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August 31, 2009

WHAT IS MULTIVARIABLE CALCULUS?

“Space. . . the final frontier”

Multivariable calculus extends one variable calculus into situations where the do-main and/or the range of functions becomes multidimensional. The key operationsof differentiation and integration will be extended into this new setting. Some newsituations require a substantial reworking of your earlier concepts, all built uponyour prior calculus study and earlier work on space, coordinates, vectors, and ma-trices.

Why extend calculus to higher dimensions?The main reasons to develop calculus ideas and concepts in several dimensions are:

1. Motion is usually in two or three dimensional spaces, whether a baseball, aplanet, an electron, or a raindrop.

2. Several variables often enter a problem, such as in chemistry (pressure, vol-umes, chemical concentrations, reaction rates, temperature, and others) oreconomics (multiple items with prices, demands, costs, rates of return, risk,and others).

3. It is step along a more sweeping generalization of calculus known as the cal-culus of variations, first considered by Newton, Leibniz, and the Bernoullis.The calculus of variations helps to minimize over paths and thereby explainsthe classical version of bending light by minimizing time of travel and helpsengineers plan the best path for a space ship flight or to optimize design inairplanes or cars.

The extension of domain and/or range to several dimensions relies on the funda-mental model for two and three dimensional spaces that come to us from phys-ical space, so we shall begin by discussing the algebra and geometry of physi-cal space. Coordinates and distance are reviewed. From there the general notionof vectors and matrices emerge and examples of vector language that are not re-lated to physical space arise. In many of these more general settings, the distinctdimensions and coordinates may be combined from disparate types of numbers,such as temperature, humidity, wind speed and barometric pressure in meteorol-ogy, or a set of prices of products in economics. In those settings, rotating in thetemperature-humidity plane or the Euclidean distance between price vectors maybe a mathematical operation with limited physical meaning. In discussing physicalspace and time, unless specifically stated in that immediate context, the Newtonian(pre-relativistic) setting will be assumed. Multivariable calculus is a fundamen-tal tool for understanding meteorology, economics, relativity (special and general)

1

and many other subjects. Some of these connections will be touched upon in theextended problems at the end of each chapter.

Some good news: It turns out that going from one dimension to two is typicallymore difficult than generalizing from two to three or more dimensions. In the textitself, the case of general dimensions will be avoided; students interested in thatextension will find exercises and problems that provide the key ideas and intro-duce the notational difficulties. Two and three dimensions will also permit easiervisualization (it isn’t always going to be easy, just easier).

In most textbooks, multivariable generalizations are done in three distinct steps:first the range is multidimensional while the domain has dimension one, then thedomain becomes multidimensional while the range has dimension one, and finallyboth domain and range become multidimensional. While this approach has manybenefits, it also delays the deployment of valuable tools and ideas. Many newconcepts and ideas come at the end of the course when those ideas are being tiedtogether in the multivariable versions of the Fundamental Theorem of Calculus. Inthis book, matrix and vector algebra will be used from the start. There are then twostages, first when the domain is one-dimensional, so your previous calculus skillsof differentiation and integration can be applied with a few small modifications,followed by the extension of differentiation and integration to multidimensionaldomains. As well, functions with multidimensional domains are useful even inthe early stages of the course, so the discussion of continuity and graphing, whichis based on the nature of higher dimensional spaces as domains, appears in thefirst chapter. This is particularly useful when differentiation of functions of severalvariables is developed in Chapter Three.

CHAPTER 1

Tools: Coordinates, Vectors, Ma-trices, and Functions

This chapter explores coordinates, vectors, matrices, and some basic functions.The geometry of 2-D and 3-D space involves notions of distance, direction, angleand can be described in various systems of coordinates. Vector notation allows usto use efficient algebra to reason and compute without reference to a choice of coor-dinates. Polar coordinates in 2-D and cylindrical and spherical coordinates in 3-Dare particularly important examples of alternative coordinate systems. The gener-alization of continuity and limit sets the stage for later chapters on differentiationand integration.

The themes in this chapter all center around the relation between functions involv-ing several variables and the calculus of one variable. There will be similaritiesof course, but also differences. The differences arise because in several variablesthere is more freedom to move around, more operations to incorporate, difficultiesin displaying graphs, a richer set of domains, and a large zoo of possible subsets ofvarious dimensions. Orientation and dimension will be recurring themes. Particu-lar emphasis will be placed on planes and lines since they will be the appropriateversions of the tangent line or linear approximation so crucial to understanding onevariable calculus.

Exciting picture goes here

1.1 Coordinates, Vectors, and Space 3

1.1 Coordinates, Vectors, and Space

Overview: This section considers the usual descriptions of two and three dimen-sional space in terms of coordinates and then in vector form. Higher dimensionalspaces arise as domains or as ranges for many functions. Geometric consider-ations and physical science and engineering applications require that alternativecoordinates be considered. The geometric motions of rotation and translation aredescribed in coordinates. Polar coordinates simplify rotation about the origin intwo dimensions while three dimensional rotations about the origin have a morecomplex structure. Two useful theorems of Euler concerning three dimensionalrotations are described.

1.1.1 Space, Axes, and Coordinates inR2

The basic two dimensional space is denotedR2 and is the setting for all the graphsyou plotted in one variable calculus and all the plane geometry you learned. Re-call that coordinate axes were specified for two coordinates, most often calledxandy and the graph ofy = f(x) displayed ordered pairs of numbers(x, y). Theset of possiblex-values (inputs) formed the domain of the function while the cor-respondingy values were the range. In geometry, thex andy coordinates wereused to describe locations and to create equations describing various objects suchas circles, lines, ellipses, hyperbolas, and parabolas.

