Can you help me with this problem?  Find the center of mass of a lamina whose region R is given by the inequality:   and the density in the point (x,y) is :    Any help?  Thanks
123804
Can you help me with this problem?  Find the center of mass of a lamina whose region R is given by the inequality:   and the density in the point (x,y) is :    The region r is this one:    Is this the proper way to set up the integral for m:  $$\int_{-1}^{1}\int_{-x-1}^{x+1} \ e^{x+y} \ dy \ dx$$  Any help?  Thanks
63211
I am pretty sure that this is such a stupid, stupid question, but how do you prove that $d_2(x,y)={|x-y|\over {1+|x-y|}}$ satisfying the third condition to be a metric, which is the triangle inequality.   I know that I should really be able to do this, but I just can't, I keep seeing a contradiction. Please help me.
49185
I read Griffiths EM today and it says something very interesting but a little bothering to me. It states for an atom, the position of center of mass of an electron cloud lies in the center of the proton.  But my question is why? What will happen if this is not true?
90558
The full question is,     If X is a Geometric random variable with parameter p, what value of k ∈ {1, 2, . . .}   maximizes P(X = k) = (1 − p)^(k-1) * p? This is referred to as the mode of the distribution.   Through trial and error (plugging in random values), I found that p = 1/2 yields the highest value of P(X=k).  Is there a way to figure this out without trial and error?
104533
Hi guys I have a midterm tommorow and I was doing this practice problem that I need help on.  So any hints or solutions would be appreciated. Thank you for your time  Problem  The head of timpani is constituted by some kind of elastic membrane which is stretched over a circular bowl; in a mathematical language, we are studying the displacement $u$ of the membrane as a function $u(x, y, t)$ deﬁned in $D = \{(x,y,t) \ |\ x^2 + y^2 \leq R, \ -\infty &lt; t &lt; \infty\}$, where $R$ is the radius of the bowl. The function u satisﬁes the two-dimensional Wave Equation  $u_{tt} = c^2(u_{xx} + u_{yy})$  with Dirichlet boundary conditions $u = 0$ on the boundary. Prove conservation of energy E(t), which is deﬁned as  $E(t) = \frac{1}{2} \iint\limits_D u_{t}^2 + c^2 (u_{x}^2 + u_{y}^2) dxdy$  [Hint: Proceed as we did for the one-dimensional case and use the divergence theorem to “integrate by parts” ] which I don't understand  Assume now that we take into account friction between air and the membrane; the displacement of the membrane now satisﬁes the damped Wave Equation, which reads:  $u_{tt} - c^2 (u_{xx} + u_{yy}) = -vu_{t}$  with the same boundary conditions. Show that, in this case, energy is always a strictly decreasing function except if $u(x, y, t) = 0$, for which it is constant equal to $0$. [Hint: What condition must $u_t$ satisfy in order for E to be non-decreasing? What PDE will u satisfy? Conclude using the maximum principle.  My attempt:  So first I found the derivative of E(t) and if the derivative of E(t) = 0 then I know the energy is conserved and I used integral by parts in 3 dimension to solve that   to get $\iint 2u_{t}u_{tt} = 0$ because it is decreasing. Since the derivative is 0 E is constant over time  therefore the energy is concerved.  But I do not understand the last part please any help or hints would be greatly appreciated.  Thank you
92467
My problem is we want to minimize $1/2 \cdot \text{Norm}(w)^2$ under suitable constraints $y(i){[w^Tx_i+b]-1}$.  We write the Lagrangian and get the above equations.  I have a problem in differentiating with respect to $w$ can somebody elaborate on that? How do we get first equation?
70972
Find the area of the shaded region in the figure below:    I am completely stuck on how to start off this question. Please help on some guidance on how to start it off.
