17-2 - Math 200, Spring 2010 Handout #29 Section 17-2...

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Math 200, Spring 2010 Handout #29 Section 17-2 Stokes’ Theorem Curl and Divergence Suppose 1, 2, 3 F F F = F is a vector field on 3 with partial derivatives of 1, 2, 3 and F F F all existing, and define the del operator , , x y z ∇ = . Then (1) the curl of F is the vector field defined by ( ) 3 3 2 1 2 1 1 2 3 curl , , F F F F F F x y z y z z x x y F F F = ∇× = i j k F F = and (2) the divergence of F is the scalar field defined by ( ) 3 1 2 1 2 3 div , , , , F F F F F F x y z x y z = ∇ + + F F = = i i Note: Suppose the vector field F represents the velocity field of the fluid. In hydrodynamics, we are concerned with two particular rates of change; these can be viewed as generalizations of the notion of the derivative to vector fields: 1. Circulation , or rate of rotation, of a fluid around a point. 2. "et flux , or rate of flow, of a fluid out of a volume at a point. The first is measured by curl and the second is measured by divergence . 1. Find (a) the curl and (b) the divergence of the vector field ( ) 3 5 z x y y e = + + F i j k 2. Compute the curl of (a) ( ) , , x y z x y = + F i j and (b) ( ) , , x y z y x = − + F i j and interpret each graphically. Graph of , x y Graph of , y x 3. Compute the divergence of (a) ( ) , , x y z x y = + F i j and (b) ( ) , , x y z y x = − + F i j and interpret each graphically.
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-10 -5 0 5 10 x - axis -10 -5 0 5 10 y - axis -10 -5 0 5 10 z - axis Interpretation of Curl and Divergence If F is the velocity field of a flowing fluid, div F represents the net rate of change of the mass of the fluid flowing from the point ( , , ) x y z per unit volume. This can also be thought of as the tendency of a fluid to diverge from a point . If div 0 = F , then the field F is called incompressible . Likewise, curl F represents the tendency of particles at the point ( , , ) x y z to rotate about the axis that points in the direction of curl F . If curl = F 0 , then the field F is called irrotational . Find the divergence and curl of the vector field ( , , ) x y z xy yz zx = + + F i j k at the points (0, 0, 0) and (5, 5, 5) and interpret the results.
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Conservative Vector Fields Theorem: If ϕ is a function of three variables that has second-order partial derivatives, then ( ) curl = 0 . This says that, if
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17-2 - Math 200, Spring 2010 Handout #29 Section 17-2...

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