The systemx andy are calledrectangular coordinates. Take a moment to con-sider the many things you have done in this setting before reading on.

Example / picture goes here of usual coordinate axes picture

Some notation: The pointP with coordinates(x, y) will be written P (x, y) orsimply asP . The origin, with coordinates(0, 0) will be denoted byO. The directedline segment fromP to Q will be denoted by

−−→PQ while the line segment will be

denoted byPQ.

Example 1.1 Example goes here

Distance formula: The Euclidean distance between two points with coordinates,sayP1 = (x1, y1), P2 = (x2, y2) comes from the Pythagorean theorem:

d(P1, P2) =√

(x2 − x1)2 + (y2 − y1)2.

Note that this is a function of all four coordinates which is non-negative and therange is therefore[0,∞) since any distance can be achieved. Recall as well thatthis distance is also the length of the line segment (either directed or not), whichwe will denote using vertical bars:

d(P1, P2) = |P1 P2| = |−−−→P1 P2|

4 Chapter One: Tools – draft August 31, 2009

The meaning of rectangular coordinates:When we sayP has coordinates(3, 1),this says that the pointP is 3 units to the right and one unit up from the origin inthe usual coordinate grid. If we only know the first coordinate, which is typicallywritten asx = 3, what do we know? Being3 units right puts us on theverticalline with equationx = 3. The correspondinghorizontal line with equationy = 1describes all points one unit up from the origin. The intersection of these two linescorresponds to the simultaneous equationsx = 3 andy = 1, which is precisely thepointP .

x0 1 2 3 4

y

0

1

2

3

4

Figure 1:The two slices defined by x = 3 (red) and y = 1 (blue) appear in widerformat than the other grid lines with similar coloring.

Polar coordinates: One of the important alternative coordinate systems ispolarcoordinates. Polar coordinates use the distancer from the origin and the angleθ that the directed segment

−−→OP makes with the positivex-axis. This leads to

the famous pairs of equations that relate the rectangular coordinates to the polarcoordinates of the same point andvice-versa:

x = r cos θ, y = r sin θ ⇔ r =√

x2 + y2, θ = tan−1(y/x)

where the inverse tangent must be chosen in the proper quadrant. The origin isan exceptional point in thatθ is not well-defined there; at all other points inR2, θis defined, but only up to multiples of2π. Note also thatθ, measured usually inradians for our purposes, has no length scale even ifx, y andr do, as in physicalspace.

The meaning of polar coordinates:This coordinate system is not a rectangularsystem but rather a circular one laid down upon the plane. Geometrically the setswherer is constant are circles of given radius and the sets whereθ is constant arehalf-lines (rays) with fixed angle. As we look at a zoomed-in picture, the radial andangular directions look flatter and seem to be perpendicular. You should know thisfrom geometry: the tangent line to the circle is perpendicular to the radial line. You


likely revisited this in your calculus course. In Chapter 3, we will use derivativesin several variables while later in this chapter the unit steps in radial and angulardirections are described in polar and rectangular coordinates.

Example 1.2 Example / picture goes here

K3 K2 K1 0 1 2 3

K3

K2

K1

1

2

3

Figure 2:The two slices defined by each polar coordinate value are plotted usingr = 2 (blue) and θ = π/3 (red).

A close-up look at polar coordinates reveals the geometry that comes from thepolar coordinate piece resolving into perpendicular “line-like” objects:

0.90 0.95 1.00 1.05 1.10

1.64

1.66

1.68

1.70

1.72

1.74

1.76

1.78

1.80

1.82

Figure 3: The two slices defined by each polar coordinate value are plotted upclose, and appear “line-like” though they remain r = 2 (blue) and θ = π/3 (red).

A particular use of polar coordinates is to reduce dimensions when there iscircu-lar symmetry. Many times in multivariable calculus we will want to revert to oursingle variable calculus skills. One common instance of this is when circular sym-metry is used with polar coordinates to create functions which only depend on the


radial distance variabler. Sometimes the single variable will be the angular vari-ableθ. Rotation around the origin in polar coordinates is simply addition of angles,which is another way they are valuable. At several times during this course, polarcoordinates will reappear, so make sure you become comfortable with them now ifthat didn’t happen in precalculus. The exercises will help.

Distance formula in polar coordinates: While the distance from the origin to apoint is much simpler in polar coordinates, the distanced between two points isnot simpler, but reverts to thelaw of cosinesyou learned in trigonometry. Usethe algebraic conversion back to rectangular coordinates to find the square of thedistance between pointsP1 andP2 with polar coordinates(r1, θ1) and(r2, θ2):

d2(P1, P2) = (x1 − x2)2 + (y1 − y2)2

= (r1 cos θ1 − r2 cos θ2)2 + (r1 sin θ1 − r2 sin θ2)2

= r21 + r2

2 − 2 r1 r2(cos θ1 cos θ2 + sin θ1 sin θ2)= r2

1 + r22 − 2 r1 r2 cos(θ1 − θ2)

which is precisely the law of cosines!

Transformations that preserve distance:The three main transformations main-taining the distance between all pairs of points combine to create what are known astheEuclidean group of motions. The three basic changes are: translation (shiftingthe origin), rotating around the origin by a fixed angle, and reflecting about a linethrough the origin. It is a result of geometry that these generate all motions pre-serving Euclidean distances. The reflections reverse orientation while translationsand rotations maintain the same orientation.

Translations: Translations are familiar to you from precalculus: Moving the pointat the origin to(h, k), the newx andy coordinates of all points are shifted byh andk respectively. The new location has coordinates related to the old ones by simpleaddition (or subtraction), with new coordinates(x + h, y + k).


