What is volume element: Definition and 23 Discussions
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
d
V
=
ρ
(
u
1
,
u
2
,
u
3
)
d
u
1
d
u
2
d
u
3
{\displaystyle dV=\rho (u_{1},u_{2},u_{3})\,du_{1}\,du_{2}\,du_{3}}
where the
u
i
{\displaystyle u_{i}}
are the coordinates, so that the volume of any set
B
{\displaystyle B}
can be computed by
Volume
(
B
)
=
∫
B
ρ
(
u
1
,
u
2
,
u
3
)
d
u
1
d
u
2
d
u
3
.
{\displaystyle \operatorname {Volume} (B)=\int _{B}\rho (u_{1},u_{2},u_{3})\,du_{1}\,du_{2}\,du_{3}.}
For example, in spherical coordinates
d
V
=
u
1
2
sin
u
2
d
u
1
d
u
2
d
u
3
{\displaystyle dV=u_{1}^{2}\sin u_{2}\,du_{1}\,du_{2}\,du_{3}}
, and so
ρ
=
u
1
2
sin
u
2
{\displaystyle \rho =u_{1}^{2}\sin u_{2}}
.
The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the area element, and in this setting it is useful for doing surface integrals. Under changes of coordinates, the volume element changes by the absolute value of the Jacobian determinant of the coordinate transformation (by the change of variables formula). This fact allows volume elements to be defined as a kind of measure on a manifold. On an orientable differentiable manifold, a volume element typically arises from a volume form: a top degree differential form. On a non-orientable manifold, the volume element is typically the absolute value of a (locally defined) volume form: it defines a 1-density.
This is probably a stupid question but,
## \frac{d\partial_p}{d\partial_c}=\delta^p_c ##
For the notation of a 4D integral it is ##d^4x=dx^{\nu}##, so if I consider a total derivative:
##\int\limits^{x_f}_{x_i} \partial_{\mu} (\phi) d^4 x = \phi \mid^{x_f}_{x_i} ##
why is there no...
For me is not to easy to understand volume element ##dV## in different coordinates. In Deckart coordinates ##dV=dxdydz##. In spherical coordinates it is ##dV=r^2drd\theta d\varphi##. If we have sphere ##V=\frac{4}{3}r^3 \pi## why then
dV=4\pi r^2dr
always?
I am currently having trouble deriving the volume element for the first octant of an isotropic 3D harmonic oscillator.
I know the answer I should get is $$dV=\frac{1}{2}k^{2}dk$$.
What I currently have is $$dxdydz=dV$$ and $$k=x+y+z. But from that point on, I'm stuck. Any hints or reference...
I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using
$$dxdydz = \left (\frac{\partial x}{\partial r}dr +...
Can we say that a volume element can be represented by a vector, or is there some hidden complication that makes this inadvisable?
For some background, the stress-energy tensor has been described as the density of energy and momentum, in for instance MTW. So if one says that the represention...
Homework Statement
Show that the surface and volume element on a deformed sphere are
\sigma = \frac{\rho^2 \sin \theta}{\cos \gamma} d\phi d\theta,
dV = \rho^3 \sin \theta d\phi d\theta,
if \gamma is the angle between normal vector and radius vector.
Homework Equations
n\cdot r = \cos...
In a change of coordinate system we have ##dx^\mu = (\partial x^\mu / \partial \xi^{\kappa})d \xi^{\kappa}##, where the term in round brackets is the Jacobian. That notation implies a sum over all values that ##\kappa## can take. This don't tell us that it's an alternating sum for the case of...
Homework Statement
Hi everyone. Here's my problem. I know that the volume element in spherical coordinate is ##dV=r^2\sin{\theta}drd\theta d\phi##. The problem is that when i have to compute an integral, sometimes is useful to write it like this:
$$r^2d(-\cos{\theta})dr d\phi$$
because...
Homework Statement
This is part of the online tutorial I'm reading: http://farside.ph.utexas.edu/teaching/em/lectures/node49.html
I'm so confused about the notation of Dirac Delta. It's said that 3-dimensional delta function is denoted as \delta^3(x, y, z)=\delta(x)\delta(y)\delta(z) in...
