|
|
| Line 1: |
Line 1: |
| In [[mathematics]], a '''volume element''' provides a means for [[Integral|integrating]] a [[function (mathematics)|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
| | The author is known as Phil. Curing individuals is my working [http://www.fizzlive.com/member/1215044/blog/view/3450967 spilleautomat på nett] day occupation now but I strategy on changing it. One of the extremely [http://www.purevolume.com/fannylaroseavotou/posts/6635252/What+You+Don%27t+Know+About+Spilleautomater+Norske+Could+Be+Costing+To+More+Than+You+Think free norrman spilleautomater] best [http://journals.fotki.com/dieterswanson/dieterswanson/entry/stsbsfdkbstdd/ spilleautomat på nett] things in the world for her is to solve puzzles and [http://journals.fotki.com/chancegreenberg/chancegreenberg/entry/stdrkwqfffwtr/ Det er på spilleautomater med progressive jackpotter] now she is trying to make money with it. [http://www.fizzlive.com/member/1215044/blog/view/3501726 orgesspill] Kansas is where my home [http://journals.fotki.com/kieran05jumbkqngo/kieran05jumbkqngo/entry/stskggwsfqsft/ orgesspill] is. If you want to discover out more [http://www.purevolume.com/fannylaroseavotou/posts/6893145/Beste+Spilleautomater+P%C3%A5+Nett+Har+Funnet+Kandidater+Iphone+Apps beste spilleautomater på nett] check out my website: http://www.[http://photo.net/gallery/tag-search/search?query_string=purevolume purevolume].com/abbyaxvdulfl/posts/6929320/Why+Everyone+Is+Dead+Wrong+About+Slotmaskiner+Er+Blant+De+Mest+Popul%C3%A6re+Casinospilleneet+And+Why.<br><br> |
| :<math>dV = \rho(u_1,u_2,u_3)\,du_1\,du_2\,du_3</math> | |
| where the <math>u_i</math> are the coordinates, so that the volume of any set <math>B</math> can be computed by
| |
| :<math>\operatorname{Volume}(B) = \int_B \rho(u_1,u_2,u_3)\,du_1\,du_2\,du_3.</math>
| |
| For example, in spherical coordinates <math>dV = u_1^2\sin u_2\,du_1\,du_2\,du_3</math>, and so <math>\rho = u_1^2\sin u_2</math>.
| |
|
| |
|
| 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 integral]]s. Under changes of coordinates, the volume element changes by the absolute value of the [[Jacobian determinant]] of the coordinate transformation (by the [[integration by substitution#Substitution for multiple variables|change of variables formula]]). This fact allows volume elements to be defined as a kind of [[measure (mathematics)|measure]] on a [[manifold]]. On an [[orientability|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 [[density on a manifold|1-density]].
| | Also visit my blog [http://journals.fotki.com/preciouscorbett/preciouscorbett/entry/sttfrbsrwbqgg/ norske spilleautomater villig nett] post: [http://www.purevolume.com/abbyaxvdulfl/posts/6929320/Why+Everyone+Is+Dead+Wrong+About+Slotmaskiner+Er+Blant+De+Mest+Popul%C3%A6re+Casinospilleneet+And+Why. norske spilleautomater kungen nett] |
| | |
| ==Volume element in Euclidean space==
| |
| In [[Euclidean space]], the volume element is given by the product of the differentials of the Cartesian coordinates
| |
| :<math>dV = dx\,dy\,dz.</math>
| |
| In different coordinate systems of the form <math>x=x(u_1,u_2,u_3), y=y(u_1,u_2,u_3), z=z(u_1,u_2,u_3)</math>, the volume element [[integration by substitution|changes by the Jacobian]] of the coordinate change:
| |
| :<math>dV = \left|\frac{\partial (x,y,z)}{\partial (u_1,u_2,u_3)}\right|\,du_1\,du_2\,du_3.</math>
| |
| For example, in spherical coordinates
| |
| :<math>\begin{align}
| |
