Calculus from an Adv Viewpoint
Calculus from an Adv Viewpoint MATH 6102
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Chapter 7 R3 Euclidean Three Space and above With each ordered triple of numbers yz we can associate a point in Euclidean three space There are two different ways of representing this that are accepted First consider the usual zy plane with the horizontal axis representing the rst or x coordinate and the vertical axis representing the second or y coordinate There must be a third axis which is often considered as the line perpendicular to the plane of the page We will consider the part above77 the page as the positive z axis and the part below the page as the negative Z77 axis The second way to perceive of the three mutually perpendicular lines would be as in the associated gure Figure 71 We can de ne a one to one correspondence between ordered triples of real numbers 1 2 3 and the points in space Construct a plane perpendicular to the Z rst axis through the point 1 a plane perpendicular to the second axis through 2 and a plane perpendic ular to the third axis through 3 The point at which these three planes intersect is the point associated with the ordered triple 1 2 3 Some thought about this construction will convince you that this procedure es tablishes a one to one correspondence between ordered triples of reals and points in space The number 1 x is called the rst coordinate of the point 2 is called the second coordinate of the point and x3 is called the third coordinate of the point Again the point corre sponding to 000 is called the origin and we speak of the point zlx2z3 when we actually mean the point which corresponds to this ordered triple The three axes so de ned is called a coordinate system for three space and the Figure 71 87 88 CHAPTER 7 R3 EUCLlDEAN THREE SPACE AND ABOVE three numbers L y and 27 where Ly7 z is the triple corresponding to the point P7 are called the coordinates of P The coordinate axes are sometimes given labels most commonly7 perhaps7 the rst axis is called the x axis7 the second axis is called the y axis7 and the third axis is called the z axis We will often designate this space by 3 This is generalizable to the one to one correspondence between ordered n tuples of real numbers 1727 7x and points in n space7 R In R3 the distance between P A 17y1721 and 0 000 is given by 77777777777777777 an121 m y To see this note that the segment from the origin to P is the hy 3 potenuse of the right triangle with legs the segment from the origin to Q zhyl 0 and the segment from Q to P The segment QP has length 21 We need X11Zi Z T to nd the length of OQ That segment x1 lies in the plane and is the hypotenuse of 061050 the right triangle with legs of length 1 y thlp and y1 so it has length x Thus the length of the segment OP is Figure 72 2 M 21 2 96 21 Now7 it is a rather easy stretch to see that if Q 271322 then the distance from P to Q is dP7 Q 1 2 11 7322 21 222 This is a metric on the space of 3 tuples The generalization to R is that the distance between two points P 17 2 7x and Q 21172127 7 is dP7Q 951 912 952 922 39 We are interested in the connection between the geometry and algebra We want to see what different equations in the variables Ly7 2 will represent in R31 Consider for example the equation 2 y2 22 1 1The question is equally interesting in R but more dif cult to visualizel MATH 6102 090 Spring 2007 89 This is simply the set of all points Whose distance from the origin is exactly 1 Novv7 you might want to describe this point by point7 but that will not be ef cient or effective In three space this describes a sphere of radius 1 How do you want to visualize the sphere Figure 73 is one way to do it using Maple II 0 quotO W Ml III II S 9 39I o i I a 41 I v 1 Iquot l l Figure 73 Maple gure Figure 74 Maple vvirefrarne Figure 75 WmPlot gure Figure 76 DP Graph gure Figures 747 757 and 76 are other manners and programs that Will represent this gure One of the hardest parts of the study of functions of several variables is the visualization of the gures The advent of computer technologies help this quite a bit MATH 6102 090 Spring 2007 90 CHAPTER 7 R3 EUCLIDEAN THREE SPACE AND ABOVE That only gets us to dimension 3 though Taking the methods that we used prior to the use of easy graphing programs and applying