I found this matrix in the wiki:
https://fr.wikipedia.org/wiki/Vitesse_ar%C3%A9olaire#.C3.89valuation_en_coordonn.C3.A9es_cart.C3.A9siennes
I think that it is very interesting because it express d²A not trivially as dxdy. So, I'd like of know if exist a matrix formulation for volume...
When we way that
\frac{d^3p}{p_0}=\frac{d^3p}{\sqrt{m^2+\vec{p}^2}}
is the invariant volume element, is that with respect to all Lorentz transformations or just proper orthochronous Lorentz transformations?
Me again, (this is what happens when you make a designer do mechanics, and give her a lecturer she doesn't understand :P ).
I'm stuck on a combined loadings question. This is the question:
"The 60 mm diameter rod (fixed at end C on the wall) is subjected to the loads shown.
1) Transform...
So, the upper light cone has a Lorentz invariant volume measure
dk =\frac{dk_{1}\wedge dk_{2} \wedge dk_{3}}{k_{0}}
according to several sources which I have been reading. However, I've never seen this derived, and I was wondering if anyone knew how it was done, or could point me towards...
Hello!
Homework Statement
I'm revising my quantum mechanics course, but I don't get this normalizing-problem. \psi = N r cos \theta e^{-r/a_0}
To begin with, this is how my teacher solves it in the solutions manual:
1=N^2\int_0^\infty r^2 e^{-r/a_0} r^2 dr \int_0^{\pi} \int_0^{2 \pi}...
I've already post this, but I've done it in the wrong section!
So here I go again..
I've a doubt on the way the infinitesimal volume element transfoms when performing a coordinate transformation from x^j to x^{j'}
It should change according to dx^1dx^2...dx^n=\frac{\partial...
Very simple question:
Let x^0,x^1,...,x^n be some fixed coordinate system, so that the infinitesimal volume element is dV=dx^0dx^1...dx^n.
Then any change to a new (primed) coordinate system x^{0'},x^{1'},...,x^{n'} transforms the volume to dV=\frac{\partial (x^0,x^1,...,x^n)}{\partial...
Given two orthonormal bases v_1,v_2,\cdots,v_n and u_1,u_2,\cdots,u_n for a vector space V, we know the following formula holds for an alternating tensor f:
f(u_1,u_2,\cdots,u_n)=\det(A)f(v_1,v_2,\cdots,v_n)
where A is the orthogonal matrix that changes one orthonormal basis to another...
Hi,
I'm reading Sean Carroll's text, ch2, and believe I understood most of the discussion on Integration in section 2.10. However in equation 2.96, he states that
\epsilon\equiv\epsilon_{\mu_1\mu_2...\mu_n}dx^{\mu_1}\otimes dx^{\mu_2}\otimes...\otimes dx^{\mu_n}...
In Ray d'Inverno's book on general relativity, he defines the volume element in a way which makes it a scalar density of weight -1, meaning it transforms with the inverse of the Jacobian. Every other source I have looked at seems to say it should transform with the Jacobian, making it a scalar...
I need some help understanding a definition:
This is supposed to be an explanation of what the author did on the page before. He had just described how to construct a (complex) Hilbert space from a (real) smooth manifold with a smooth nowhere vanishing volume element, and then moved on to...
Hi all,
I am solving a problem for N classic harmonic oscillators. I have the Hamiltonian
H = sum(i=1,3N)(p_i^2/(2m) + m*o^2/2 *q_i^2
where p is momentum and q I presume is scaled coordinates. I am given the following hint that the volume in phase space can be written as
V(E,N) =...
i'm at a loss about how to do this type of integration. can some one show me how to evaluate the integral of (d^3)k exp[ik*(x1-x2)]/[(k^2+m^2)(2pi)^3], where "*" is the dot product between the 3 vector k and (x1-x2), which are both 3 vectors. this come from the energy equation used to get the...