Published on METEO 300: Fundamentals of Atmospheric Science (https://www.e-education.psu.edu/meteo300)

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Lesson 1: Getting Started

Overview

Overview

Meteorologists devote their lives to one purpose: to forecast the weather. But many other scientists work to build our understanding of the atmosphere, which is the basis for better prediction that meteorologists can use. Atmospheric science is the foundation upon which all meteorology is based.

Just as with all other physical sciences, mathematics is the language of atmospheric science. If you had our first-year meteorology course, you learned about weather forecasting and some of the physical basis behind the forecasts. This course will introduce you to even more physical concepts of atmospheric science and some of the mathematics that describe and quantify those physical concepts.

You have learned some of the mathematics that you need for this course and will be learning more about vectors and vector calculus soon. In this lesson, we will practice some of the mathematics. You will also prepare an Excel workbook that will help you with some of this course’s assessment problems.

Learning Objectives

By the end of this lesson, you should be able to:

  • correctly use significant figures, dimensions, and units
  • solve simple problems using integral and differential calculus
  • prepare and use a course Excel workbook for course calculations 

Questions?

If you have any questions, please post them to the Course Questions discussion forum. I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.

1.1 The atmosphere is …

1.1 The atmosphere is …

The atmosphere is amazing, awe-inspiring, frightening, deadly, powerful, boring, strange, beautiful, and uplifting – just a few of thousands of descriptions. So much of our lives depend on the atmosphere, yet we often take it for granted.

Clouds over the Arctic Ocean at sunrise
Clouds over the Arctic Ocean at sunrise
Credit: W. Brune

Atmospheric science attempts to describe the atmosphere with physical descriptions using words, but also with mathematics. The goal is to be able to write down mathematical equations that capture the atmosphere’s important physical properties (predictability) and to use these equations to determine the atmosphere’s evolution with time (prediction). Predicting the weather has long been a primary focus, but, increasingly, we are interested in predicting climate.

We know quite a lot about the atmosphere. It has taken decades, if not centuries, of careful observation and insightful theory that is based on solid physical and chemical laws. We have more to learn. You could help to advance the understanding of the atmosphere, but you must first understand the physical concepts and mathematics that are already well known. That is a primary purpose of this course – to give you that understanding.

What follows, below, is a series of pictures and graphical images. Each one depicts some atmospheric process that will be covered in this course. Look at these images; you will see them again, each in one of the next ten lessons. Of course, in each observation there are many processes going on simultaneously. In the last lesson, you will have the opportunity to look at an observation and attach the physical principles and the mathematics that describe several processes that are causing the phenomena that you are observing.

Discussion Activity: Finding Errors

We have offered Meteo 300 online several times but use of the Canvas assessment platform is relatively recent. There may still be errors despite our best efforts.

As we go through the course, if you find an error or typo, post it in the Finding Errors Discussion Forum in Canvas. If you are the first to find an error, you will be awarded 0.1 additional points on your final grade. If you aren't the first one to post, you receive no credit for your find, so it's a good idea to read each lesson at the beginning of each week. A student can earn up to 1 point total (i.e., by being the first to find an error ten times). Now, I hope that we have very few errors, but ...

1.2 You won’t believe what you can do with math!

1.2 You won’t believe what you can do with math!

You’ve been told many times that meteorology is a math-intensive field. It is. But for this course, you already know much of the math, and what you haven’t seen, you will see in vector calculus. To get ready for the meteorology and atmospheric science in this course, you will need to refresh your ability to solve simple math problems, including solving simple problems in differential and integral calculus. At the same time, we will remind you about the importance of correctly specifying significant figures and units in your answers to the problems. The goal of this first lesson is to boost your confidence in the math you already know.

How many figures should be in my answer?

Suppose you are asked to solve the following word problem:

In the radar loop, a squall line is oriented in the north-south direction and is heading northeast at 57 km hr-1. In the last frame of the loop, the line is 17 km west of the Penn State campus. You are out running and know that you can make it back to your apartment in 25 minutes. Will you get back to your apartment before you get soaked?