them with the graphing technology can help us understand what geometric and visual properties higher dimensional gures may have Consider for example the equation 32 y2 z2 1 This is a hyperboloid of one sheet whose graph looks like the following Figure 78 DP Graph gure Figure 77 Maple gure How can we describe the graph of this gure One method is the method of slicing the surface There are c slices7 y slices7 and z slices These are slices through the gure by planes perpendicular to the Caxis7 the yaxis7 and z axis7 respectively These are called level sets First7 if we look at the yz plane the 6 slice when 6 0 we have a hyperbola on the y axis7 cf7 Figure 79 Note that the y slices will look the same since then we are looking at 2 k2 z2 1 The z slices are more informative Each slice gives us a circle 32 92 1 k2 and the smallest circle in the collection has radius 1 These curves are known as the level curves of the graph The graph of the level curves is given in Figure 710 Let S be the graph of Z2 y2 2 1 When we look at the level curve z C we have c2 92 2 1 or 2 y2 C2 1 Note that we have the same curve for z C and z C This tells us that the graph is symmetric about the plane z 0 That way we can just that part of the graph that is above the asy plane These curves are concentric circles of radius 02 1 centered at the origin There are no real level sets MATH 6102 090 Spring 2007 Figure 79 x slices Figure 710 Level curves Figure 711 Level curves for 101 lt 17 and for c 1 or 71 the level set is a single point7 the origin Figure 712 Level curves Figure 713 Level Set z 0 Next7 slice with the planes z 0 and y 0 to get a better idea of what this thing looks like For x 07 we see 22 7 y2 17 a hyperbola7 of Figure 713 Now putting this all together we see that we need to take the hyperbola in Figure 713 and rotate it around the level curves of Figure 712 This gives us the following gure This also is called a hyperboloid This is a hyperboloid of two sheets7 while the previously described hyperboloid is a hyperboloid of one sheet MATH 6102 090 Spring 2007 92 CHAPTER 7 R3 EUCLIDEAN THREE SPACE AND ABOVE Iill illll39nlguaa Figure 714 Hyperboloid of Two Sheets 71 Vectors There is another way to think about the points in the plane or n space We de ne a directed line segment in space to be a line segment together with a direction The directed line segment from the point P to the point Q is different from the directed line segment from Q to P We say that two segments have the same direction if they are parallel and their directions are the same We de ne two directed segments L and M to be equivalent L g M if they have the same direction and have the same length An equivalence class containing a segment L is the set of all directed segments equivalent to L These equivalence classes of directed line segments are called vectors The members of a vector V are called representatives of V Given a directed segment u the vector which contains u is called the vector determined by u The length or magnitude of a vector V is de ned to be the common length of the representatives of V This length is designated by The angle between two vectors u and V is the angle between the directions of representatives of u and V which we may choose to have a common origin Vectors turn out to be the right mathematical objects to describe certain con cepts in physics Velocity provides a ready example Saying the car is traveling 50 mileshour doesn7t tell the whole story you must specify in what direction the car is moving Thus velocity is a vectoriit has both magnitude and direction Such physical concepts abound force displacement acceleration etc The real numbers are called scalars as will be clear later but could be considered vectors of dimension77 one Thus we have a new mathematical object What do we have to do We have to MATH 6102 090 Spring 2007 71 VECTORS 93 see how to add subtract multiply and maybe divide these things That is we need to introduce an arithmetic or algebra of vectors First we tackle addition We must de ne what we mean by the sum of two vectors u and V Choose a representative of u and a representative of V Place the tail of V at the nose of u The vector which contains the directed segment from the tail of u to the nose of V is de ned to be the sum of the vectors u V The geometric representation