Examples of flux in the following topics:
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- The divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.
- More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface.
- The divergence theorem can be used to calculate a flux through a closed surface that fully encloses a volume, like any of the surfaces on the left.
- It can not directly be used to calculate the flux through surfaces with boundaries, like those on the right.
- Apply the divergence theorem to evaluate the outward flux of a vector field through a closed surface
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- If we wanted, we could obtain a general expression for the volume of blood across a cross section per unit time (a quantity called flux).
- Therefore, the total flux $F$ is written as:
- Once we have an (approximate) expression for $v(r)$, we can calculate the flux from the integral.
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- The flux is defined as the quantity of fluid flowing through $S$ in unit amount of time.
- This illustration implies that if the vector field is tangent to $S$ at each point, then the flux is zero, because the fluid just flows in parallel to $S$, and neither in nor out.
- This also implies that if $\mathbf{v}$ does not just flow along $S$—that is, if $\mathbf{v}$ has both a tangential and a normal component—then only the normal component contributes to the flux.
- Based on this reasoning, to find the flux, we need to take the dot product of $\mathbf{v}$ with the unit surface normal to $S$, at each point, which will give us a scalar field, and integrate the obtained field as above.
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- More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.