You reason that the line is moving northeast, and thus, at an angle of 45o relative to the east. Therefore, the eastward motion of the squall line is just the velocity times the cosine of 45o. That gives you the eastward speed. You decide to divide the distance by the eastward speed to get the amount of time before the line hits campus. You plug the numbers into your calculator and get the following result:

time= 17 km  ( 57 km/h) ⋅cos( 45º ) =0.42178 hours =25.3070 minutes MathType@MTEF [1]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabshacaqGPbGaaeyBaiaabwgacqGH9aqpdaWcaaqaaiaaigdacaaI3aaabaGaaGynaiaaiEdacqGHflY1ciGGJbGaai4BaiaacohadaqadaqaaiaaisdacaaI1aaacaGLOaGaayzkaaaaaiabg2da9iaaikdacaaI1aGaaiOlaiaaiodacaaIWaGaaG4naiaaicdacaqGGaGaaeyBaiaabMgacaqGUbGaaeyDaiaabshacaqGLbGaae4Caaaa@501D@

According to your calculation, you will make it back with 0.3 minutes (18 seconds) to spare. But can you really be sure that the squall line will strike in 25.3070 minutes? Maybe you should figure out how many significant figures your answer really has. To do that, you need to remember the rules:

Significant Figures Rules

  1. Non-zero numbers (1,2,3,4,5,6,7,8,9) are ALWAYS significant.
  2. Zeroes are ALWAYS significant:
    1. between non-zero numbers
    2. SIMULTANEOUSLY to the right of the decimal point AND at the end of the number
    3. to the left of a written decimal point and part of a number ≥ MathType@MTEF [1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzImlaaa@2FDF@ 10
  3. In a calculation involving multiplication or division, multiply numbers as you see them. Then the answer should have the same number of significant figures as the number with the fewest significant figures.
  4. In a calculation involving addition and subtraction, the number of significant figures in the answer depends on the number of significant figures to the right of the decimal point when all the added or subtracted numbers are put in terms of the same power-of-ten. Add or subtract all the numbers. The answer has the same number of significant figures to the right of the decimal point as the number with the least number of significant figures to the right of the decimal point.
  5. The number of significant figures is unchanged by trigonometric functions, logarithms, exponentiation, and other related functions.
  6. Exact numbers never limit the number of significant figures in the result of a calculation and therefore can be considered to have an infinite number of significant figures. Common examples of exact numbers are whole numbers and conversion factors. For example, there are exactly 4 sides to a square and exactly 1000 m in a km.
  7. For multi-step calculations, any intermediate results should keep at least one extra significant figure to prevent round-off error. Calculators and spreadsheets will typically keep these extra significant figures automatically.
  8. When rounding, numbers ending with the last digit > 5 are rounded up; numbers ending with the last digit < 5 are rounded down; numbers ending in 5 are rounded up if the preceding digit is odd and down if it is even.
Examples
Number(s) Answer Number of Significant Figures Reason
25+.3 25 2 25 has only 2 significant figures
25·0.3 8 1 25·0.3=7.5 , round to 8 because 0.3 has only 1 significant figure
1.5 10 3 +3.24 10 2 1.8 10 3 2 1.5 10 3 +0.324 10 3 =1.824 10 3 , then drop 2 to get 1.8 10 3
1.5 10 3 +3.86 10 2 1.9 10 3 2 1.5 10 3 +3.86 10 2 =1.886 10 3 , round up then drop 2 to get 1.9 10 3
57.3+6.41 15.6 4.08 3 63.71 15.6 =4.0840 , trim to 3 significant figures to get 4.08
200 3.142 600 1 200. has 3 significant figures; 200 (no decimal point) has 1 but is ambiguous
152 e −.52 90 2 number in exponent has only 2 significant figures

Check out this video (11:23): Unit Conversions & Significant Figures for a brief (1 minute) explanation of those rules! Start watching at 9:14 for the most relevant information. Note a minor error starting at 9:50 in which "60" should actually have a decimal point following the zero.

Unit Conversions and Significant Figures
Click here for transcript of the Significant Figures video.