Rotations: Rotation about the origin is also familiar to you from precalculus, butperhaps with less comfort: rotation by an angleα in the usual counterclockwisedirection, the newx andy coordinates are combinations of the old ones mixedwith the sine and cosine of the angle of rotation. The new polar coordinates of thelocation that began at(r, θ) is now (r, θ + α). The new rectangular coordinatesdenoted(x′, y′) relate to the old ones(x, y) so basic trigonometric identities yieldthe coordinates(x′, y′) of the rotated location by a direct calculation:

x′ = r cos(θ + α) = r cos θ cosα− r sin θ sinα = x cosα− y sinα

y′ = r sin(θ + α) = r cos θ sinα + r cos θ sinα = x sinα + y cosα

Rotation is simpler in polar coordinates than in rectangular coordinates!


Picture/example goes here


Reflections: Reflection about they axis takes a pair of coordinates(x, y) andchangesx to its negative−x while leaving they coordinate unchanged:(x, y) 7→(−x, y). Reflection about thex axis takes a pair of coordinates(x, y) and changesy to its negative−y while leaving thex coordinate unchanged::(x, y) 7→ (x,−y).Reflection about the liney = x takes a pair of coordinates(x, y) and uses thevaluex as the second coordinate while changing the first coordinate to the valuey:(x, y) 7→ (y, x). Reflection through the origin takes both coordinates to their oppo-site sign version: :(x, y) 7→ (−x,−y). The algebra shows this is the compositionof the reflection in thex andy axes done in either order. It is also the rotation byangleπ, so it preserves orientation!


Alternative coordinates for unchanged position:In many physics and engineer-ing and computer graphics settings, a location is unmoved but an observer is eithermoving or using other coordinate axes perhaps with a different origin. The coor-dinate changes described above used a fixed coordinate system. The two ideas arerelated (the much over-used word isduality ) and you are asked to explain somecases in the exercises. Moving coordinate systems with no relative accelerationplayed a key part in Einstein’s development of special relativity.

Coordinate changes:Rotating the coordinate axes in 2-D leads to new axes thatare still the same length, still perpendicular, and with similar orientation. The newcoordinates are combinations of the origin coordinates involve sines and cosines ofthe angle of rotation. Using the subscript “new” on the variables for them, a rotatedset of coordinates for rotation of axes counterclockwise byα is:

xnew = x cosα + y sinα

ynew = x sinα− y cosα

Please note the minus sign difference between moving the point in fixed coordi-nates versus describing the point unmoved but now in rotated coordinates.

Example goes here. Picture goes here.

Other coordinate changes might involve combiningx andy, stretching or shrink-ing lengths in or both directions (as in changing units), or using functions ofx ory or both, as in the integrals you did in one variable calculus by substitution. Mul-tivariable calculus tools will help us understand all such coordinate changes givenby functions.

Linear coordinate changes:A general linear coordinate grid consists of two fam-ilies of lines representing the generalization of the standard rectangular coordi-nate grid. The vertical linesx = 0,±1,±2, . . . and the horizontal linesy =


0,±1,±2, . . . (or those with more general even spacing) can be replaced by twofamilies of lines. Axes whose directions make acute or obtuse angles with one an-other create parallelogram tiles or diamonds as a pattern of grid lines. This will bea very important picture for understanding many facts about multivariable calculusand will appear often. In a highly skewed grid, it may take large coordinate changesto move a small distance geometrically. Geometrically the original rectangular gridbased on perpendicular axes has had each axis rotated and the corresponding num-ber line possibly stretched in scale. The two axes may have rotated and/or stretchedby different amounts. This topic will be revisited later in the chapter when we havemore tools.

Example 1.5 Picture goes here

1.1.2 Space, Axes, and Coordinates inR3

The basic three dimensional space is denotedR3 and is an extension ofR2 whichadds a third dimension and therefore a third coordinate. Coordinate axes which arefixed, orthogonal and right-handed were specified for three coordinates, most oftencalledx, y andz. These are again calledrectangular coordinates. Take a momentto consider the many things you have done in this setting before reading on.

Some notation: The pointP with coordinates(x, y, z) will be written P (x, y, z)or simply asP . The origin, with coordinates(0, 0, 0) will be denoted byO. Thedirected line segment fromP to Q will be denoted by

−−→PQ. Note that most of this

notation is similar to the version in 2-D. It should always be clear from the contextwhen such notation appears whether the underlying space isR2 orR3.


Distance formula: The Euclidean distance between two points with coordinates,sayP1 = (x1, y1, z1), P2 = (x2, y2, z2) comes from the Pythagorean theorem:

d(P1, P2) =√

(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2.

Note that this is a function of all six coordinates which is non-negative and therange is therefore[0,∞) since any distance can be achieved. Again, the notationis identical to the 2-D case, so careful reading is needed. This is also true whenwriting segments and their lengths, as in: the length of the line segment (eitherdirected or not), which we will denote using vertical bars:

d(P1, P2) = |P1 P2| = |−−−→P1 P2|

The meaning of rectangular coordinates:When we sayP has coordinates(3, 1, 2),this says that the pointP is three units to the right, one units forward, and two units


upward from the origin in the usual coordinate grid. If we only know the first coor-dinate, which is typically written asx = 3, what do we know? Being3 units rightputs us on thevertical plane with equationx = 3. The other coordinatesy andzare unspecified. The correspondingvertical plane with equationy = 1 describesall points one units forward from the origin. The intersection of these two planescorresponds to the simultaneous equationsx = 3 andy = 1, which is precisely theline with z still unspecified. To get to the pointP , a third,horizontal plane of theform z = 2 is also specified.

