Lesson plan
Volume element
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 \mathrm{d}V = \rho(u_1,u_2,u_3)\,\mathrm{d}u_1\,\mathrm{d}u_2\,\mathrm{d}u_3 where the u_i are the coordinates, so that the volume of any set B can be computed by \operatorname{Volume}(B) = \int_B \rho(u_1,u_2,u_3)\,\mathrm{d}u_1\,\mathrm{d}u_2\,\mathrm{d}u_3. For example, in spherical coordinates \mathrm{d}V = u_1^2\sin u_2\,\mathrm{d}u_1\,\mathrm{d}u_2\,\mathrm{d}u_3 , and so \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.
Article summary
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 \mathrm{d}V = \rho(u_1,u_2,u_3)\,\mathrm{d}u_1\,\mathrm{d}u_2\,\mathrm{d}u_3 where the u_i are the coordinates, so that the volume of any set B can be computed by \operatorname{Volume}(B) = \int_B \rho(u_1,u_2,u_3)\,\mathrm{d}u_1\,\mathrm{d}u_2\,\mathrm{d}u_3. For example, in spherical coordinates \mathrm{d}V = u_1^2\sin u_2\,\mathrm{d}u_1\,\mathrm{d}u_2\,\mathrm{d}u_3 , and so \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.
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