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Unformatted text preview: (0,0) , (1,0) , (1,1) , and (0,1). for a particle moving counterclockwise around the curve C: Now let’s evaluate the line integral directly to show it equals negative two. Path 1: y=0 dy = 0 dx 0<x<1 Path 2: x=1 dx = 0 dy 0<y<1 Path 3: y=1 dy = 0 dx 1>x>0 Path 4: x=0 dx = 0 dy 1>y>0 Summing up the values = -2
Use Green’s Theorem to evaluate the line integral where C is represented in the following picture Example:
Use Green’s Theorem to evaluate the line integral where C is the closed curve bounded by Notice we can use Green’s Theorem to show that a conservative vector field on a closed curve will have Green’s Theorem can also be used to come up with an Area Formula with a line integral. Example:
Find the area of the ellipse: We need to parameterize the curve:
Thus Section 14.5 Curl and Divergence
Let’s first start with their definitions and work from there… What happens if is a conservative vector field? So we have a way to see if 3D vector fields are conservative – If
There is a technicality to this though – the first partial derivatives must be continuous in for This is called the Laplacian of f . When we set it equal to zero we have what is called Laplace’s Equation: This is equation pops up in all sorts of applications. Example:
Hydrodynamics: Motion of Compressible Fluid
We will look at the motion of a fluid in a region R having no sources or sinks. Ie: no points at which fluid is produced or
disappears. The concept of fluid state covers liquid, gases and vapors. Compressibility means the density (mass per
unit volume) depends on the point in space and possibly on time.
We consider the flow through a small rectangular parallelepiped W of dimensions
where the edges of W
parallel the coordinate axes. Thus the volume of W is
. Let the velocity vector of motion be
and let which represents the movement of the mass of the fluid We now want to investigate the FLUX across the boundary of W. Meaning the NET CHANGE in mass leav...
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