4

-4-4

2

-2

-2

0

x

z

00

2

y

4

2-2

4-4

Figure 4: The three planes defined by each coordinate value are plotted. x = 3(red), y = 1 (green), and z = 2 (blue)

1.1.3 Two versions of polar coordinate systems in 3-D

In 3-D, there aretwo versions of polar coordinates. One simply uses the polarcoordinate description in thex− y plane and leaves the third coordinate alone.

Cylindrical coordinates in 3-D Polar coordinates have two distinctive ways inwhich they become three-dimensional. The first is to add a rectangular variablez to the polar coordinates for the rectangular coordinatesx andy. This choice isknown ascylindrical coordinates. Cylindrical coordinates have one polar coordi-nate pair and correspond to a system where ther variable refers todistance fromthez axis and the theta variable refers to a rotation around thez axis and of coursez refers to distance along thez axis. Going from cylindrical coordinates to rect-angular and back is summarized by the formulas for polar coordinates and a thirdtransformation which is to leavez unchanged, which is written strangely asz = z:

x = r cos θ

y = r sin θ

z = z

and conversely

r =√

x2 + y2

θ = tan−1(y/x)z = z

Notice that the variabler is distance from the origin in theprojection onto anx − y plane, which is also distance from the line which is thez axis. Writing


z = z seems somewhat silly, but reminds us that this is a three-dimensional changeof coordinates, even if we only change two of them.

The meaning of cylindrical coordinates: Specifying the coordinater fixes thedistance from thez-axis, which yields the surface of an infinite cylinder of fixedradius. Specifying the coordinateθ specifies a half-plane making fixed angle withthe positivex-axis. Specifyingz describes a plane of fixed height above or belowthex− y coordinate plane.

-4

-2

0

2

4

-4-2

02

4-4

-2

0

2

4

Figure 5:The three slices defined by each cylindrical coordinate value are plotted.

A close-up look at cylindrical coordinates reveals the geometry that comes fromthe polar coordinate piece simplifying into perpendicular “plane-like” objects:

1.2

1.3

1.4

1.5

1.6

1.21.3

1.41.5

1.61.9

1.95

2.0

2.05

2.1

Figure 6:The three slices defined by each cylindrical coordinate value are plottedup close, and appear “plane-like”.

Spherical coordinates in 3-D:A second extension of polar coordinates involvestwo polar coordinate choices in successionand thereforetwo angular variables.This is known asspherical coordinates.


Spherical coordinates from cylindrical coordinates:Measure the angle betweenthe positivez-axis and the pointP and call itφ. Conversion ofr andz to polar co-ordinatesρ andφ creates the second angular variable and leaves only one distancevariable. This coordinate system is best used when rotations about the origin arethe fundamental symmetries. Many physical models involve functions that dependonly upon the distance from the origin. Two warnings: physicists often use oppo-site names for the angular variables; many writers will reuser to mean the distancefrom the origin, which we are callingρ.

Going from cylindrical coordinates to spherical coordinates and back is summa-rized by the formulas:

z = ρ cosφ

r = ρ sinφ

θ = θ

and conversely

ρ =√

r2 + z2

φ = tan−1(r/z)θ = θ

where the inverse tangent must be chosen in the first two quadrants (0 ≤ φ ≤ π).Combining this with the earlier conversion from cylindrical coordinates to rect-

angular coordinates and back by composition leads to a set of formulas convertingspherical coordinates to rectangular ones and back:

x = ρ sinφ cos θ

y = ρ sinφ sin θ

z = ρ cosφ

and conversely

ρ =√

x2 + y2 + z2

φ = tan−1(√

x2 + y2/z)

θ = tan−1(y/x)

Using the intermediate step of cylindrical coordinates was useful since the twoparts each involved only two of the three variables, while the linkage betweenrectangular and spherical coordinates involves all three coordinates at once (exceptin the expressions forz andθ).

Note: In finding the inverse mappings, the order of composition gets reversed. Tosee why this is a basic property of composition of functions or operations, thinkabout putting on and then taking off your shoes and socks!

The meaning of spherical coordinates:Specifying the coordinateρ fixes thedistance from the origin, which yields the surface of a sphere of fixed radius. Spec-ifying the coordinateθ again specifies a half-plane making fixed angle with thepositivex-axis. Specifyingφ describes a cone of angle with the positivez-axis.Note that the two angles do not generate similar sets when one is specified.

A close-up look at spherical coordinates reveals the geometry that comes from thepolar coordinate pieces resolving into perpendicular “plane-like” objects:

Both of these coordinate systems are frequently useful in applications. The sym-metry of the problem dictates which choice is more useful. During this course we


-4

-2

0

2

4

-4

2 0 -2 -44

-2

0

2

4

Figure 7:The three slices defined by each spherical coordinate value are plotted.

0.98

1.18

1.38

0.981.081.181.281.381.480.78

0.88

0.98

1.08

1.18

Figure 8:The three slices defined by each cylindrical coordinate value are plottedup close, and appear “plane-like” and orthogonal to each other.


will have many chances to experiment and learn how to use each of these systems.Later in this chapter we will examine the directions that small increases in sin-gle coordinates make. In Chapter 3, we will find derivatives of these coordinatesviewed as functions of each other or as functions of the rectangular coordinates.

Coordinates changes that preserve distance:The three main changes of coordi-nates maintaining the distance between all pairs of points combine to create whatare known as theEuclidean group of motions. The three basic changes are:trans-lations (shifting the origin), rotating around the origin by a fixed angle, andreflecting about a plane through the origin. It is a result of Euclidean geometrythat these transformations and their compositions generateall motions preservingdistance.

Translations: The shifts of origin are familiar to you from precalculus: with atranslation by(h, k,m), the newx, y andz coordinates are shifted byh, k andmrespectively. The new location has coordinates related to the old ones by simpleaddition (or subtraction), with new location(x + h, y + k, z + m).

