Book07vectors - Weather Observation and Analysis John Nielsen-Gammon Course Notes These course notes are copyrighted If you are presently

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Weather Observation and Analysis John Nielsen-Gammon Course Notes These course notes are copyrighted. If you are presently registered for ATMO 251 at Texas A&M University, permission is hereby granted to download and print these course notes for your personal use. If you are not registered for ATMO 251, you may view these course notes, but you may not download or print them without the permission of the author. Redistribution of these course notes, whether done freely or for profit, is explicitly prohibited without the written permission of the author. Chapter 7. BITS OF VECTOR CALCULUS 7.1 Vector Magnitude and Direction Consider the vector shown in the diagram. The vector is drawn pointing toward the upper right. The origin of the vector is, literally, the origin on this x-y plot. Suppose we want to know the magnitude of this vector. In high school you probably learned about computing vector lengths by starting ATMO 251 Chapter 7: Vector Calculus page 1 of 21
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with the magnitudes of the components of the vector, computing the squares of the lengths, summing them, and then taking the square root. Well, we’ll use that technique eventually, but that’s way too complicated for most weather analysis applications. Instead, let’s keep things simple. The length of the vector is proportional to its magnitude, so once we know what a given vector length corresponds to, we can just measure the vector and convert it to a magnitude. Since the figure in this case has a grid background, we’ll start by asking what the magnitude of a vector would be if it were exactly one grid box side long. Then we can see how many grid boxes the vector covers, and convert that to a vector magnitude. Enough hypotheticals, let’s do this for real. Let’s say the vector is the horizontal wind. The magnitude of the wind is called the wind speed. Now suppose each grid box corresponds to a wind speed of one meter per second (1 m s -1 ). If we take a ruler to the page, we find that each grid box is half an inch wide. So a vector that’s ½ inch long on this particular graph would have a magnitude of 1 m s -1 . A vector that’s an inch long would be 2 m s -1 , a vector that’s 1 ½ inches long would be 3 m s -1 , and so forth. If we measure the vector, we find that the vector is 1.6 inches long. So the wind speed is just a little bit more than 3 m s -1 . Specifically, it’s 3.2 m s -1 . For some of you, it may be obvious where that answer came from. If not, this is like any conversion problem, and solving conversion ATMO 251 Chapter 7: Vector Calculus page 2 of 21
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problems is a necessary skill, so let’s work it out in detail. There is one conversion here: 1 m s -1 = 0.5 graph inches In this case, we know the length of the vector in graph inches. Divide the conversion equation by the side with the units that have been measured: 1 m s -1 / 0.5 graph inches = 0.5 graph inches / 0.5 graph inches The right hand side is unity: a number divided by itself. Divide and simplify the left hand side so that its denominator is unity too:
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This note was uploaded on 04/02/2008 for the course ATMO 251-501/50 taught by Professor Alcorn during the Fall '07 term at Texas A&M.

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Book07vectors - Weather Observation and Analysis John Nielsen-Gammon Course Notes These course notes are copyrighted If you are presently

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