For a volume charge distribution ( such as charge within a bulk metal ) where gives the charge per unit volume at position, and is an infinitesimal element of volume,
12.
For a linear charge distribution ( a good approximation for charge in a wire ) where gives the charge per unit length at position, and is an infinitesimal element of length,
13.
For a surface charge distribution ( a good approximation for charge on a plate in a parallel plate capacitor ) where gives the charge per unit area at position, and is an infinitesimal element of area,
14.
From equilibrium of forces on the infinitesimal element, the magnitudes of the normal stress \ sigma _ \ mathrm { n } and the shear stress \ tau _ \ mathrm { n } are given by:
15.
Nelson's formulation is made more accessible for the lay-mathematician by leaving out many of the complexities of meta-mathematical logic that were initially required to justify rigorously the consistency of number systems containing infinitesimal elements.
16.
The model consists of an " infinite series " of the infinitesimal elements shown in the figure, and that the values of the components are specified " per unit length " so the picture of the component can be misleading.
17.
With a horizontal movement or two, we can find a way to the surface, so we might as well integrate over z after all . ( The container is not symmetrical, but all that fluid mechanics requires is local symmetry : each infinitesimal element of fluid is symmetrical .)
18.
For a volume charge distribution ( such as charge within a bulk metal ) where \ rho ( \ mathbf { r ^ \ prime } ) gives the charge per unit volume at position \ mathbf { r ^ \ prime }, and dV ^ \ prime is an infinitesimal element of volume,
19.
For a linear charge distribution ( a good approximation for charge in a wire ) where \ lambda ( \ mathbf { r ^ \ prime } ) gives the charge per unit length at position \ mathbf { r ^ \ prime }, and dl ^ \ prime is an infinitesimal element of length,
20.
For a surface charge distribution ( a good approximation for charge on a plate in a parallel plate capacitor ) where \ sigma ( \ mathbf { r ^ \ prime } ) gives the charge per unit area at position \ mathbf { r ^ \ prime }, and dA ^ \ prime is an infinitesimal element of area,