Example 1.7 Example of translation goes here

Rotations about the origin 3-D: A rotation about one of the coordinate axes isan extension of the two-dimensional case: with a new orientation of the axes byrotation by an angleα in the usual counterclockwise direction about thez axis,the newx andy coordinates are combinations of the old ones mixed with the sineand cosine of the angle of rotation while thez coordinate is unchanged. The newcoordinates denoted(x′, y′, z′) relate to the old ones(x, y, z) by simple additioninthe angle in cylindrical coordinates, with new angleθ + α, so basic trigonometricidentities yield the new coordinate locations(x′, y′, z′) by the same calculationdone in 2-D:

x′ = x cosα− y sinα

y′ = x sinα + y cosα (1)

z′ = z

(2)

This rotation is simpler in cylindrical coordinates than in rectangular coordinates!But now there are three different axes to start from and our cylindrical coordinatesonly referred to one of them. As well, you can imagine rotating about any direc-tion (known as theaxis of rotation). This creates a difficulty in describing allrotations!

Euler’s Rotation Theorem: Euler showed thatany continuous geometric mo-tion which moved the original x − y − z axis system rigidly while fixing theorigin must be a rotation of some angle about some axis!So in some senserotation about one of the axes is typical.



Euler angles and rotations in 3-D:Euler also showed that rotations about thecoordinate axes can combine if done sequentially to create any rotation and thatit takes three such rotations at most. One of the projects at the end of the chapterexplores this theorem. You can try to convince yourself of the validity of thisstatement by playing with a globe.

Example 1.8 What is the coordinate change for rotation by an angleβ in they−zcoordinate plane? How about angleγ in thez − x coordinate plane?

Cyclic Rotation: An important tool for understanding coordinate formulas andorientation is thecyclic rotation: (x, y, z) 7→ (z, x, y) which corresponds to send-ing thex-axis to they-axis, they-axis to thez-axis, and thez-axis to thex-axissimultaneously (what a mouthful, wasn’t the algebra formula much cleaner andless confusing!). Applying this cyclic rotation twice results in(x, y, z) 7→ (y, z, x)and applying it three times cycles back and sends each point to itself! This trans-formation sends the positive coordinate axes into each othercyclically. A physicalexperiment with fingers or a model of the axes will convince you that this can bedone. Euler’s theorem on rotations then says that this will be a rotation about someaxis, which is unmoved by the rotation. Can you decide what the axis is? Whatshould the angle of rotation be? Looking down the axis from some distance awayfrom the origin, what do the positive coordinate axes look like? See the WERQproblem at the end of the section.

The cyclic rotation will be very useful in saving effort by completing calculationsof objects after they are done in one component by appealing to the cyclic rotationto generate the corresponding terms. This will become clearer when we discussvector products in the next two sections.

Reflections:Reflection using rectangular coordinate planes takes a triple of coor-dinates(x, y, z) and changes one of them to its negative while leaving the othertwo coordinates unchanged:(x, y, z) 7→ (−x, y, z) represents the reflection in they − z plane. (x, y, z) 7→ (x,−y, z) and(x, y, z) 7→ (x, y,−z) represent reflec-tions about which planes? Switching coordinates, as in(x, y, z) 7→ (y, x, z), willbe some kind of reflection. What reflection will it be?

Reflection about the planey = x takes a pair of the three coordinates(x, y, z)and uses the valuex as the second coordinate while changing the first coordinateto the valuey: (x, y, z) 7→ (y, x, z). Reflection through the origin takes all threecoordinates to their opposite sign version:(x, y, z) 7→ (−x,−y,−z).

Reflection about a plane will always reverse orientation. For the coordinate axes,it involves one multiplication by−1 in only one coordinate. In the plane, weobserved that the so-called reflection through the origin was also a rotation byangleπ, and so it preserved orientation. What do you think happens in 3-D when



we reflect through the origin? Is orientation reversed or preserved? Why do youthink that?


Other coordinate changes:Other coordinate changes might involve combiningx,y andz, stretching or shrinking lengths in one or two or three directions (as inchanging units), or using functions ofx or y or z or any subset, as in the integralsyou did in one variable calculus by substitution. Multivariable calculus tools willhelp us understand all such coordinate changes given by functions.

Linear coordinate changes: A general linear coordinate grid consists of threefamilies of planes representing the generalization of the standard rectangular co-ordinate grid. The planesx = 0,±1,±2, . . ., y = 0,±1,±2, . . ., and z =0,±1,±2, . . . (or those with more general even spacing) can be replaced by threefamilies of planes. Axes whose directions make acute or obtuse angles with oneanother create parallelepiped boxes as a grid pattern. Linear coordinate changeswill be a very important picture for understanding many facts about multivariablecalculus and will appear often. In a highly skewed grid, it may take large coordi-nate changes to move a small distance geometrically. Geometrically the originalrectangular grid based on perpendicular axes has had each axis rotated and thecorresponding number line possibly stretched in scale. The three axes may haverotated and/or stretched by different amounts and the rotations may have been indifferent directions. This topic will be revisited later in the chapter when we havemore tools.


1.1.4 Vectors inR2

Location inR2 specified by coordinates(x, y) or (r, θ) can also be viewed in termsof aposition vectorand then written invector notation. Vectors will be written inboldface type, such asu andv although other books and handwriting often uses anarrow over the vector, as in−→u . At first the use of vector language seems merely tobe a device to save writing, but eventually you will appreciate the simpler algebraand clearer conceptual understanding that comes with thinking in terms of vectors.

The fundamental virtue of using vectors is that the vector concept is not tiedto coordinate choices. This algebraic tool can be used independent of coordinatechoice, a very important idea in mathematics, physics and engineering. As well, insolving problems or calculating, where coordinate choices are usually unavoidable,such choices can be deferred or made in useful and flexible ways.