Now to the magic of figuring out how many sig figs your answer should have. There are two simple rules for this. If it's addition or subtraction it's only the number of figures after the decimal point that matters. The number with the fewest figures after the decimal point decides how many figures you can have after the decimal in your answer. So 1,495.2+1.9903 you do the math. First you get 1,497.1903 and then you round to the first decimal, because that first number only had one figure after the decimal. So you get 1,497.2. And for multiplication, just make sure the answer has the same sig figs as your least precise measurement. So 60 x 5.0839 = 305.034, but we only know two sig figs so everything after those first two numbers is zeroes: 300. Of course then we'd have to point out to everyone that the second zero but not the third is significant so we'd write it out with scientific notation: 3.0 * 10^2. Because science! Now I know it feels counterintuitive not to show all of the numbers that you have at your fingertips, but you've got to realize: all of those numbers beyond the number of sig figs you have? They're lies. They're big lying numbers. You don't know those numbers. And if you write them down people will assume that you do know those numbers. And you will have lied to them. And do you know what we do with liars in chemistry? We kill them! Thank you for watching this episode of Crash Course Chemistry. Today you learned some keys to understanding the mathematics of chemistry, and you want to remember this episode in case you get caught up later down the road: How to convert between units is a skill that you'll use even when you're not doing chemistry. Scientific notation will always make you look like you know what you're talking about. Being able to chastise people for using the wrong number of significant digits is basically math's equivalent of being a grammar Nazi. So enjoy these new powers I have bestowed upon you, and we'll see you next time. Crash Course Chemistry was filmed, edited, and directed by Nick Jenkins. This episode was written by me, Michael Aranda is our sound designer, and our graphics team is Thought Bubble. If you have any questions, comments or ideas for us, we are always down in the comments. Thank you for watching Crash Course Chemistry.

Credit: Crash Course [2]

What are the typical types of variables?

There are two types of variables – scalars and vectors. Scalars are amount only; vectors also have direction.

Dimensions and units are your friends.

Most variables have dimensions. The ones used in meteorology are:

  • L, length
  • T, time
  • Θ, temperature
  • M, mass
  • I, electric current

Some constants such as π have no units, but most do.

The numbers associated with most variables have units. The system of units we will use is the International System (SI, from the French Système International), also known as the MKS (meter-kilogram-second) system, even though English units are used in some parts of meteorology.

We will use the following temperature conversions:

K=o C+273.15

5 9 o F−32 = o C

We will use the following variables frequently. Note the dimensions of the variables and the MKS units that go with their numbers.

Variables With Associated Dimensions and MKS Units
Type Variable Dimensions MKS Units Common Unit Name
Scalar length (x or ...) L m
area (A) L2 m2
volume (V) L3 m3
speed (u, v, w) L/T m/s
energy (E) ML2/T2 kg m2/s2 J = Joule
power (P) ML2/T3 kg m2/s3 W = Watt
density (ρ) M/L3 kg/m3
pressure (p) M/LT2 kg/ms2 Pa = Pascal
electrical potential ML2/T3A kg m2/s3A V = Volt
temperature (T) Θ K
Vectors velocity (v) L/T m/s
momentum (mv) ML/T kg m/s
acceleration (a) L/T2 m/s2
force (F) ML/T2 kg m/s2 N = Newton

Pressure is used for many applications.

p = (normal force)/area = (mass x acceleration)/area = ML/T2L2 = M/LT2

1 Pa = 1 kg m–1 s–2; 1 hPa = 100 Pa = 1 mb = 10–3 bar (hPa = hecto-Pascal)

1013.25 hPa = 1.01325 x 105 Pa = 1 standard atmospheric pressure = 1 atm

Wind speed is another frequently used variable.

The knot (kt) is equal to one nautical mile (approximately one minute of latitude) per hour or exactly 1.852 km/hr. The mile is nominally equal to 5280 ft and has been standardized to be exactly 1,609.344 m.

Thus, 1 m/s = 3.6 km/hr ≈ 1.944 kt and 1 kt ≈ 1.151 mph.

surface winds are typically 10 kts ~ 5 m/s

500 mb winds are ~50 kts ~ 25 m/s

250 mb winds are ~100 kts ~ 50 m/s

Temperature is a third frequently used variable.

Kelvin (K) must be used in all physical and dynamical meteorology calculations. Surface temperature is reported in oF or (oC for METARS) and in oC for upper air soundings.

Water vapor mixing ratio is another frequently used variable.

w =  mass H 2 O mass dry air MathType@MTEF [1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabEhacaqGGaGaaeypaiaabccadaWcaaqaceaadbGaaeyBaiaabggacaqGZbGaae4CaiaabccacaqGibWaaSbaaSqaaiaaikdaaeqaaOGaae4taaqaaiaab2gacaqGHbGaae4CaiaabohacaqGGaGaaeizaiaabkhacaqG5bGaaeiiaiaabggacaqGPbGaaeOCaaaaaaa@47B5@

Usually the units for water vapor mixing ratio are g kg-1. In the summer w can be 10 g kg-1; in the winter, it can be 1-2 g kg-1.

Dimensions truly are your friend. Let me give you an example. Suppose you have an equation ax + b = cT, and you know the dimension of b, x (a distance), and T (a temperature), but not a and c. You also know that each term in the equation – the two on the left-hand side and the one on the right-hand side – must all have the same units. Therefore, if you know b, you know that the dimensions of a must be the same as the dimensions of b divided by L (length) and the dimensions of c must be the same as the dimensions of b divided by Θ.

