Unit 10 - Physics 225 Relativity and Math Applications...

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Physics 225 Relativity and Math Applications Spring 2010 Unit 10 Vector Fields and their Friends Div & Curl N.C.R. Makins University of Illinois at Urbana-Champaign © 2010
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Physics 225 10.2 10.2
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Physics 225 10.3 10.3 Unit 10: Vector Fields and their Friends Div and Curl Remember our jargon-busting from last week? We took the mystery out of the word “ field ” and defined it to be what it is: a function in 2D or 3D space. A field is just a map of some interesting quantity that varies with position in some region of space. We also noted that there are two kinds of fields: A scalar field is a field that describes a scalar quantity, i.e., one without direction . Example: the temperature throughout this room is a scalar field T ( x , y , z ). A vector field is a field that describes a vector quantity, i.e., a quantity with direction . Example: the velocity of the air currents in this room is a vector field ! v ( x , y , z ) Last week, we got lots of practice visualizing and working with scalar fields . We also learned how to take partial derivatives of such scalar fields and combine them into the gradient operator. Today we’ll develop that same level of intuition about vector fields : how to visualize them, how to take their partial derivatives, and how to combine those partials into the divergence and curl operators. Exercise 10.1: Vector Fields (a) Quick recap from last week: come up with three examples of a vector field, all of which have some physical meaning and could be measured in this room. Vector fields are rather easy to plot. Last week we studied three different ways to represent a scalar field graphically. For vector fields, there are only two such representations, and we’ll skip the second one as it is too subtle for our introduction. The easiest way to draw a vector field is a simple sampling method: just draw little arrows at a sample of points in your space. The size and direction of the arrows shows the size and direction of the field at the arrow’s location. Nothin’ to it. (b) At right is a picture of a vector field ! E ( x , y ) in 2D space. What field is this? Take your best guess and write down a mathematical expression for ! E ( x , y ) .
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Physics 225 10.4 10.4 (c) For any scalar function V ( x , y , z ) , the gradient ! ! V ( x , y , z ) is a vector function with a special meaning . To make sure you remember that meaning, here’s a matchup exercise for you. On the left are contour maps of four scalar fields V ( x , y , z ) , with darker colors indicating larger values; on the right are their gradient fields ! ! V ( x , y , z ) . Which goes with which?
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Physics 225 10.5 10.5 (d) One more guess-the-function: what is this vector field? (e)
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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 10 - Physics 225 Relativity and Math Applications...

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