87029
I was asked in a exam: does there exist a function(need not be continous) $f:\mathbb{R}\rightarrow \mathbb{R}$ whose graph is dense in $\mathbb{R}^2$?  I proved that  but did not provide explicit example, could any one give me one such? thank you
112499
I just started learning about Brownian motion and I am struggling with this question:  Suppose that $X_t = B_t + ct$, where $B$ is a Brownian motion, $c$ is a constant. Set $H_a = \inf \{ t: X_t =a \}$ for $ a &gt;0$. Show that for $c \in \mathbb{R}$, the density of $H_a$ is  \begin{equation} f_{H_a} (t) = \frac{ a \exp \Big\{ \frac{- (a-ct)^2}{2t} \Big\} }{\sqrt{2 \pi t^3}}. \end{equation}
105165
Does the Grashof number lead to the answer?  The Wikipedia article () yields an equation for vertical plates $$Gr_L = \frac{g\beta(T_s-T_\infty L^3)}{\nu^2}$$ Could I just solve for $L$, with $Gr_L$ equal to $10^8$ (upper boundary for laminar flow)?
52820
The first question that comes into my mind here is whether any cylinder that touches(at 4 pts) the circumference of the sphere and does not go out of it, has equal volume?   Second, how do i mathematically limit the volume of the cylinder to be less than that of a sphere? Squeeze theorem?   Please help, thanks!
71524
Find the MLE of the unknown parameter $\theta$ when $X_1,X_2,...,X_n$ is a sample from the distribution whose density function is: $$f_X(x) = \frac12e^{-|x-\theta|}, -\infty&lt;x&lt;\infty$$ What I did so far: $$\text{maximize }(\frac12)^ne^{-|x_1-\theta|-|x_2-\theta|-...-|x_n-\theta|} \\ \text{thus we want to minimize }|x_1-\theta|+|x_2-\theta|+...+|x_n-\theta|$$ Since $|x_i-\theta|$ is positive for all $i$, we want to solve for $\theta$ so: $$|x_1-\theta|+|x_2-\theta|+...+|x_n-\theta|=0$$ What do I do from here?
637
My friend and I have been wracking our heads with this one for the past 3 hours... We have 2 point masses, $m$ and $M$ in a perfect world separated by radius r.  Starting from rest, they both begin to accelerate towards each other. So we have the gravitational force between them as:  $$F_g ~=~ G\frac{Mm}{r^2}$$  How do we find out at what time they will collide?  What we're having trouble with is this function being a function of $r$, but I have suspected it as actually a function of $t$ due to the units of $G$ being $N(m/kg)^2$.  I've tried taking a number of integrals, which haven't really yielded anything useful.  Any guidance?  No, this is not an actual homework problem, we're just 2 math/physics/computer people who are very bored at work :)
61474
I would like to ask if someone could help me with solving the following task.  Find the area ratio between ABC and the hatched triangle.    Thank you in advance.
58154
We have a complex plane and there are inequality on it: $2|z| &gt; |1 + z^2|$. What is the locus of such set of $z$?  I tried to solve it by using polar coordinates and got: $r^4 - 2r^2\cos(2\phi ) - 1 = 0$.  As well I tried solve it using $z = x + iy$ and got: $4(x^2 + y^2) = (x^2 - y^2 + 1)^2 + 4x^2y^2$  By I still can`t figure out what is the explicit name or at least descritpion for such locus of points. Will be very grateful for any help.
46486
The lemniscates $r^2 = a^2\cos2\theta$ revolves about a tangent at the pole. What is the volume generated by it ?  Please explain in detail. I found a couple of answers on finding surface areas, volumes but didn't understood it.  Can someone solve this. I'm stuck in the middle with integrations.
81573
Specifically, define the sub-gaussian norm for a r.v. X as  $$ \|X\|_{\psi_2}=\inf\{t&gt;0:e^{X^2/t^2}\leq 2\}. $$  How do we prove that the centering inequality with $C=1$ does not hold in general? i.e. $$ \|X-\mathbb{E}X\|_{\psi_2}\leq C\|X\|_{\psi_2}.  $$ A counter example would work too. Thanks in advance!