of the Triangle lnequality shows us that Hu vll lt vv uv lt This is known as the parallelogram form for vector addition It should be clear that the it does not matter which representative of u and V you choose and that u V V u It should not be too hard to yourself also that addition is associative uVWuVW Since it does not matter where the parentheses occur it is traditional to omit them and write simply u V W The difference u 7 V of two vectors is de ned to be the vector you add to V to get u From our parallelogram the other diagonal is the difference of the vectors What do we mean by u7 u We de ne a special vector with 0 length called the zero vector and denoted 0 We may think of 0 as the collection of all degenerate line segments or points Note that the zero vector is special in that it has no direction If you are going 0 mileshour the direction is not importantl To make our algebra of vectors nice we make the zero vector behave as it should uiu0 and u0 ufor all vectors u Next we de ne the product of a scalar k E R with a vector u The product ru is de ned to be the vector with length and direction the same as the direction of u if r gt 0 and direction opposite the direction of u if r lt 0 We leave it as an exercise to show that 1 rsurusu 2 ruV rurV 3 0u 0 and 4 u71Vu7V This last property shows that it is safe to write 7u to stand for 71u A vector whose length is 1 is called a unit vector If V is a nonzero vector then the vector is the unit vector in the same direction as V Constructing a unit vector from a nonzero vector V is called normalizing a vector MATH 6102 090 Spring 2007 94 CHAPTER 7 R3 EUCLlDEAN THREE SPACE AND ABOVE Now how do we see that these vectors77 correspond to what we are studying We need to de ne a one to one correspondence between vectors and points in n space This will then establish a one to one correspondence between vectors and ordered n tuples of real numbers Take a representative of the vector u and place its tail at the origin The end or nose of this representative is at a point in n space and we associate that point with u We handle the vector with no representatives by associating the origin with the zero vector The fact that the point with coordinates a1 a2 an is associated with the vector u in this manner is indicated by writing u J lta1a2 agt The numbers a1a2 an are called the coordinates or components of u We frequently do not distinguish between points and vectors and indiscriminately speak of a vector lta1 a2 agt or of a point u Let us consider 3 space for the moment What we learn here generalizes immedi ately to n space Suppose u lta b cgt and V ltzy The following are true Whom uVltaxbycdgt u7Vlta7b7ycidgtand r5905 ru ltra rb rcgt Let i be the vector corresponding to the point lt1 0 0gt letj be the vector corre sponding to lt010gt and let k be the vector corresponding to lt0 0 1gt Any vector u can now be expressed as a linear combination of these special coordinate vectors u ltzyzgt iyjzk The set ijk is called the standard basis of R3 Likewise i lt10gt andj lt01gt are the standard unit vectors in R2 Vectors i andj in that order de ne an orientation of R2 Let V ai bj be a nonzero vector in R2 Then 31 0 and a b VHVH i1iJ HVH HVH Now aHvH cos0 and sin 0 where 6 is the angle that V is the angle that the representative of V that initiates at the origin makes with the positive d axis 7 measured counterclockwise This means that we have another way of representing V in terms of and 0 V HvHltcos0i sindj This is called the polar form of a vector since the components of V are expressed in polar coordinates with r MATH 6102 090 Spring 2007 71 VECTORS 95 711 Parametric Equations of Lines and Planes Let V ltoli2gt be a nonzero vector in R2 Draw the line Z that contains the origin 0 and the point 0102 This means that the line Z contains77 the vector V Pick any other point P on Z and consider the vector W determined by O and P Since W is parallel to V there is a real number t so that W tV Thus for every point P on the line Z there exists a unique scalar multiple tV of V whose tip is located at that point The line Z can then be represented as the collection of all possible scalar multiples of V Z tv 1 t E R Now pick a point A 1102 and a vector V ltoli2gt and visualize V as its representative directed line segment that starts at A