Notes 2 - EE468G NOTES(2 Reading assignment Chapter 1 Contents Calculus of scalar and vector fields Spatial differentiation and integration of

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EE468G NOTES (2) Reading assignment: Chapter 1 Contents: Calculus of scalar and vector fields
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Spatial differentiation and integration of scalar and vector functions Outline: Gradient, divergence, and curl operators Volume, surface, and line integral Theorems about integrals Gradient: Operated on a scalar function Result is a vector The magnitude of the gradient is the max rate of change The direction is the direction of max rate of change In Cartesian system: gradient of scalar function (, ,) fxyz is defined as () ˆˆˆ grad ( , , ) , , xyz fff f xyz a a a ∂∂∂ =∇ = + + Contour of a scalar function Gradient: A vector Magnitude is the rate of change In the direction of max rate of change.
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Solution: (1) By definition ( ) ( ) ( ) () yz a z x a xy a z yz y x a y yz y x a x yz y x a f z y x z y x 10 ˆ 5 3 ˆ 6 ˆ 5 3 ˆ 5 3 ˆ 5 3 ˆ 2 2 2 2 2 2 2 2 + + + = + + + + + = (2) The max rate of change at P(1,2,0) is given by 71 . 6 45 0 2 10 0 5 1 3 2 1 6 2 2 2 2 2 0 , 2 , 1 = = + + + = f (3) First we need to determine the direction of P1-P2: ()( ) 22 ˆˆˆ ˆ ˆ 10 21 ˆˆ ˆ ˆ ˆ 11 2 x yz x y lx y x y la a a aa al l =− + + = + == + + = + r rr The rate of change in the direction of P1-P2 is: [] 2 5 3 2 6 2 / ˆ ˆ ˆ 10 ˆ 5 3 ˆ 6 ˆ 2 2 2 2 z x xy a a a yz a z x a xy a F y x z y x l + + = + + + + = (0,1,2) 30 52 601 ˆ 10 2 l fa ⋅+ ⋅⋅ ∇⋅ = + = Example : given ,, 3 5 f xyz xy y z =+ , (1) Calculate f . (2) Calculate the max rate of change of f at point P(1,2,0). (3) Calculate the rate of change of f along the direction from point P1(0,1,2) to P2(1,2,2) evaluated at P(1,2,0).
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The gradient operation in cylindrical system: ˆˆ ˆ z fff fa a a z ρφ ρρ φ ∂∂∂ ∇= + + ∂∂ The gradient operation in sphereical system: ˆ sin r ff f a a rr r θφ θθ + + Solution: In Cartesian system: y z y x a z f a y f a x f a f ˆ ˆ ˆ ˆ = + + = In cylindrical system: () ,, s i n fz ρ = sin sin ˆˆˆ ˆ sin cos a a a φφ + = + In spherical system: s i n c o s fr r θ = sin cos sin cos sin cos sin ˆ sin cos cos cos sin r r rrr a a r aa a + + =+ Example : given fxyz y = , Calculate f in Cartesian, cylindrical, and spherical systems:
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Divergence: Operated on a vector function The result is a scalar function.
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This note was uploaded on 01/19/2012 for the course EE 461 taught by Professor Cambron during the Spring '11 term at Ohio State.

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Notes 2 - EE468G NOTES(2 Reading assignment Chapter 1 Contents Calculus of scalar and vector fields Spatial differentiation and integration of

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