Note there is no reason why z must be taken as the

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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: Let 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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This note was uploaded on 03/05/2014 for the course MATH 101c taught by Professor Loukianoff,v during the Spring '08 term at Ohlone.

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