Also, if you invert a messy equation and you're not sure that you didn’t make a mistake, you can check the dimensions of the individual terms and if they don’t match up, it’s time to look for your mistake. Or, if you have variables multiplied or divided in an exponential or a logarithm, the resulting product must have no units.

Always write units down and always check dimensions if you aren’t sure. That way, you won’t crash your spacecraft on the back side of Mars [3]. View the following video (2:42).

When NASA Lost a Spacecraft Because it Didn't Use Metric
Click here for transcript of the NASA video.

Remember when NASA lost a spacecraft because it's simultaneously used Imperial and metric measurements on the same mission? The Mars Climate Orbiter disappeared 15 years ago this month and here's a very brief recap of exactly what went wrong. The Mars Climate Orbiter launched on December 11, 1998 on a mission to orbit Mars. This first interplanetary weather satellite was designed to gather data on Mars' climate and also serve as a relay station for the Mars Polar Lander, a mission that launched a few weeks later. But you can't just launched a spacecraft towards Mars and trust that it's going to get where it's going. You to have to monitor its progress. Many spacecraft have reaction wheels to keep them oriented properly and navigation teams behind interplanetary spacecraft that constantly monitor the angular momentum and adjust trajectory to make sure it gets exactly where it needs to go. In the case of the Mars Climate Orbiter, monitoring its trajectory and angular momentum involved a few steps. First, data from the spacecraft was transferred to the ground by telemetry. There it was processed by a software program and stored in an angular momentum desaturation file that process data was what scientists used to adjust the trajectory. Adjustments that were made by firing the spacecraft's thrusters. Every time the thrusters were fired, the resulting change in velocity was measured twice once by software program on the spacecraft and once by software program off the ground. And here's where the problem comes in. It turned out that the two systems the processing software on the spacecraft and the software on the ground we're using two different units of measurements. The software on the spacecraft measured impulse, or the changes by thrusters in newton seconds a commonly accepted metric unit of measurement, while the processing software on the ground use the Imperial pound seconds. And it was unfortunately the ground computer's data that scientists used to update the spacecraft trajectory and because one pound of force is equal to 4.45 Newton's every adjustment was off by a factor of 4.45. For a spacecraft traveling tens of millions of miles to destination a number of seemingly small errors really add up. During the Mars Climate Orbiters nine-month cruise to Mars seven errors were introduced into its trajectory that meant that when it reached the red planet it was 105 miles closer to the Martian surface than expected. This turned out to be an unsurvivably low altitude for its Mars encounter when the spacecraft fire its main engine for the orbit insertion burn that was designed to put it into an elliptical orbit nothing happened. NASA lost contact quite abruptly with the spacecraft. So while we know the root cause of just what went wrong we'll never know exactly what happened to the Mars Climate Orbiter. The loss of the Mars Climate Orbiter very sadly happened in space. Leave your spacey questions and comments below, and don't forget to subscribe.

Credit: Scientific American Space Lab [4]

Quiz 1-1: Significant figures, dimensions, and units.

Now it's time to to take a quiz. I highly recommend that you begin by taking the Practice Quiz before completing the graded Quiz. Practice Quizzes are not graded and do not affect your grade in any way (except to make you more competent and confident to take the graded Quizzes : ).

  1. In Canvas, find Practice Quiz 1-1. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz.
  2. When you feel you are ready, take Quiz 1-1. You will be allowed to take this quiz only once. This quiz is timed, so after you start, you will have a limited amount of time to complete it and submit it. Good luck!

1.3 If you thought practice makes perfect, you could be right.

1.3 If you thought practice makes perfect, you could be right.

Calculus is an integral part of a meteorologist’s training. The ability to solve problems with calculus differentiates meteorologists from weather readers. You should know how to perform both indefinite and definite integrals. Brush up on the derivatives for variables raised to powers, logarithms, and exponentials. We will take many derivatives with respect to time and to distance.

Need Extra Practice?

Visit the Khan Academy website that explains calculus with lots of examples [5], practice problems, and videos. You can start with single variable calculus, but may find it useful for more complicated calculus problems. 

