Unit 9 - Physics 225 Relativity and Math Applications...

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Physics 225 Relativity and Math Applications Spring 2010 Unit 9 Differential Calculus … in 3D N.C.R. Makins University of Illinois at Urbana-Champaign © 2010
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Physics 225 9.2 9.2
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Physics 225 9.3 9.3 Unit 9: Differential Calculus … in 3D We’re done with Special Relativity and we’re moving on. Our overarching goal for the second half of the course is to study electromagnetism in the way it was meant to be studied: using 3D differential calculus . Along the way, we will encounter many mathematical tools, introducing them always in a physical context. Our destination is to see Maxwell’s equations in their most powerful form — differential form — and learn how to use them. When we reach our goal, I promise you: you will be so familiar with differential calculus that you will be able to read these beautiful equations like English sentences. Exercise 9.1: Fields Death to jargon! “Jargon” means “a technical term that is used to conveniently represent an idea in some specific area of expertise”. To experts in that area, jargon is great: everyone knows what it means, so you can use one word of jargon to convey a complex idea that would otherwise take a whole sentence to explain. But until you become an expert, jargon is miserable and can be very dangerous! When an expert-sounding person permeates their speech with jargon whose meaning is unclear to you, you must ask . “What does word X mean?” All too frequently, people (not just students!) are intimidated by jargon. People are generally scared of sounding ignorant, so they don’t ask what word X actually means because they think it’s supposed to be obvious. Pretty soon, they’ve heard word X enough to figure out roughly what it means, so they start constructing sentences using word X (basically by copying what they’ve heard other people say). Before long, this becomes a terrible, terrible habit. You can find yourself far down the road of your studies and routinely using terms whose meaning was never clear to you in the first place. Building your knowledge on jargon you don’t really understand is like building a house on sand: it gets shakier and shakier as it grows, and that is one very uncomfortable house to live in. You know what? Most of the time, the fancy-sounding jargon that makes folks sound so expert and seems so intimidating isn’t complicated at all . You just have to ask … and ask early! Here is our first piece of Jargon Assassination for today: fields . Electric fields, force fields, gravitational fields, magnetic fields … Fields start popping up all over the place in 212 and they sound like something very deep and profound. We’ll they’re not. A field is a function that depends on more than one spatial coordinate , That’s it: a field is a function in 2D or 3D space. f ( x ) is an ordinary 1- dimensional function, while f ( x , y ) is a field in 2D space, and f ( x , y , z ) is a field in 3D space. A field is very much like a map : it’s a map of some interesting quantity within a region of 2D or 3D space.
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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 9 - Physics 225 Relativity and Math Applications...

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