Thus if you are dealing with a complicated surface

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Unformatted text preview: roblem 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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