Simple Integrals and Derivatives That are Frequently Used to Describe the Behavior of Atmospheric Phenomena

1. 

da dt =−ka da a =−kdt ∫ a o a 1 da a =− ∫ t o t 1 kdt ln( a 1 )−ln( a 0 )=−k( t 1 − t 0 ) ln( a 1 / a 0 )=−k( t 1 − t 0 ) a 1 / a 0 = e ( −k( t 1 − t 0 ) ) =exp( −k( t 1 − t 0 ) ) a 1 = a 0 e ( −k( t 1 − t 0 ) ) = a 0 exp( −k( t 1 − t 0 ) ) MathType@MTEF [1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@B6C6@

2.  p= p o e ( −z/H )  ;  ∫ 0 ∞ pdz= ?     (Do the definite integral.) MathType@MTEF [1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9iaadchadaWgaaWcbaGaam4BaaqabaGccaWGLbWaaWbaaSqabeaadaqadaqaaiabgkHiTiaadQhacaGGVaGaamisaaGaayjkaiaawMcaaaaakiaaywW7caGG7aGaaGzbVpaapehabaGaamiCaiaadsgacaWG6bGaeyypa0JaaGjbVlaab+dacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGebGaae4BaiaabccacaqG0bGaaeiAaiaabwgacaqGGaGaaeizaiaabwgacaqGMbGaaeyAaiaab6gacaqGPbGaaeiDaiaabwgacaqGGaGaaeyAaiaab6gacaqG0bGaaeyzaiaabEgacaqGYbGaaeyyaiaabYgacaqGUaGaaeykaaWcbaGaaGimaaqaaiabg6HiLcqdcqGHRiI8aaaa@615B@

∫ 0 ∞ p dz= −H p o e −z/H | 0 ∞ =−H p o ( 0−1 )= p o H MathType@MTEF [1]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapehabaGaamiCaiaaykW7caWGKbGaamOEaiabg2da9maaeiaabaGaeyOeI0IaamisaiaadchadaWgaaWcbaGaam4BaaqabaGccaWGLbWaaWbaaSqabeaacqGHsislcaWG6bGaai4laiaadIeaaaaakiaawIa7aaWcbaGaaGimaaqaaiabg6HiLcqdcqGHRiI8aOWaa0baaSqaaiaaicdaaeaacqGHEisPaaGccqGH9aqpcqGHsislcaWGibGaamiCamaaBaaaleaacaWGVbaabeaakmaabmaabaGaaGimaiabgkHiTiaaigdaaiaawIcacaGLPaaacqGH9aqpcaWGWbWaaSbaaSqaaiaad+gaaeqaaOGaamisaaaa@55F1@

3.  p= p 0 e ( − z H ) ; 1 p dp dz =? MathType@MTEF [1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9iaadchadaWgaaWcbaGaaGimaaqabaGccaWGLbWaaWbaaSqabeaadaqadaqaaiabgkHiTmaalaaabaGaamOEaaqaaiaadIeaaaaacaGLOaGaayzkaaaaaOGaai4oamaalaaabaGaaGymaaqaaiaadchaaaWaaSaaaeaacaWGKbGaamiCaaqaaiaadsgacaWG6baaaiabg2da9iaac+daaaa@3F8A@

dp dz =− 1 H p 0 e −z H =− 1 H p; 1 p dp dz =− 1 H MathType@MTEF [1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaamiCaaqaaiaadsgacaWG6baaaiabg2da9iabgkHiTmaalaaabaGaaGymaaqaaiaadIeaaaGaamiCamaaBaaaleaacaaIWaaabeaakiaadwgadaahaaWcbeqaamaalaaabaGaeyOeI0IaamOEaaqaaiaadIeaaaaaaOGaeyypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGaamisaaaacaWGWbGaai4oamaalaaabaGaaGymaaqaaiaadchaaaWaaSaaaeaacaWGKbGaamiCaaqaaiaadsgacaWG6baaaiabg2da9iabgkHiTmaalaaabaGaaGymaaqaaiaadIeaaaaaaa@49A9@

4.  dln(ax) dt =?                 dln(ax) dt =  1 ax d(ax) dt =  1 ax adx dt = 1 x u , where u = velocity MathType@MTEF [1]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@6C7F@

5.  d(cos(x))=?               d(cos(x))=−sin(x)dx       MathType@MTEF [1]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgacaGGOaGaci4yaiaac+gacaGGZbGaaiikaiaadIhacaGGPaGaaiykaiabg2da9iaac+dacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaWGKbGaaiikaiGacogacaGGVbGaai4CaiaacIcacaWG4bGaaiykaiaacMcacqGH9aqpcqGHsislciGGZbGaaiyAaiaac6gacaGGOaGaamiEaiaacMcacaWGKbGaamiEaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaaaa@59D0@

You have the power.

