I am studying complex variables with Brown and Churchill. In it, they define the principal value of z^c, with both variables complex, to be e^{c\; \text{Log }z} , where \text{Log} is the principle value branch of the complex logarithm.
Now, suppose z = i and c = 3 . We know that \text{Arg...
Yes, that is what I mean.
In any case, I thought about it, and the determination of the probabilities isn't as hard as I thought it would be. For example, the probability of getting at least one 2 is the complement of getting no twos.
The probability of getting no twos is:
\frac{{1...
Ah, you forgot to divide out a (1/4) in order to get the summand into the form x r^{x-1} . In particular:
\sum_{x=0}^\infty x r^{x}
Divide out an r, dropping the power in the summand down by 1:
r \sum_{x=0}^\infty x r^{x-1}
Recognize that this is the derivative of the series with...
I am working on correcting an exam so that I may study for my probability final. Unfortunately, I don't have the correct answers, so I was hoping that someone here might be able to check my thought process.
1) Pick three numbers without replacement from the set {1,2,3,3,4,4,4}. Let T be the...
Ah, I think I see part of my confusion now. It is necessary for the stress tensor field to be divergenceless for equilibrium, not the tensor being zero...
Perhaps you could answer this to solidify my understanding: Say you have some object that you are accelerating straight up in the z...
Alright, this sort of helps...
I am trying to get a better understanding of the stress tensor mainly because we are reviewing the Maxwell stress tensor in my E&M class. The professor talks about the stress tensor as representing "momentum flow", i.e, the amount of momentum crossing a surface...
I understand all that. I guess my question is, if I look at a point in the medium, and there is a force from the left pointing toward to point, and a force from the right also pointing towards the point, do the forces cancel such that T_xx is zero...
I have a couple of questions about the stress tensor. I am not an engineering student, so this is the first time I have dealt with internal forces, stress, shears, and such.
It is my understanding that the entries in the stress tensor are forces per unit area. I assume this means the total...
Homework Statement
Find the open circuit voltage across A & B:
Homework Equations
V = IR
The Attempt at a Solution
If I0 is the current leaving the voltage source and passing through the first resistor, I1 is the current through the left resistor in parallel, and I2 through...
Going from cylindrical coordinates to cartesian coordinates, we can use the transformations x = r \cos(\theta) , y = r \sin(\theta) , and, of course, z = z_{\text{cyl}}.
Now, going from cartesian to cylindrical isn't as obvious, but is still simple. For example, the radius, r, is simply r...
I think you are on the right track. What I would do, instead of minimizing the distance, is to take advantage of the geometry.
Draw a vector \vec{p} from the origin to the point. Draw another vector from the origin to the parameterized vector function: \vec{r}(t) = (3t-1, 2-t, t).
Now, let...
I am working on a problem that uses the notation:
\sum_{i,j=1}^n A_{i,j}
Where A is an (n x n) matrix. I am a little unsure of what the summation is over, due to the odd notation "i,j = 1". My first guess is that this is shorthand for
\sum_{i=1}^n \sum_{j=1}^n A_{i,j}
But I...
I am an undergraduate physics and applied math major in his junior year, and it is about time for me to start seriously researching graduate programs.
I have had some experience in the lab, enough to know that I would rather not be an experimentalist in some field like condensed matter. At the...
I am encountering two-particle systems for the first time, and I am a little confused about something.
Consider a infinite square well with two noninteracting particles, both fermions (though, disregard spin degrees of freedom, this is a strictly 1-d scenario).
So I go to write down the...
So what gives with your examples? In short, I believe it is because you are dealing with charge distributions that involve singularities. The Laplacian doesn't behave well because of this.
Take the analogous case of the divergence of the E-field of a point charge:
\vec{E} = \frac{1}{4\pi...
If I understand your question correctly, then no, in general the Laplacian of the scalar potential is not 0. Remember that:
\vec{E} = -\vec{\nabla} \phi
And that...
\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0}
Therefore
\frac{\rho}{\epsilon_0} = \vec{\nabla} \cdot...
Bam!, I think I have it.
A is diagonalizable, so...
A = U^\ast \Lambda U
And so:
A^\ast = U^\ast \Lambda ^\ast U
But the eigenvalues are all real, so big lambda is self-adjoint, and we can say that:
A^\ast = U^\ast \Lambda U = A
I think that does it. Cool beans.