We often do this naturally without much thought. For example, in my neighbor-hood the roads curve and houses are rarely lined up in East-West-North-South fash-ion. When I am inside a house (all my neighbors have rectangular house plans), Inaturally think in terms of front and back and sides, and I presume the builders didtoo. When navigating, I tend to think in terms of the street grid, which is curved.Building the house at different times required different uses of coordinates. Thefoundation was oriented relative to a N-S and E-W grid, but then the buildersworked from their usual blueprints in thelocal orientation connected with thathouse. Mathematics, science and engineering problems are often simpler whenviewed in some special orientation of coordinates that relates to the problem athand and sometimes use more than one. Vector language takes care of that natu-rally and efficiently.

Formally, avector quantity in R2 has amagnitude (length) anddirection (ex-cept the zero vector). This contrasts withscalars, which belong toR and haveno direction. Points are associated to the vector that goes from the origin to thepoint. Polar coordinates are perfect for this description sincer is the length andθdescribes the direction. But often the original rectangular coordinates are preferredfor other reasons.

The Zero Vector: The vector with magnitude zero has no unique sensible way todefine its direction and is called thezero vector. It will be denoted by0.

Notation for Length of a Vector: The length of a vector is denoted by the symbol:|v|.

Visualization: Vectors are plotted asdirected arrows showing length and direc-tion. The rectangular coordinate form uses two special vectors which are of length1 in the direction of the increasing coordinates. These are usually denotedi andjrespectively. Then coordinates(x, y) correspond to the vectorxi + yj, which sug-gests a need for the algebraic operation ofvector addition. Later in this chapterspecial vector-valued functions are considered. In that situation, the plotted arrowis usually located so that its “tail” is the location of the input values. Otherwise weare free to place our arrows in any location when convenient.


Vector Addition: Based upon the example of moving in space from one locationto another, moving to a third location from the second will generate our conceptof vector addition. A graphical way of interpreting such an addition is to place thevector for the second movement at the end of the first vector. This creates a paral-lelogram which represents vector addition. The alternate sides of the parallelogramdepict the movement done in the reverse order. The symbol for the vector sum isnaturallyu + v. Subtraction is the inverse of addition as usual, sou − v is thevector that yieldsu when added tov.

Properties of vector addition: Vector addition is commutative and associative.The geometric reasoning is left as exercise.



Triangle Inequality: The basic inequality for the length of the sum is:

|u + v| ≤ |u|+ |v|.

Geometrically it says that the direct route is the shortest route between two pointsand making an intermediate may make the trip longer or not, but it can never makeit shorter.

Scalar Multiplication: The rescaling of a vector is called scalar multiplication.For any numberc, the vector multiplied by the scalarc is described as follows: Thenew magnitude is the absolute value ofc times the original vector magnitude whilethe direction is unchanged for positivec values and reversed for negativec values.The notation for scalar multiplication of the vectorv by c is: cv.

Properties of Scalar Multiplication: Scalar multiplication is commutative andassociative. The combination of addition and scalar multiplication has the distribu-tive property. The geometric reasoning is left to the exercises.


Many times vector operations will be viewed in coordinates. This means that somechoice of axes or more generally some system of vector directions has been made.Once such a choice is prescribed, all vectors can the found as vector sums of vectorsdirected along the specified special directions. The special directions are calledbasis vectorsand coordinates can be given in the terms of those basis vectors.The coordinates are usually called thecomponentsof the vector. When usingcomponents to describe the vector, our notation will use different end brackets toalert us. Thus2i − 5j =< 2,−5 > saves writing. If no basis is named explicitly,the default basis is{i, j}.

Basis Restrictions: To be a basis, the two vectors must not only be non-zerobut also point in different directions so that their scalar multiples form a two-dimensional grid. In linear algebra this is calledlinear independenceof the vec-tors. This also means that the components of a given vector in a given basis isunique.

The most common choice of a basis of vectors inR2 is tied to the usual rectangularcoordinates system and are thereforei andj. But other choices are valid and useful,so more generally we consider two vectorsb1 andb2 and then writev = c1b1 +c2b2. Clearly the components are tied to the choice of basis.

Vector Addition in Components: If u = a1b1 + a2b2 andv = c1b1 + c2b2,thenu+v = (a1 +c1)b1+(a2 +c2)b2, which is not surprising and uses the basic


properties (commutativity, associativity) of vector addition. Written in componentnotation this becomes:

< a1, a2 > + < c1, c2 >=< a1 + c1, a2 + c2 > .

Scalar Multiplication in Components: If u = a1b1 + a2b2, then for somescalarc, cu = (ca1)b1 +(ca2)b2, which is not surprising and again uses the basicproperties of scalar multiplication. Written in component notation this becomes:

c < a1, a2 >=< ca1, ca2 > .

Length of Vector in Components: If u = xi + yj, then|u| =√

x2 + y2. For ageneral basis, the expression for length of a vector in components is an importantproblem we will return to when more tools are available.

Sometimes vector algebra is so effective you can also use it to generalize from twodimensions to three or to some arbitrary positive integern without changing thealgebra symbols much if any while the meaning is shifted. Watch as this is done insetting up vectors inR3.

1.1.5 Vectors inR3

Location inR3 specified by coordinates(x, y, z) or (r, θ, z) or (ρ, φ, θ) can alsobe viewed in terms of aposition vector and then written invector notation. Vec-tors will be written in boldface type, such asu andv although other books andhandwriting often uses an arrow over the vector, as in−→u .

