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Unformatted text preview: the dependent variable, we could use y=f(x,z) or z=f(y,z) and project
the surface into either the xz-plane or yz-plane.
Calculating Mass of a Surface
Given the mass density per unit surface area:
be a small patch of area on the surface of the object. Then
would be given by: . So the total mass of the object Example:
Find the mass of the surface where and Example:
Find the mass of the thin metal covering given by the surface
and the density is given by: Scratch: where We will now revisit Flux. We dealt with flux through a rectangular box which was easy since our surfaces were parallel
to the coordinate axes so the movement of the mass was perpendicular to the surface. Now we want to investigate the
movement of mass through an arbitrary surface in space. Thus we will need the normal vector for each point on the
surface. (Let’s assume the flow is steady, ie: not dependent on time and our density is constant throughout the fluid)
So our velocity vector field will be given by: ThtThe mass moving through the Where
Thus Flux of can be approximated by (unit normal vector) across S = doesn’t always have to be a velocity field. It could have been an electric field or a gradient field for a temperature
function. In order to use the above formula we need to find the unit normal vector . We will find the Outward Unit
Normal vector which is the vector that points upward from the surface.
We could use the same idea as above to look at the net amount of fluid
leaving S per unit time. This is a 2D curve that we partition into . We have exactly the same derivation as above
The only difference is we are in 2D and have
Thus Flux of . across C = If the integral is positive, the amount of fluid in S tends to decrease (or there is a source) – more is leaving across the
boundary than is coming in
If the integral is negative, the amount of fluid in S tends to increase (or there is a sink) – more is coming in than is leaving
If the integral is zero, there is no net gain or loss of fluid in S The word FLUX is used to denote the net rate at which fluid flows across a boundary. Remember that the Gradient is perpendicular to a Level S...
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