Often in meteorology and atmospheric science you will need to manipulate equations that have variables raised to powers. Sometimes, you will need to multiply variables at different powers together and then rearrange your answer to simplify it and make it more useful. In addition, it is very likely that you will need to invert an expression to solve for a variable. The following rules should remind you about powers of variables.

Laws of Exponents

a x a y = a x+y ab x = a x b x a x y = a xy a −x = 1 a x a x a y = a x−y a 0 =1 a b x = a x 1 b x = 1 a −x b −x = b a −x

If a= b x , then raise both sides to the exponent 1 x to move the exponent to the other side: a 1 x = b x 1 x = b x x =b

If a x b y , and you want to get an equation with a raised to no power, then raise both sides to the exponent 1 x : a x b y 1 x = a x 1 x b y 1 x =a b y x =new constant

This brief video (7:42) sums up these important rules: 

Rules of Exponents
Click here for transcript of the Rules of Exponents.

In this video we're going to be talking about all of the basic rules of exponents. And remember, when we're talking about exponents we can have an exponent here like X to the fourth where x is the base what we call the base and four is the exponent this small number in the upper right hand corner. It means that we're going to multiply X by itself four times or it means we have four factors of X multiplied together. So, if we expand this out its x times X times X times X. if we collapse it its X to the fourth. So, what happens when we do addition, subtraction, multiplication, and division of exponents? Well, in all cases we have to be really careful about like terms. For example, when we add terms that have exponents in them together both the bases and the exponents have to be the same in order for us to add them together. So, if we look at this first example 3x squared plus 2x squared the base here is X and the base here is X so the bases are the same which is good because we need that. and the exponents we have 2 and 2 which is good because we also need the exponents to be the same in order to add these together. So basically we have 3x squared added to 2x squared is going to give us five of them, 5x squared. So, if you're going to do addition and subtraction the bases and the exponents have to be the same. In this case we have X to the third plus x squared our bases X are the same but our exponents are different we have three and two. These are not like terms, so we can't add these together we can't simplify this at all. What happens when we do subtraction well again we're looking for similar basis so we have X and X for our base and then we have exponents of four and four. So because the bases and the exponents of the scene we can combine these like terms. We have six of them were subtracting and applied one of them which is going to leave us with five of them. So 5 times X to the fourth, but in this problem despite having the same base they will have a base of X we have different exponents we have a 4 and a 3 and because we're doing subtraction we can't combine these. We can't simplify this at all. What happens when we multiply two values together where exponents are involved? Well, here in order to simplify all we care about is that the bases are the same. The exponents do not have to be the same. So here we have base X and base X and we know already that's all we need to multiply these together it doesn't matter that the exponents are also the same we just add them. So we have three times to these are coefficients on our x squared terms. We multiply those together. So three times two is six, so that's going to be the first part and then we have x squared times x squared. And if we look at that x squared times x squared what we're going to do is add the exponents together. And the reason is because if we expand these out we know that x squared is two factors of X multiplied together. We're multiplying that by another x squared, so we're multiplying that by two more factors of X multiplied together. All together this is X to the fourth. Which we know because this essentially becomes the rule x to the a x x to the B is X to the a plus B. We just add the exponents together. So two plus two is four we get X to the fourth. Here's another example we have X to the third times x squared remember there's an implied one coefficient in front of both of these when we multiply 1 x 1 we get one so there will be a implied one coefficient on our final answer. x cubed + x squared. We just care that the bases are the same and they both have a base X so we know will be able to multiply them together. We have X to the third times x squared and remember that is going to be X to the three plus two so when we simplify we get X to the fifth and that should make sense because we have 3 factors of x x 2 factors of X adding them all up we get five factors of X so X to the fifth. The quotient rule for exponents tells us that in the same way as when we multiplied we didn't have to have the same exponent. When we divide we also don't have to have the same exponent we only care about the bases so here we have like basis. We have base X for both of these the exponents happened to be the same but that doesn't matter we're just going to leave this six and our final answer, so we'll get six here. And then what we're going to do is subtract the exponent in the denominator from the exponent in the numerator so the result is going to be X to the 4 minus 4. This is the four from the numerator this is the four from the denominator. 4-4 is 0 so we get 6 x 20 x to the 0 is 1 so this is 6 times 1 or just six. Even if we have different numbers again we only care about the bases both of these have the same base of X so again we'll just keep our two and our final answer and then we'll have X to the 4-3 because we say numerator exponent minus denominator exponent. That's going to give us 2 times X to the 4 minus 3 is 1. so X to the first which is of course just equal to 2x. What about a power raised to another power or an exponent raised to another exponent? Well, just like before in this example here when we said X to the fourth means multiply X by itself four times here we're saying multiply x squared by itself three times. So this is going to be equal to x squared times x squared times x squared and now we're really just back at this right here for multiplying like bases together and we add the exponents. So, this is just the same as X to the two plus two plus two. Two plus two plus two is six so we get x to the sixth power. What we realize then is that we can expand this and then add the exponents together using this rule over here or we can just multiply these two exponents together. Two times three gives us six and so we can do it that way as well. We can even do this when we have a negative base. So this problem here is telling us multiply 3 factors of negative x squared together so this is going to be negative x squared times negative x squared times negative x squared. We can deal with the negatives separately. Remember we can cancel every two negatives and they become a positive so negative and negative become a positive we're just left with this single negative sign here. so our answer will be negative and then x squared times x squared times x squared we know is X to the sixth. You can also think about it this way when you have this negative sign inside the parentheses. It's the same thing as saying negative 1 times x squared all raised to the third power and then you can apply this exponent to the negative 1 negative 1 times negative 1 times negative 1 is going to give you negative 1 which is this part right here. And then x squared to the third is going to be X to the 60 you get this X to the sixth and when you multiply them together you get negative x to the sixth. So those are just some of the most basic exponent rules that you need to know.