Formally, avector quantity in R3 has amagnitude (length) anddirection (exceptthe zero vector). This contrasts withscalars, which belong toR and have nodirection. Points are associated to the vector that goes from the origin to the point.Spherical coordinates are perfect for this description sinceρ is the length andφand θ describe the direction. But often the original rectangular coordinates arepreferred for other reasons.

The zero vector: The vector with magnitude zero has no sensibly unique way todefine its direction and is called thezero vector. It will be denoted by0.

Visualization: Vectors are plotted asdirected arrows showing length and direc-tion. The rectangular coordinate form uses three special vectors which are of length1 in the direction of the increasing coordinates. These are usually denotedi, j andk respectively. Then coordinates(x, y, z) correspond to the vectorxi + yj + zk,which suggests a need for the algebraic operation ofvector addition. Later inthis chapter special vector-valued functions are considered. In that situation, theplotted arrow is usually located so that its “tail” is the location of the input values.Otherwise we are free to place our arrows in any location when convenient.



Vector Addition: Based upon the example of moving in space from one locationto another, moving to a third location from the second will generate our conceptof vector addition. A graphical way of interpreting such an addition is to placethe vector for the second movement at the end of the first vector. This creates aparallelogram which represents vector addition. The notation is unchanged fromthe 2-D case. Subtraction is defined similarly as well.

Properties of vector addition: Vector addition is commutative and associative.The geometric reasoning is again left as exercise.


Triangle Inequality: The basic inequality for the length of the sum is:

|u + v| ≤ |u|+ |v|.The geometric reasoning is analogous to the 2-D case.Scalar Multiplication: The rescaling of a vector is called scalar multiplication.For any numberc, the vector scaled byc is described as follows: The new mag-nitude is the absolute value ofc times the original vector magnitude while thedirection is unchanged for positivec values and reversed for negativec values. Thenotation is againcv.

Properties of Scalar Multiplication: Scalar multiplication is commutative andassociative. The combination of addition and scalar multiplication has the distribu-tive property. The geometric reasoning is again left to the exercises.


Vector operations inR3 will often be viewed in coordinates. This means that somechoice of axes or more generally some system of vector directions has been made.Once such a choice is prescribed, all vectors can the found as vector sums of vec-tors directed along the specified directions. The special directions are calledbasisvectorsand coordinates can be given in the terms of those basis vectors. The coor-dinates are usually called thecomponentsof the vector. When using componentsto describe the vector, our notation will again use different end brackets to alert us.

Basis Restrictions: To be a basis, the three vectors must not only be non-zerobut also point in different directions so that their scalar multiples form a three-dimensional grid. In linear algebra this is calledlinear independenceof the vec-tors. This also means that the components of a given vector in a given basis isunique.

The most common choice of a basis of vectors inR3 is tied to the usual rectangularcoordinates system and are thereforei, j andk. But other choices are valid and


useful, so more generally we consider three vectorsb1, b2 andb3 and then writev = c1b1 + c2b2 + c3b3. Clearly the components are tied to the choice of basis.

Vector Addition in Components: If u = a1b1 + a2b2 + a3b3 andv = c1b1 +c2b2 + c3b3, thenu + v = (a1 + c1)b1 + (a2 + c2)b2 + (a3 + c3)b3, which isnot surprising and uses the basic properties (commutativity, associativity) of vectoraddition. Written in component notation this becomes:

< a1, a2, a3 > + < c1, c2, c3 >=< a1 + c1, a2 + c2, a3 + c3 > .

Scalar Multiplication in Components: If u = a1b1 + a2b2 + a3b3, then forsome scalarc, cu = (ca1)b1 + (ca2)b2 + (ca3)b3, which is not surprising andagain uses the basic properties of scalar multiplication.

Written in component notation this becomes:

c < a1, a2, a3 >=< ca1, ca2, ca3 > .

Length of Vector in Components: If v = xi+yj+zk, then|v| =√

x2 + y2 + z2.For a general basis, the expression for length of a vector in components is an im-portant problem we will return to when more tools are available.

Vector Addition and Free Body Diagram: In many applications where a forcevector is involved, thenet force on some object or at some location is the vectorsum of all the forces. For objects which are not in motion, there must be zeronet force. A sketch of force vectors for a motionless object is called afree bodydiagram and is often used to deduce the forces of tension or extension in a rope orbeam or bungee cord.

Picture goes here (Acme safe company?).

Vectors are so fundamental to this subject many books and many courses have thetitle “vector calculus”. Many of our key results will involve vectors so you mustbecome fully comfortable and confident in basic vector operations.


1.1.6 EXERCISES

Writing: Explaining, Reacting, Questioning

1. Does vector addition have all the properties you expect when you say “addi-tion”? If not, which properties are missing? If so, list all the properties youthought of and checked.

2. Does scalar multiplication have all the properties you expect when you say“multiplication”? If not, which properties are missing? If so, list all theproperties you thought of and checked.

3. The length of a vector has notation similar to the absolute value of a realnumber. In what ways are these concepts related? How do they differ?

4. Moving coordinate systems relative to one another play a major role in for-mulating physics. In driving a car along a straight section of road, describecarefully the distinction between your following distance (distance to the carin front of you) and your distance driven as measured by the odometer.

5. If we rotate the usualx − y coordinate axes by a counterclockwise rotationof π/2 radians, how are the new coordinate axes (the rotated vectors of theoriginal ones) related to the original ones? What does this say about the newcoordinates measured relative to the new axes for the same location? Howis this different from rotating the plane by the same angle while freezing thecoordinate axes?

6. On the circle of radius 5, how is the length of an arc along the circle tied tothe angular measurementθ? What if the radius of the circle is 10? Generalizeto any fixedr.

7. Around the equator, the 24 time zones happen over a distance of around24, 901.55 miles. Assume the earth is spherical. Recalling that latitude ismeasured as angles from the equatorial plane, what is the distance of a timezone at latitude30 deg? Compare that to the equator. What is the time zonedistance at a general latitude? Bonus: what time zone is it at the North Pole?

