Today the inexact differential is usually denoted with δ, but in a text by a Russian author I found a dyet (D-with stroke, crossed-D) instead:
In response to my question to the author about this deviation from normal usage, he stated that this was a suggestion from von Neumann. (Which of course...
Hi,
for ease of reference this posting is segmented into :
1. Background
2. Focus
3. Question
1. Background:
Regarding (one, linear, second-order, homogeneous, ordinary, differential) equation describing the force in a non-driven, damped oscillation:
F = m.a = -k.x - b.v
F =...
I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using
$$dxdydz = \left (\frac{\partial x}{\partial r}dr +...
Let's say we have ##df=2xy^3dx + 3x^2y^2dy## - this is an exact differential.
In integrating, to find f, can we write ## f = \int 2xy^3 \, dx + \int 3x^2y^2 \, dy = 2x^2y^3 + C ##
Or am I getting it wrong?
From my interpretation of this problem (image attached), the force applied to the point charge is equal and opposite to the repulsive Coulomb force that that point charge is experiencing due to the presence of the other point charge so that the point charge may be moved at a constant velocity. I...
Hello,
let $$M^n \subset \mathbb{R}^N$$ $$N^k \subset \mathbb{R}^K$$
be two submanifolds.
We say a function $$f : M \rightarrow N$$ is differentiable if and only if for every map $$(U,\varphi)$$ of M the transformation
$$f \circ \varphi^{-1}: \varphi(U) \subset \mathbb{R}^N \rightarrow...
Hi! I'm asked to find all the non-zero Christoffel symbols given the following line element:
ds^2=2x^2dx^2+y^4dy^2+2xy^2dxdy
The result I have obtained is that the only non-zero component of the Christoffel symbols is:
\Gamma^x_{xx}=\frac{1}{x}
Is this correct?
MY PROCEDURE HAS BEEN:
the...
Homework Statement
Solve the following differential equations/initial value problems:
(cosx) y' + (sinx) y = sin2x
Homework Equations
I've been attempting to use the trig ID sin2x = 2sinxcosx.
I am also trying to solve this problem by using p(x)/P(x) and Q(x)
The Attempt at a Solution...
Homework Statement
Solve the following differential equations/initial value problem:
y^(4) - y'' - 2y' +2y = 0 Hint: e^-x sinx is a solution
Homework Equations
I was attempting to solve this problem by using a characteristic equation.
The Attempt at a Solution
y'''' -y'' -2y' + 2y = 0 -->...
I have seen how to solve the heat equation:
$$ \frac{ \partial^2 u(x,t) }{\partial x^2} = k^2 \frac{ \partial u(x,t) }{\partial t} $$
With boundary conditions.
I use separation variables to find the result, but i dont know how to solve the equation plus a...
I was doing the calculations for this: http://fermi.la.asu.edu/PHY531/cylinder/index.html
But I can't figure out how to go from \frac{d\sigma}{d\phi} to the total cross section. My guess was that you did the integral from \phi=0 to \phi=2\pi, but that's not helping since I can't tell either how...
Homework Statement
$$y = x.\log_e {\sqrt{x}}$$
Homework Equations
f(x) = g(x) h(x)
f ' (x) = g ' (x) . h (x) - h ' (x) . g(x)
The Attempt at a Solution
$$y = x .\log_e {\sqrt {x}}$$
$$y '(x) = 1.ln \sqrt{x} + \frac{1}{2} $$
the right answer is
$$ y ' = \log_{10} {\sqrt{x}} + \frac{1}{2} $$...
So I've seen in several lectures and explanations the idea that when you have an equation containing a relation between certain expressions ##x## and ##y##, if the expression ##x## approaches 0 (and ##y## is scaled down accordingly) then any power of that expression bigger than 2 (##x^n## where...
I need to solve:
\dot{\mathbf{r}}=-kv\hat{r} - \dot{\mathbf{r}_s}
However, I do not know how to deal with the fact that there is a unit vector. How can this be done? \dot{\mathbf{r}_s} is a constant vector.
I tried using the Bernoulli equation to solve this. I took two points at the surface of the water in both the containers and formed this equation:
gh_{b}=\frac{1}{2}v^2+gh
This is assuming that the velocity of the water in the large tank is approximately zero and using the fact that both the...