Credit: Krista King [6]

Are you ready to give it a try? Solve the following problem on your own. After arriving at your own answer, click on the link to check your work. Here we go:

Check Your Understanding

x=a y b

What does y equal?

Click for answer.

x 1/b = ( a y b ) 1/b = a 1/b ( y b ) 1/b = a 1/b y y= x 1/b / a 1/b = ( x a ) 1/b MathType@MTEF [1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@5F4C@

Quiz 1-2: Solving integrals and differentials.

Now it's time to to take another quiz. Again, I highly recommend that you begin by taking the Practice Quiz before completing the graded Quiz, since it will make you more competent and confident to take the graded Quiz : ).

  1. Go to the Canvas and find Practice Quiz 1-2. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz.
  2. When you feel you are ready, take Quiz 1-2. You will be allowed to take this quiz only once. This quiz is timed, so after you start, you will have a limited amount of time to complete it and submit it. Good luck!

1.4 Are you ready to get with the programming?

1.4 Are you ready to get with the programming?

Meteorologists and atmospheric scientists spend much of their time thinking deep thoughts about the atmosphere, the weather, and weather forecasts. But to really figure out what is happening, they all have to dig into data, solve simple relationships they uncover, and develop new ways to look at the data. Much of this work is now done by programming a computer. Many of you haven’t done any computer programming yet, and for those of you who have, congratulations – put it to good use in this class. For those who are programming novices, we can introduce you to a few of the concepts of programming by getting you to use Excel or another similar spreadsheet program.

To help you learn and retain the concepts and skills that you will learn in this course, you will solve many word problems and simple math problems. For several activities, we give you the opportunity to practice solving particular types of problems enough times until you gain confidence that you can solve those same types of problems on a quiz. That means that you will be solving some types of problems several times and only the numbers for the variables will change. The simplest way for you to solve these problems is to program a spreadsheet to do that repetitive math for you.

screenshot of spreadsheet. See link in caption for text description
Spreadsheet Screenshot
Click for a text description of the spreadsheet screenshot.

Screenshot shows an Excel Spreadsheet

A text box says "put activity number in row 1" and an arrow points to cell A1.

A second text box says "put variable names in row 2" with an arrow pointing to cells A2 and B2

A third text box says "start numbers for variables in row 3" with an arrow pointing to cells A3 and B3

A final text box says "calculations follow variable numbers" with an arrow pointing to C3.

Let’s do a simple example. Suppose we have several boxes, some with different shapes and sizes, and we want to calculate the volume of the boxes and find the total volume. I have put in the names of the variables (with units!) and then the numbers for the length, width, and height of each box type and the total number of each box. To calculate the volume of each box, click on E3 and put an “= a3*b3*c3” in the equation line. Hit enter and it will do the calculation and put the answer in E3. A small square will appear in the lower right corner of E3. Click on this square with the mouse and pull down over the next three rows. Excel will automatically do the calculations for those rows. To calculate the total volume, go to F3 and enter “=d3*e3,” and hit “enter.” Grab the small box and pull down to get the total volume of each type of box. To get the total volume, click on F7, click on “Formulas,” and then “AutoSum,” and finally “Sum.” Excel will show you which cells it intends to sum. You can change this by adjusting the edges of the box it shows. Finally, pay attention to significant digits in your calculated volumes. You can adjust the number of decimal places by highlighting a cell or group of cells and then by clicking on the appropriate icon on the Excel tool bar. Note that an incorrect number of significant figures is displayed in some of the answers--see if you can figure out where.