8. What does it mean when we say a plane is a two-dimensional subset in athree-dimensional space? What does it mean when we say that the plane isflat? Is the surface of a sphere flat or not? Explain what you mean.

9. How could you guarantee that a quantity or property was independent ofhow the origin of the coordinate system was chosen? Independent of theorientation of the axes with a given origin? What would yield evidence ofsuch independence? How would it relate to particular coordinate systems?

10. Cal Clueless goes to find the polar coordinates of the point(−1,−1) on hiscalculator. He getsr =

√2 andθ = π/4. What did he do wrong? For

additional reading, look for the functionatan2 online.


11. Randy M. Algebrist writes|u−v| ≤ |u|−|v|. Give an example to show thisis not correct, but then formulate a correct relation between the two sides ofthe inequality.

12. Consider the usual triplei, j, k, of unit vectors along thex, y andz axes re-spectively. In this section, the rigid motion that takes them cyclically aroundwas claimed to be a rotation. What is the axis of rotation (hint: it staysfixed by the motion)? What is the angle of rotation? What do the posi-tive coordinate axes look like when viewed as a 2-D projection in the planeperpendicular to the axis of rotation? If you are puzzled, try looking up atceilings in corners of rectangular rooms. It works!

13. Use coordinates in the usual basis forR2 to attempt to prove the triangle in-equality. If successful, congratulations. Whether successful or not, describethe algebra issues you confronted.

Calculational ExercisesFor problems 1 to 16, find the conversion from the given coordinates to the speci-fied others:

1. From rectangular to polar:x = 4, y = 4

2. From rectangular to polar:x = 1, y = −2

3. From rectangular to polar:x = 0, y = −5

4. From rectangular to polar:x = −7, y = 7

5. From polar to rectangular:r = 4, θ = π/3

6. From polar to rectangular:r = 1, θ = −π/6

7. From polar to rectangular:r = 4, θ = π/3

8. From polar to rectangular:r =√

2, θ = 11π/4

9. From rectangular to cylindrical:x = 3, y = −3, z = 2

10. From rectangular to cylindrical:x = 1, y = 0, z = 12

11. From rectangular to cylindrical:x = −1, y = −√3, z = 7

12. From rectangular to cylindrical:x = 5√

2, y = −5√

2, z = −5

13. From cylindrical to spherical:r = 6, θ = π/3, z = 6√

3

14. From cylindrical to spherical:r = 5√

2, θ = 7π/6, z = −5√

2

15. From cylindrical to spherical:r = 1, θ = 5π/3, z = −1


16. From cylindrical to spherical:r = 17, θ = 17π, z = 17

For problems 17 to 26, find the given quantity for the specified vector incomponents and when requested, draw a visualization (plot):

17. < 2, 4 > + < 1,−2 > with plot of parallelogram.

18. < −2, 3 > + < 1, 0 > with plot of parallelogram.

19. < 2, 4 > −5 < 1,−2 > .

20. < 7,−1, 4 > + < −1,−2, 3 > with plot of parallelogram.

21. | < 2, 4 > + < 1,−2 > | compared to| < 2, 4 > |+ | < 1,−2 > |.22. | < 2,−6 > + < 1,−3 > | compared to| < 2,−6 > |+ | < 1,−3 > |.23. | < 2, 4 > −5 < 1,−2 > | compared to| < 2, 4 > |+ 5| < 1,−2 > |24. | < 7,−1, 4 > + < −1,−2, 3 > | compared to| < 7,−1, 4 > | + | <

−1,−2, 3 > |.25. | < 1, 2, 2 > + < 3, 6, 6 > | compared to| < 4, 8, 8 > |26. | < 1, 3,

√6 > +t < 1, 3,

√6 > compared to|(1 + t) < 1, 3,

√6 > | for

any real value oft.

27. Consider the linear function3x + 4y and rewrite it in polar coordinates.Then consider a rotated set of polar coordinates that hasθ′ defined so thatthe maximum of the function on the unit circler = 1 occurs whenθ′ = 0.What does this say about the values of the function? Converting back torectangular coordinates, with the new rotation built in, say(x′, y′), showthat the function becomesAx′ for some constantA and then find the valueof A.

28. Generalize the previous problem to considerax+ by for any constantsa andb with a2 + b2 > 0. Then think about the 3-D caseax+ by + cz for a bit anddescribe why that was more difficult (unless you found it easy, which makesyou very talented).

29. A picture is hung on the wall where on the back there is a wire that runsacross and is located some distance from the top of the frame. If the wireis hung 4 inches below the top, the picture is 24 inches wide and 36 inchesacross and weighs 40 lbs. and is hung on one hook in the wall, find thebalance of forces needed. What happens if the picture is hung with twohooks? What happens if the wire is hung 8 inches down instead?

30. Oscillations: Amplitude and phaseIn many situations, a combination ofthe basic trig functions is found in the form:A cos t + B sin t. Interpret thevariablet (often time) as an angleθ and rewrite the combination in rectan-gular coordinates. Also consider viewingθ as shifted fromt (rotated coor-dinates) and show that every expressionA cos t + B sin t can be rewritten as


C cos(t + δ) with C > 0 unlessA = B = 0. C is called theamplitude andδ is called thephaseof the oscillation. In many situations, this is a betterway to see the function. For example, economists watch the economy go upand down and say that employment is a lagging indicator. The phase wouldbe the time to wait for such information to show in the employment data.The amplitude would be the change in employment.

what is multivariable calculus? - …math.gmu.edu/~rsachs/math215/textbook/math215ch1sec1.pdfmany...

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