Homework Statement
Suppose at time zero, the bob was drawn upward four units from the equilibrium position, let C=2, K=2, m=1 lbm, initial speed=2 unit/sec find an expression for body's position.
and in the solution it says: y''+6y'+5y=0
my question is: from where does the numbers (6) and (5)...
Homework Statement
How does one show that q(t) is indeed a solution?
Homework Equations
The Attempt at a Solution
My current idea is that i should come up with any form of solution, like q = Acos(ωt), and slot it in the RHS.
Reason being that if q is indeed a solution, the result of the...
So in differential calculus we have the concept of the derivative and I can see why someone would want a derivative (to get rates of change). In integral calculus, there's the idea of a definite integral, which is defined as the area under the curve. Why would Newton or anyone be looking at the...
Homework Statement
It is the driven series RLC circuit. It is given in the following images.
It is from the section 12.3 in this note.
Homework Equations
The differential equation, as given by 12.3.3, is ##L \frac{d^2 Q}{d t^2} + R \frac{d Q}{d t} + \frac{Q}{C} = V_0 \sin{(\omega t)}##...
Homework Statement
http://imgur.com/a/k7fwG
Find the vector magnetic potential at point P1.
Homework Equations
Vector magnetic potential given by:
$$
d \bar{A} = \frac{\mu I d\bar{l'}}{4 \pi | \bar{r} - \bar{r'} | }
$$
The Attempt at a Solution
I split up the problem in 3 parts,
first...
Homework Statement
We assume from ODE theory that given a smooth A: I → gl(n;R) there exists a
unique smooth solution F : I → gl(n;R), defined on the same interval I on which
A is defined, of the initial value problem F' = FA and F(t0) = F0 ∈ gl(n;R) given.
(i) Show that two solutions Fi : I...
Homework Statement
The Frenet frame of a curve in R 3 . For a regular plane curve (and more generally for a regular curve on a 2-dimensional surface - e.g. the 2-sphere above) we could construct a unique adapted frame F. This is not the case for curves in higher dimensional spaces. Besides the...
Homework Statement
Let γ : I → R3 be an arclength parametrized curve whose image lies in the 2-sphere S2 , i.e. ||γ(t)||2 = 1 for all t ∈ I. Consider the “moving basis” {T, γ × T, γ} where T = γ'.
(i) Writing the moving basis as a 3 × 3 matrix F := (T, γ × T, γ) (where we think of T and etc...
Homework Statement
Find the values of a and b that make f a differentiable function.
Note: F(x) is a piecewise function
f(x):
Ax^2 - Bx, X ≤ 1
Alnx + B, X > 1
Homework Equations
The Attempt at a Solution
Made the two equations equal each other.
Ax^2 - Bx = Alnx + B
Inserting x=1 gives,
A -...
Homework Statement
Please bear with the length of this post, I'm taking it one step at a time starting with i)
Let A: I → gl(n, R) be a smooth function where I ⊂ R is an interval and gl(n, R) denotes the vector space of all n × n matrices.
(i) If F : I → gl(n, R) satisfies the matrix ODE F'...
Homework Statement
[/B]
Summarizing: two civilizations hate each other, one of the civilizations throws a curse at the second. The second civilization succumbs to chaos and has a change in Population each week of ΔP= -√P. That is:
Pn = Pn-1-√Pn-1
Homework Equations
[/B]
Considering that the...
Homework Statement
Prove the formula for inertia of a ring (2D circle) about its central axis.
Homework Equations
I = MR^2
Where:
M: total mass of the ring
R: radius of the ring
The Attempt at a Solution
- So I need to prove the formula above.
- First, I divide the ring into 4...
Homework Statement
How exactly they combined equation1 and equation2 and got that system? I don't get that part.
Homework Equations
A*(dy/dt)= -k*y eq1
A*(dz/dt)=ky-kz eq2
The Attempt at a Solution
I tried substituting the 1st ky in the 2nd equation and then differentiating but I don't...
Hi folks,
I was wondering if there are books that explain how to solve differential equations using infinite series. I know it is possible to do it since Poincaré used that method.
Do you know which ones are the best?
I find books on infinite series but they talk just about series...