Equivalent/Hydraulic Diameter

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 cross-sectional flow area shown with an equivalent diameter based upon a closed rectangle (entire perimeter)...
5. ### Heat Transfer in 1D

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)...
6. ### Second Moment of Area

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...
7. ### Shaft Problem

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...
8. ### Pressurizer Problem

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...
9. ### Using Matlab - plotting frequency responses

Does this help at all? http://www.mathworks.com/help/toolbox/signal/freqz.html [Broken]
10. ### Fourier Series

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...
11. ### Convolution Integral

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...
12. ### Linear Algebra - Eigenvectors

For A-5I 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 A-0I
13. ### Linear Algebra - Eigenvectors

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 __________________________________________ A-5I...
14. ### Material Indices

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 yield-before-break...
15. ### Convergence of a series

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} |...
16. ### Trig Integration

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 u-substitution. \int sin^3(2x) dx u = 2x du = 2dx dx = du/2 So the new integral looks as such...
17. ### Area of a parallelogram with vectors

Thank you, I got it.
18. ### Area of a parallelogram with vectors

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...
19. ### Electromagnetic Intensity

Homework Statement A helium-neon 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...
20. ### Taylor Series question

So is it: \sum\fract(-1)^n^+^2 3x^n /{n}
21. ### Taylor Series question

Homework Statement Determine the Taylor Series for f(x) = ln(1-3x) about x = 0 Homework Equations ln(1+x) = \sum\fract(-1)^n^+^1 x^n /{n} The Attempt at a Solution ln(1-3x) = ln(1+(-3x)) ln(1+(-3x)) = \sum\fract(-1)^n^+^2 x^3^n /{n} Is that right?
22. ### Power Series

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) =...
23. ### Power Series

Is it just Σ(5^n)*(x^n)*10 ?
24. ### Power Series

Homework Statement Determine the series of the given function: f(x) = 10 / (1-5*x) Homework Equations Power series of 1/(1-x) = Σ from n=0 to n=infinity of (x^n) The Attempt at a Solution f(x) = 10/(1-5x) = 10*(1/1-5x) = 10 * Σ(5x)^n = 10 * Σ(5^n)*(x^n) = Σ (50^n)*(x^n) <--- Not...
25. ### Torque on a charged particle moving in a circle in a uniform B field

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...
26. ### First Order Linear Differential Equation

I just realized that too. Thank you
27. ### First Order Linear Differential Equation

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...
28. ### Semicircular Dielectric

I'll try the parallel attempt first
29. ### Semicircular Dielectric

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...
30. ### Differential Equation

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...