screenshot of spreadsheet. See link in caption for text description
Click for a text description of the spreadsheet example part 1.
length (m) width (m) height (m) number of boxes box volume (m3) total volume (m3)
0.72 0.52 0.52 3 =A3*B3*C3
0.52 0.52 0.52 4
1.2 0.85 1.01 1
0.62 0.62 0.95 2
screenshot of spreadsheet. See link in caption for text description
Click for a text description of the spreadsheet example part 2.
length (m) width (m) height (m) number of boxes box volume (m3) total volume (m3)
0.72 0.52 0.52 3 0.19 0.58
0.52 0.52 0.52 4 0.14 0.56
1.2 0.85 1.01 1 1.03 1.03
0.62 0.62 0.95 2 0.37 0.73

Hopefully this example is a refresher for most of you. For those who are totally unfamiliar with Excel, please click on the question mark in the upper right of the screen and type in the box “creating your first workbook.” You can also visit Microsoft's help page for additional step-by-step instructions for how to Use Excel as Your Calculator [7]. The best way to learn, after the introduction, is by doing. The Keynote Support website [8] also lists helpful summaries of instructions.

Activity 1-3: Setting up your Meteo 300 Excel workbook.

Please follow the instructions above for setting up an Excel workbook. You will be using this workbook to do calculations, plot graphs, and answer questions on quizzes and problems for the rest of the course.

Use the same headings and the same numbers in the first four columns. Enter your own equations and find the box volumes and total volumes.

This assignment is worth 15 points. Your grade will mostly depend upon showing that you set up the workbook, but some additional points will be assigned contingent upon how well you follow the instructions. When your Excel workbook is complete, please do the following:

  1. Make sure that the file for your workbook follows this naming convention: Workbook_your last name (i.e., Smith)_your first name_(i.e., Eileen).xlsx.
  2. In Canvas, find Activity 1-3: Setting up your  Meteo 300 Excel workbook. Upload your Excel workbook there.

Summary and Final Tasks

Summary

There is a very good reason that you are taking this class and I am teaching it – all of us are fascinated by the weather, awed by the atmosphere’s power, and passionate about learning more about it. Quite honestly, I can’t imagine a more rewarding career than the one that you are embarking upon or the one that I have. Nothing could be more rewarding than saving lives by making the atmosphere more predictable or by making the perfect prediction. Nothing.

But, do you know what? The best forecasters are the ones who can not only read weather maps, but who also know physically what the atmosphere is doing. The best forecasters know how to translate the physics into mathematics so that hand-waving can be turned into usable numbers. This course will start to make all of these connections between observations and physical cause-and-effect and help us find numerical solutions to questions.

For those of you who are in related disciplines, this course will give you a solid basic understanding of the atmosphere that you can apply in your studies and career, whether it be civil engineering, mechanical engineering, environmental engineering, chemistry, hydrology, or many other fields. 

We have now reviewed some important concepts like significant figures and dimensions and units. You will continue to gain confidence in using the differential and integral calculus that you already know. As you go through the course, I want you to look back at the pictures of the atmosphere and imagine which equations are governing the processes that are causing your observations.

Reminder - Complete all of the Lesson 1 tasks!

You have reached the end of Lesson 1! Make sure that you have completed all of the tasks in Canvas.


Source URL: https://www.e-education.psu.edu/meteo300/node/3

Links
[1] mailto:MathType@MTEF
[2] https://www.youtube.com/channel/UCX6b17PVsYBQ0ip5gyeme-Q
[3] http://en.wikipedia.org/wiki/Mars_Climate_Orbiter
[4] https://www.youtube.com/channel/UCbRTUq5LGqbn4ngTTi9AnMQ
[5] https://www.khanacademy.org/math/calculus-1
[6] https://www.youtube.com/channel/UCUDlvPp1MlnegYXOXzj7DEQ
[7] https://support.office.com/en-au/article/Use-Excel-as-your-calculator-df26bd14-e1a8-4618-b411-92349dd777a7
[8] http://www.keynotesupport.com/excel-basics/excel-math-formulas.shtml#basic