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# Unit 12 - Physics 225 Relativity and Math Applications...

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Physics 225 Relativity and Math Applications Spring 2010 Unit 12 Multidimensional Integration N.C.R. Makins University of Illinois at Urbana-Champaign © 2010

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Physics 225 12.2 12.2
Physics 225 12.3 12.3 Unit 12: Multidimensional Integration Exercise 12.1: Recap of curvilinear coordinate systems A coordinate system is simply a way of uniquely labeling points in space . So far, all our work in multidimensional calculus has been restricted to the Cartesian coordinate system, where spacepoints are labeled by the coordinates ( x , y , z ). In your lecture this week, you learned the essential features of curvilinear coordinate systems , which include angles as coordinates. The two systems in common use are spherical ( r , θ , φ ) and cylindrical coordinates ( s , φ , z ). Now what exactly do you need to know about these (or any other) coordinate systems in order to use them? Four things. Let’s recap them from lecture. 1. The transformation equations to/from Cartesian coordinates. Cartesian coordinates are the bedrock on which all other coordinate systems rest. Every coordinate system is defined in terms of Cartesian coordinates. Fortunately, you do not need to memorize these transformation equations … you need to memorize the meaning of ( r , θ , φ ) and ( s , φ , z ), then Draw! A! Sketch! Here are those sketches the drawings that define the spherical and cylindrical coordinate systems: (a) Using only the sketches, write down the transformation equations that get you back and forth from Cartesian to/from spherical and cylindrical. I’ve given you a couple to get you started: SPHERICAL CYLINDRICAL x = r = x 2 + y 2 + z 2 x = s = x 2 + y 2 y = θ = y = φ = z = r cos ! φ = z = z z = z z x y φ θ r SPHERICAL CYLINDRICAL z x y φ s z

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Physics 225 12.4 12.4 Before we continue with the essential features of our new coordinate systems, let me highlight in a big box the one feature of curvilinear coordinate systems that makes them fundamentally different — and more complicated — than Cartesian coordinates: The unit vectors ˆ r , ˆ s , ˆ ! , and ˆ ! are position-dependent . Do you remember this point from lecture? It’s important, don’t forget … Now, onward: 2. The position vector ! r The generic vector of position ! r appears in countless physics formulas. It represents the “driving directions” that get you from the origin to any spacepoint ( x , y , z ) , ( r , ! , " ) or ( s , ! , z ) of your coordinate system. In Cartesian, it’s simple: ! r = x ˆ x + y ˆ y + z ˆ z . In words, this means: “Start at the origin, then move a distance x in the ˆ x direction, then a distance y in the ˆ y direction, then a distance z in the ˆ z direction … et voila! You are now at the point ( x , y , z ) .” (b) What is the position vector ! r in spherical and cylindrical coordinates? You only need the sketches to figure it out, but think carefully … and remember what’s in the box above … 3. The line element d !
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Unit 12 - Physics 225 Relativity and Math Applications...

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