Physics
Volume Integral
A volume integral is a mathematical tool used to calculate the total value of a function over a three-dimensional region. It involves dividing the region into small volumes and summing up the function values over each volume. This concept is commonly used in physics to calculate quantities such as mass, charge, and energy.
Written by Perlego with AI-assistance
Related key terms
1 of 5
3 Key excerpts on "Volume Integral"
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
Surface integrals resulting in vectors occur less frequently. An example is afforded, however, by the total resultant force experienced by a body immersed in a stationary fluid in which the hydrostatic pressure is given by p ( r ). The pressure is everywhere inwardly directed and the resultant force is F = − S p d S , taken over the whole surface. 3.6 Volume Integrals • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Volume Integrals are defined in an obvious way and are generally simpler than line or surface integrals since the element of volume dV is a scalar quantity. We may encounter Volume Integrals of the forms V φ dV , V a dV . (3.12) Clearly, the first form results in a scalar, whereas the second form yields a vector. Two closely related physical examples, one of each kind, are provided by the total mass of • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 13 Probably the most familiar is Gauss’s theorem, which can be written as S E · d S = − 1 0 ∑ i q i for a system of charges q i in a vacuum that are contained within a surface S . 14 Use this result to find an expression for the solid angle enclosed by a cone of half-angle α . 148 Line, surface and Volume Integrals a fluid contained in a volume V , given by V ρ ( r ) dV , and the total linear momentum of that same fluid, given by V ρ ( r ) v ( r ) dV where v ( r ) is the velocity field in the fluid. As a slightly more complicated example of a Volume Integral we may consider the following. Example Find an expression for the angular momentum of a solid body rotating with angular velocity ω about an axis through the origin.
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
We may encounter Volume Integrals of the forms V φ dV, V a dV. (12.12) Clearly, the first form results in a scalar, whereas the second form yields a vector. Two closely related physical examples, one of each kind, are provided by the total mass of a 512 Line, surface and Volume Integrals fluid contained in a volume V , given by V ρ ( r ) dV , and the total linear momentum of that same fluid, given by V ρ ( r ) v ( r ) dV , where v ( r ) is the velocity field in the fluid. As a slightly more complicated example of a Volume Integral we may consider the following. Example Find an expression for the angular momentum of a solid body rotating with angular velocity ω about an axis through the origin. Consider a small volume element dV situated at position r ; its linear momentum is ρ dV ˙ r , where ρ = ρ ( r ) is the density distribution, and its angular momentum about O is r × ρ ˙ r dV . Thus for the whole body the angular momentum L is L = V ( r × ˙ r ) ρ dV. Since the body is solid and rotating as a whole, the velocity of an element at position r is given by ˙ r = ω × r . Substituting this yields L = V [ r × ( ω × r )] ρ dV = V ω r 2 ρ dV − V ( r · ω ) r ρ dV. It should be noted that both integrals produce vectors; the first is necessarily positive and in the direction of ω , but the second could be in any direction. The evaluation of the first type of Volume Integral in ( 12.12 ) has already been considered in our discussion of multiple integrals in Chapter 8 . The evaluation of the second type of Volume Integral follows directly, since we can write V a dV = i V a x dV + j V a y dV + k V a z dV, (12.13) where a x , a y , a z are the Cartesian components of a . Of course, we could have written a in terms of the basis vectors of some other coordinate system (e.g. spherical polars) but, since such basis vectors are not, in general, constant, they cannot be taken out of the integral sign as in ( 12.13 ) and must be included as part of the integrand.
- Gary N. Felder, Kenny M. Felder(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
In the special case where S is a closed surface (a surface that completely surrounds a 3D region) we write ∯ S V ⋅ d A. To understand ∫∫ V ⋅ d A you need to understand three important physical quantities, all with different meanings and different units. ∙ V (x , y, z) could be the momentum density of a fluid, meaning its density times velocity, with units of (kg/m 3 )(m/s). We prefer to think of that as kg/m 2 /s because it repre- sents how much fluid is flowing through the surface around (x , y, z) per unit area per second. ∙ d A is the area vector of a surface. Its units are of course m 2 . ∙ If V is a momentum density then V ⋅ d A is a “mass flow rate.” That quantity has units of kg/s because it measures how much fluid is flowing through this surface every second. When you write V ⋅ d A you are calculating the flow rate through one tiny differen- tial surface. When you integrate it you are calculating the flow rate through a larger surface, but this does not change the meaning or units of the quantity. The word we are dancing around in this discussion is “flux.” In fluid flow “flux” does not exactly refer to any of the three quantities we have just mentioned: the flux is the component of the momentum density in the direction of the area. Like V it has units of kg/m 2 /s, and also like V , it is defined at a point rather than over a surface. What makes this particularly confusing is that the word is used quite differently in elec- tromagnetism. The local quantity (analogous to V in fluid flow) is an electric or magnetic field. The word flux is generally used for the integrated quantity: for example “magnetic flux” means “the surface integral of the magnetic field.” FIGURE 5.13 If this vector field represents the momentum density of the air, then its surface integral tells us how rapidly the container is losing air.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
