Unit 11 - Physics 225 Relativity and Math Applications...

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Physics 225 Relativity and Math Applications Spring 2010 Unit 11 The Gauss-Green-Stokes Theorem N.C.R. Makins University of Illinois at Urbana-Champaign © 2010
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Physics 225 11.2 11.2
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Physics 225 11.3 11.3 Unit 11: The Gauss-Green-Stokes Theorem Exercise 11.1: Recap of ABC = Grad, Div, Curl … Intro of GGS = Gauss, Green, Stokes Over the past two weeks, we have studied in detail the three fundamental operators of differential vector calculus: gradient , divergence , and curl . First, here are the formulas: ! ! " # # x , # # y , # # z $ % ( ) grad( V ) ! ! " V div( ! E ) ! ! " # ! E curl( ! E ) ! ! " # ! E The most important aspect of our work was determining the physical meaning of these three operations. Here is what we discovered: grad: ! ! V tells you the direction of maximum increase of a scalar field V at each point ( x , y , z ) ; its magnitude gives you the slope in this “ uphill ” direction. div: ! ! " ! E tells you how much outward flux of the field ! E emanates from each point. curl: ! ! " ! E tells you the axis around which the field twists maximally ; if ! E is a force field, and you place an object at a point ( x , y , z ) , the curl indicates the torque on that object. (a) Warmup! The plot at right shows a scalar field V ( x , y , z ) . Indicate on the plot the direction of the gradient ! ! V at a selection of points. Be sure to include at least one point in each quadrant. Is the gradient zero anywhere? (b) The next plot shows a vector field ! E ( x , y , z ) . Indicate on the plot the direction of the curl ! ! " ! E at a selection of points. Be sure to include one in each quadrant. Is the curl zero anywhere?
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Physics 225 11.4 11.4 (c) Same field, different question. Indicate on the plot the sign of the divergence ! ! " ! E at a selection of points. Include one point in each quadrant. Is the divergence zero anywhere? (d) Finally, let’s make sure your calculational skills are in shape as well. The vector field shown above is ! E ( x , y , z ) = 4 x 2 y 2 ˆ x + ˆ y . Calculate its divergence and curl, and check your work against your physically-motivated results from parts (b) and (c). If there are any discrepancies, be sure to ask your instructor we are moving on! We have a lock on the qualitative meaning of grad, div, and curl … but we don’t yet have a quantitative description of exactly what we mean by “axis of twist” (curl) and “flux emanating from a point” (div). That’s where the Gauss-Green-Stokes Theorem comes in. It is actually a class of theorems that all have this form: df ! ! = f " ! ! Here, ! denotes a region : a 1D, 2D, or 3D section of space. The symbol ! ! denotes the boundary of that region. The integrands f and df denote respectively “some field f ” and “some derivative of f” . That derivative can be a regular 1D derivative, or a gradient, divergence or curl each type of derivative makes an appearance in one of the many incarnations of the GGS theorem. To figure out how it all goes together, we must now turn to
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This note was uploaded on 10/06/2011 for the course PHYS 225 taught by Professor Makins during the Spring '10 term at University of Illinois, Urbana Champaign.

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Unit 11 - Physics 225 Relativity and Math Applications...

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