I'm not quite getting why these are relevant. The conjugate of a self-adjoint matrix A is just A transposed. Not sure what is so interesting about the diagonal matrix of eigenvalues...
Homework Statement
If A has eigenvalues 0 and 1, corresponding to the eigenvectors (1,2) and (2, -1), how can one tell in advance that A is self-adjoint and real.
Homework Equations
e=m^2
The Attempt at a Solution
I can show that A is real: it has real orthogonal eigenvectors and...
I am trying to see why exactly the momentum eignenstates for a free particle are orthogonal. Simply enough, one gets:
\int_{-\infty}^{\infty} e^{i (k-k_0) x} dx = \delta(k-k0)
I can see why, if k=k0, this integral goes to zero. But if they differ, I don't see why it goes to zero. You have...
Oh, yes, that would be it. I suppose the equations don't like it very much when you try to restrict the electric field from a point charge to a plane... :redface:
Thanks!
So I am playing around with the differential form of Gauss's Law:
\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}
Starting off simple with a point charge, the electric field is:
\vec{E} = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} \hat{r}
And the divergence, in spherical coordinates...
The Lagrangian immediately yields the equations of motion for a system. Try a harmonic oscillator for example. Then you have:
L = T - V = \frac{1}{2}m \dot{x}^2 - \frac{1}{2}k x^2
\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}} \right) = \frac{\partial L}{\partial x}...
Homework Statement
Let T:U \rightarrow V be a linear transformation, and let U be finite-dimensional. Prove that if dim(U) > dim(V), then Range(T) = V is not possible.
Homework Equations
dim(U) = rank(T) + nullity(T)
The Attempt at a Solution
I almost think there must be a typo in the...
Yes, but a centripetal force does not change the magnitude of the velocity, thus there is no change is the total kinetic energy of the object it pulls on.
The friction is transferring some of the translational kinetic energy of the ball to rotational. I have a hunch that the total kinetic...
Homework Statement
"A small ball with radius r and uniform density rolls without slipping near the bottom of a fixed cylinder of radius R. What is the frequency of small oscillations, assuming r<<R?"
http://img89.imageshack.us/img89/8614/helpwu8.png [Broken]
Homework Equations...
Yes and no.
The "real" way, in my mind, does require calculus. I consider it the real way because it comes directly from F=ma. The calculus isn't that tough, though, its a pretty simple differential equation that says that m x == m x''.
What kof refers to is this...
I'm going to bump, and at the same time add another question that might be related to my understanding of the original problem.
Suppose two people are in a field, pulling a massless rope either way with 50 Newtons of force. They and the rope are stationary. The tension in the rope is 50...
Right, I understand and can get the right answer. I understand string conservation. I just don't quite get how different parts of the same string can have different accelerations when (seemingly) the same forces are acting on them.
I had some time, anyways:
So if the right mass is much greater than the mass attached to the movable pulley, then the right mass will fall and the center pulley will be lifted. All the time, the string above the mass to the right will accelerate at the same rate as the mass, while a bit...
I recently did a problem where two masses were connected to the same massless string in an Atwood's machine, yet they had different accelerations. This wasn't really intuitive to me. If one mass had a greater magnitude of acceleration than the other mass, then it would imply that the bit of...
Coming into college, I had a vague idea that I wanted to be an engineer. After the first quarter of engineering classes, though, I realized how "industrial" the profession was. I never considered the business part of it, sitting down in an office and designing parts all day. So I dropped out of...
I don't know much about p-type semiconductors, but from what I have read a positive "hole" is created when an electron is accepted by the impurity. If all of the electrons are shoved to the back of the rod when it is accelerated, then all of the positive holes should be created back there, and...
True and true, but I managed to find the text I was looking for, and it was called the Tolman-Stewart experiment. From Knight's "Physics for Scientists and Engineers, Volume 4":
"The Tolman-Stewart experiment of 1916 was the first direct evidence that electrons are the charge carriers in...
I remember reading somewhere that it was discovered that electrons, not protons, composed current when an accelerated conductor was found to have slightly negative charge on the end opposite the direction of the acceleration. Because of this it was determined that electrons were free to move in...