Spherical polar coordinates integration volume element

FIGURE 10.8 Volume element in spherical polar coordinates. Integration over the two polar angles gives... 

This simply accounts for the fact that the total probability of finding the particle somewhere adds up to unity. The integration in Eq (2.41) extends over all space, with the symbol dr denoting the appropriate volume element. For example, in cartesian coordinates, dx = dxdydz in spherical polar coordinates, dx = sin 9 dr d9 dcj). 

Sometimes it is convenient to take a multiple integral over an area or over a volume using polar coordinates or spherical polar coordinates, and so on, instead of Cartesian coordinates. Figure 7.6 shows how this is done in polar coordinates. We require an infinitesimal element of area given in terms of the coordinates p and 0. One dimension of the element of area is dp and the other dimension is p dtj), from the fact that an arc length is the radius of the circle times the angle subtended by the arc, measured in radians. The element of area is p d(j) dp. If the element of area... 

The three most us ul equations for determining the density at the gas-liquid surface are the tirst YBG equation (4.34), the integral equation over the direct correlation function (4.S2), and the potential distribution theorem (4.69). At a planar surface with K- 0 these can be re-written in simpler forms. If the direction normal to the surface (the heig ) is that of the z-axis, then the only non-zero gradient is d/dz and we have cylindrical symmetry about this axis. The volume element can then be more usefully expressed in cylindrical or spherical polar coordinates. 

If an integral over a volume in three-dimensional space is needed and spherical polar coordinates are used, the volume element is as depicted in Figure B.4. The length... 

Problem. The differential volume element in spherical polar coordinates is 4V = sin 0 d(p d dr. Given that goes from 0 to 2 r, 9 goes from 0 to jr, and r goes from 0 to r, evaluate the triple integral... 

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