EEE224_Fall09_DivergenceCurl

EEE224_Fall09_DivergenceCurl - Del Operator ( ) is a vector...

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1 “Del” Operator ( ) is a vector operator that acts upon T or A. is not a vector that “multiplies” T or A. 3. On a vector function A , via the cross product: × A ( curl ) There are 3 ways the operator can act: 1. On a scalar function T : T ( gradient ) 2. On a vector function A , via the dot product: · A ( divergence )
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2 Gradient The gradient T points in the direction of maximum increase of the function T . The magnitude | T | gives the slope (rate of increase) along this maximal direction. Direction of gradient (Function, T is height) Gradient of T (vector quantity We want to “generalize” the notion of “derivative” to functions like T , which depend not on one , but on three variables. “How fast does T vary?”
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3 Gradient 1 ˆ ˆ ˆ r z V V V V a a a r r z φ = + + ˆ ˆ ˆ x y z V V V V a a a x y z = + + 1 1 ˆ ˆ ˆ sin R V V V V a a a R R R θ = + + Cylindrical Coordinates: ( 29 , , r z Cartesian Coordinates: Spherical Coordinates: ( 29 , , R θ φ
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4 Divergence Variations in a vector field can be represented by FLUX LINES , which are directed curves indicating the direction of vector fields. The magnitude of the field is emphasized by either density or length of the flux lines in the vicinity of a particular point. Velocity of fluid flow Electric field of a charge Strongest field region Flux: the rate of flow across a surface
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5 Divergence Consider a volume V enclosed by a surface S When the volume contains a “source”, there will be a net “outward flux ” through S. (+ divergence) When the volume contains a “sink”, there will be a net “inward flux ” through S. (- divergence)
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EEE224_Fall09_DivergenceCurl - Del Operator ( ) is a vector...

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