Homework Statement
Let B be an m×n matrix with complex entries. Then by B* we denote the n×m matrix that is obtained by forming the transpose of B followed by taking the complex conjugate of each entry. For an n × n matrix A with complex entries, prove that if u*Au = 0 for all n × 1 column...
Homework Statement
Find the solution u, via the Fourier sine/cosine transform, given:
u_{tt}-c^{2}u_{xx}=0
IC: u(x,0) = u_{t}(x,0)=0
BC: u(x,t) bounded as x\rightarrow \infty , u_{x}(0,t) = g(t)
2. The attempt at a solution
Taking the Fourier transform of the PDE, IC and BC...
Homework Statement
Two masses, m1 and m2, are connected to each other by a spring with a spring constant k. The system moves freely on a horizontal frictionless plane. Find the natural frequency of oscillation.
Homework Equations
F = -kx
F = ma
The Attempt at a Solution
Let m1 be the mass...
Homework Statement
Given u_tt = F(x,t,u,u_x, u_xx), give the finite difference approximation of the pde (ie using u_x = (u(x + dx; t) - u(x - dx; t))/(2dx) etc.)
Homework Equations
Well, clearly, u_x = (u(x + dx; t) - u(x - dx; t))/(2dx)
The Attempt at a Solution
I really have no idea how...
Homework Statement
u_t = -{{u_{x}}_{x}}
u(x,0) = e^{-x^2}
Homework Equations
The Attempt at a Solution
The initial state is a bell curve centred at x=0. The second partial derivative of u at t=0 is {4x^2}{e^{-x^2}}, which is a Gaussian function, which means nothing to me other than its...
Homework Statement
Find
\int{\int_{D}x dA}
where D is the region in Q1 between the circles x2+y2=4 and x2+y2=2x using only polar coordinates.
The Attempt at a Solution
Well, the two circles give me r=2 and r=2 cos \theta, and the integrand is going to be r2cos \theta, but I have no...
Homework Statement
Show that f(x,y) = -(x^2 - 1)^2 - (yx^2-x-1)^2 has only two critical points, and both are maxima.
The Attempt at a Solution
Set partial derivatives (wrt x and y) to zero to find critical pts.
f_x = -2(x^2 - 1)(2x) - 2(yx^2 - x - 1)(2xy - 1) = 0
f_y = -2(yx^2 - x -...
Homework Statement
Find the gradient vector of:
g(r, \theta) = e^{-r} sin \theta
Homework Equations
The Attempt at a Solution
I know how to get gradients for Cartesian - partially derive the equation of the surface wrt each variable. But I have no idea how to do it for...
Homework Statement
If the equations
x^2 - 2(y^2)(s^2)t - 2st^2 = 1
x^2 + 2(y^2)(s^2)t + 5st^2 = 1
define s and t as functions of x and y, find \partial^2 t / \partial y^2
The Attempt at a Solution
Equating the two, we get 4y^2*s^2*t = -7s*t^2. My main problem is, as simple as this...
Homework Statement
A particle is attracted towards the origin by a force proportional to the cube of its distance from the origin. How much work is done in moving the particle from the origin to the point (2,4) along the path y = x^2 assuming a coefficient of friction \mu between the particle...
Homework Statement
Find the surface area traced out when the curve 8y^2 = x^2(1-x^2) is revolved around the x-axis.
The Attempt at a Solution
x-axis means y = 0
When y = 0, x = 0, -1 or 1.
Since this curve is "the infinity symbol", the curve has symmetry at x = 0.
Isolating y,
8y^2 =...
Homework Statement
Find the surface area of the portion of the sphere x^2 + y^2 + z^2 = 3c^2 within the paraboloid 2cz = x^2 + y^2 using spherical coordinates. (c is a constant)
Homework Equations
The Attempt at a Solution
I converted all the x's to \rho sin\phi cos\theta, y's to \rho...
Homework Statement
A uniform surface charge lies in the region z = 0 for x^2 + y^2 > a^2, and z = \sqrt{a^2-x^2-y^2} for x^2 + y^2 \leq a^2. Find the force on a unit charge placed at the point (0,0,b)
Homework Equations
dF = G (\delta)(vector of point to surface) / (magnitude of surface)^3...
Homework Statement
Given a curve C that starts from the origin, goes to (1,0) then goes to (0,1), then back to the origin, find the centroid of the enclosed area D.
Homework Equations
\bar{x} = {1/(2A)}*\int_C {x^2 dy}
\bar{y} = -{1/(2A)}*\int_C {y^2 dx}
The Attempt at a Solution
Well...