| x&=\rho\cos\theta\sin\phi\\
| |
| y&=\rho\sin\theta\sin\phi\\
| |
| z&=\rho\cos\phi
| |
| \end{align}
| |
| </math>
| |
| the Jacobian is
| |
| :<math>\left |\frac{\partial(x,y,z)}{\partial (\rho,\theta,\phi)}\right| = \rho^2\sin\phi</math>
| |
| so that
| |
| :<math>dV = \rho^2\sin\phi\,d\rho\,d\theta\,d\phi.</math>
| |
| This can be seen as a special case of the fact that differential forms transform through a pullback <math>F^*</math> as
| |
| | |
| :<math> F^*(u \; dy^1 \wedge \cdots \wedge dy^n) = (u \circ F) \det \left(\frac{\partial F^j}{\partial x^i}\right) dx^1 \wedge \cdots \wedge dx^n </math>
| |
| | |
| == Volume element of a linear subspace ==
| |
| Consider the [[linear subspace]] of the ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> that is spanned by a collection of [[linearly independent]] vectors
| |
| :<math>X_1,\dots,X_k.</math>
| |
| To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the <math>X_i</math> is the square root of the [[determinant]] of the [[Gramian matrix]] of the <math>X_i</math>:
| |
| :<math>\sqrt{\det(X_i\cdot X_j)_{i,j=1\dots k}}.</math>
| |
| | |
| Any point ''p'' in the subspace can be given coordinates <math>(u_1,u_2,\dots,u_k)</math> such that
| |
| :<math>p = u_1X_1 + \cdots + u_kX_k.</math>
| |
| At a point ''p'', if we form a small parallelepiped with sides <math>du_i</math>, then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix
| |
| :<math>\sqrt{\det\left((du_i X_i)\cdot (du_j X_j)\right)_{i,j=1\dots k}} = \sqrt{\det(X_i\cdot X_j)_{i,j=1\dots k}}\; du_1\,du_2\,\cdots\,du_k.</math> | |
| This therefore defines the volume form in the linear subspace.
| |
| | |
| ==Volume element of manifolds==
| |
| On a [[Riemannian manifold]] of dimension ''n'', the volume element is given in coordinates by
| |
| :<math>dV = \sqrt{\det g}\, dx^1\cdots dx^n</math> | |
| where <math>\det g</math> is the [[determinant]] of the [[metric tensor]] ''g'' written in the coordinate system.
| |
| | |
| === Area element of a surface ===
| |
| A simple example of a volume element can be explored by considering a two-dimensional [[surface]] embedded in ''n''-dimensional [[Euclidean space]]. Such a volume element is sometimes called an ''area element''. Consider a subset <math>U \subset \mathbf{R}^2</math> and a mapping function
| |
| | |
| :<math>\varphi:U\to \mathbf{R}^n</math>
| |
| | |
| thus defining a surface embedded in <math>\mathbf{R}^n</math>. In two dimensions, volume is just area, and a volume element gives a way to determine the area of parts of the surface. Thus a volume element is an expression of the form
| |
| | |
| :<math>f(u_1,u_2)\,du_1\,du_2</math>
| |
| | |
| that allows one to compute the area of a set ''B'' lying on the surface by computing the integral
| |
| | |
| :<math>\operatorname{Area}(B) = \int_B f(u_1,u_2)\,du_1\,du_2.</math>
| |
| | |
| Here we will find the volume element on the surface that defines area in the usual sense. The [[Jacobian matrix]] of the mapping is
| |
| | |
| :<math>\lambda_{ij}=\frac{\partial \varphi_i} {\partial u_j}</math>
| |
| | |
| with index ''i'' running from 1 to ''n'', and ''j'' running from 1 to 2. The Euclidean [[metric (mathematics)|metric]] in the ''n''-dimensional space induces a metric <math>g=\lambda^T\lambda</math> on the set ''U'', with matrix elements
| |
| | |
| :<math>g_{ij}=\sum_{k=1}^n \lambda_{ki} \lambda_{kj}
| |
| = \sum_{k=1}^n
| |
| \frac{\partial \varphi_k} {\partial u_i}
| |
| \frac{\partial \varphi_k} {\partial u_j}.
