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by picking a simple surface with the same boundary. RIGHT HAND RULE FOR STOKES THEOREM!!!!! Conservative vector fields are irrotational because curl F = 0 Regions in space are bounded by surfaces rather than curves for 2D (xy-plane) Laplace’s Equation
A function that satisfies this equation in a region D is said to be Harmonic in D
Ex: f(x,y) = x^3 -3xy^2 is harmonic in the xy-plane Physical Interpretation of Divergence
div F(x0 , y0 , z0) is the flux per unit volume at the point (x0 , y0 , z0)
The term flux is used to denote the NET rate at which fluid flows across a boundary. is a flux integral and measures the new flow of fluid outward across the boundary C of the region S
Interpretation of div F at a point (x0 , y0)
Flux of F across Cr =
We are looking at an infinitesimal small circle around the point (x0 , y0).
for the entire circle. We know the area of the circle is
So the sign of the flux integral is completely dependent on the sign of
There is a source of fluid at the point
There is a sink for the fluid at the point
There are no sources or sinks at the point Let be a velocity field. For small For
and are approximately constant and
in units ( ft2 / sec , area per unit time ) ,we pick an arbitrary point (xi , yi) to represent the subsection.
is approximately linear so we can find the amount of fluid crossing Div F is a measure of the local behavior of F
If div F > 0 Then more fluid is flowing out across a small sphere centered at (x0 , y0 , z0) than is flowing in
The fluid is expanding at (x0 , y0 , z0) so this point is called a source. If div F < 0 Then more fluid is flowing in across a small sphere centered at (x0 , y0 , z0) than is flowing out
The fluid is contracting at (x0 , y0 , z0) so this point is called a sink. If div F = 0 Then the fluid flowing out across a small sphere centered at (x0 , y0 , z0) equals the amount flowing in
The fluid is incompressible at (x0 , y0 , z0) If curl F = 0 then the fluid flow is said to be IRROTATIONAL
Curl F measures the tendency of the fluid to rotate about (x0 , y0)
Curl F records the direction and magnitude of MAXIMUM circulation at a given point
The magnitude is a measure of the tendency of the fluid at (x0 y0 z0) to rotate and the direction
Is along the axis about which the fluid has the MAXIMAL tendency to rotate
It is a LOCAL PROPERTY of a vector field The motion of the fluid will cause the paddle wheel based at P0 to rotate most quickly when the axis points parallel to
curl F Proof t...
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