| |
| </math>
| |
| | |
| The [[determinant]] of the metric is given by
| |
| | |
| :<math>\det g = \left|
| |
| \frac{\partial \varphi} {\partial u_1} \wedge
| |
| \frac{\partial \varphi} {\partial u_2}
| |
| \right|^2 = \det (\lambda^T \lambda)</math>
| |
| | |
| For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2.
| |
| | |
| Now consider a change of coordinates on ''U'', given by a [[diffeomorphism]]
| |
| | |
| :<math>f \colon U\to U , \,\!</math>
| |
| | |
| so that the coordinates <math>(u_1,u_2)</math> are given in terms of <math>(v_1,v_2)</math> by <math>(u_1,u_2)= f(v_1,v_2)</math>. The Jacobian matrix of this transformation is given by
| |
| | |
| :<math>F_{ij}= \frac{\partial f_i} {\partial v_j}.</math>
| |
| | |
| In the new coordinates, we have
| |
| | |
| :<math>\frac{\partial \varphi_i} {\partial v_j} =
| |
| \sum_{k=1}^2
| |
| \frac{\partial \varphi_i} {\partial u_k}
| |
| \frac{\partial f_k} {\partial v_j}
| |
| </math>
| |
| | |
| and so the metric transforms as
| |
| | |
| :<math>\tilde{g} = F^T g F </math>
| |
| | |
| where <math>\tilde{g}</math> is the pullback metric in the ''v'' coordinate system. The determinant is
| |
| | |
| :<math>\det \tilde{g} = \det g (\det F)^2. </math>
| |
| | |
| Given the above construction, it should now be straightforward to understand how the volume element is invariant under an orientation-preserving change of coordinates.
| |
| | |
| In two dimensions, the volume is just the area. The area of a subset <math>B\subset U</math> is given by the integral
| |
| | |
| :<math>\begin{align}
| |
| \mbox{Area}(B)
| |
| &= \iint_B \sqrt{\det g}\; du_1\; du_2 \\
| |
| &= \iint_B \sqrt{\det g} \;|\det F| \;dv_1 \;dv_2 \\
| |
| &= \iint_B \sqrt{\det \tilde{g}} \;dv_1 \;dv_2.
| |
| \end{align}</math>
| |
| | |
| Thus, in either coordinate system, the volume element takes the same expression: the expression of the volume element is invariant under a change of coordinates.
| |
| | |
| Note that there was nothing particular to two dimensions in the above presentation; the above trivially generalizes to arbitrary dimensions.
| |
| | |
| ===Example: Sphere===
| |
| For example, consider the sphere with radius ''r'' centered at the origin in '''R'''<sup>3</sup>. This can be parametrized using [[spherical coordinates]] with the map
| |
| :<math>\phi(u_1,u_2) = (r\cos u_1\sin u_2,r\sin u_1\sin u_2,r\cos u_2).</math>
| |
| Then
| |
| :<math>g = \begin{pmatrix}r^2\sin^2u_2 & 0 \\ 0 & r^2\end{pmatrix},</math>
| |
| and the area element is
| |
| :<math> \omega = \sqrt{\det g}\; du_1 du_2 = r^2\sin u_2\, du_1 du_2.</math>
| |
| | |
| ==See also==
| |
| * [[Cylindrical coordinate system#Line and volume elements]]
| |
| * [[Spherical coordinate system#Integration and differentiation in spherical coordinates]]
| |
| | |
| ==References==
| |
| * {{Citation|last1=Besse|first1=Arthur L.|title=Einstein manifolds|publisher=[[Springer-Verlag]]|location=Berlin, New York|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10|isbn=978-3-540-15279-8|year=1987|pages=xii+510}}
| |
| | |
| [[Category:Measure theory]]
| |
| [[Category:Integral calculus]]
| |
| [[Category:Multivariable calculus]]
| |