Let Z denote the line that contains A and whose direction is the same as V and let P zy be a point on the line cf Figure 715 Then 15 6W where 17 ltygt and Ti lta1a2gt and W is the vector from A to P Since W is parallel to V W tV for some t E R and therefore 15 6 tV t e R This derivation is the vector form of Figure 715 a parametric equation of the line Z As the parameter t takes on different values in R the tips of the vectors describe points on the line The equation is usually written as Rt EitV t6 R or in component as lt01 1 ZU112 Iii2 t6 R or as xa1to1 ya2to2 tER Any of these forms is called a parametric equation of a line This method works in R3 as well The line Z in R3 that contains the point A a1a2a3 and whose direction is that of a vector V lto1i2oggt has parametric equation lt 55 tV lta1 t111a2 toga3 to3gt t E R Example 71 Find the equation of the line that contains 120 and 0 724 MATH 6102 090 Spring 2007 96 CHAPTER 7 R3 EUCLlDEAN THREE SPACE AND ABOVE We need to get the vector that points from A 120 to B 07 724 This is simply V lt1 7 07 2 7 720 7 4gt lt1747 74gt Then the parametric equation of the line is either Z10lt17270gttlt174774gtlt1t724t774tgt7t6 R or 21tlt077274gttlt174774gtltt772 4m 7 4tgt t6 R While these appear to be different7 they do describe the same set of points Let V and W be nonzero7 nonparallel vectors7 visualized as directed line segments starting at the same point A The point A and the tips of V and W determine a unique plane called the plane through A spanned by V and W Now7 lets see what the parametric equations might be Pick a point P Lyn on the plane and let P 7 A ltx 7 my 7 122 7 13gt Write this vector as a linear combination of V and W7 PA tVSW7 571 E R This means thatp EitV5W7 571 E R This is an example of how to use vectors to nd where the medians of a triangle intersect We need to nd scalars s and It so that la la 6tlt b7a slta b76 Simplifying gives us This means that we must have s t 0 d7770 an 2 2 s t 2 2 Otherwise7 5i and 8 would be nonzero scalar multiples of one another7 which would mean they have the same direction It follows that st7 3 This is the result that we would expect 72 Dot or Scalar Product We have seen how to add and subtract vectors We would like to nd a product of vectors Should the product of two vectors be another vector This should be the MATH 6102 090 Spring 2007 72 DOT OR SCALAR PRODUCT 97 case but it doesnt have to be We will rst de ne a different product called the scalar product u V If V lt11121ngt and W ltw1w2 wngt then de ne V 39 W i1 Clearly in R2 we have that i j 0 and i i j j 1 In R3 we have that iijjkkandijikjk0 We have the following properties for the dot product VW WV uVW uVuW av WaVW VaW OV 0 V39VllVll2 VW if V and W are parallel in the same direction V W illvll HwH if V and W are parallel in the opposite direction Each choice of a pair of vectors in R will give us a plane thinking of taking the representatives starting at the same point Let 6 be the smaller of the two angles determined by V and W so that 0 S 6 S 7139 If V and W are parallel then 6 0 or 6 7139 Theorem 71 Let V and W be vectors then V39W HVHHWHCOS where 6 is the angle between V and W PROOF If V 0 or W 0 then both sides are 0 If 6 0 or 6 7139 then we have seen that these are true from above So lets assume that V 31 0 W 31 0 and 0 lt 6 lt 7139 The vectors V W and V 7 W form a triangle with 6 the angle opposite V 7 W By the Law of Cosines HV WHZ M2 HWH2 2M MW 0039 Now expand the left hand side of the equation Hviwllz V7WV7WV7W VV7WV7VWWW Hvll272VWlel2 Simplifying from above we get 72V W 72llvll cos6 and VW HvHHchos6 MATH 6102 090 Spring 2007 98 CHAPTER 7 R3 EUCLlDEAN THREE SPACE AND ABOVE Theorem 72 Let V and W be nonzero vectors then V W 0 if and only ifV and W are orthogonal De nition 71 Vectors V1V2 Vk where h 2 2 in R n 2 2 are said to form an orthonormal set if the are of unit length and each vector in the set is orthogonal to the others Theorem 73 Angle between Two Vectors LetV andW be to nonzero vectors then V W cos 6 7 HVH W H where 6 is the angle between V and W This is really just a corollary of Theorem 71 If V is a vector in the plane R2 then the directional angles are the angles given by the directional cosines Vi z cosai HVl xx2y2 COSBQL H V962 112 Note that cos 04 is the cosine of the angle that the vector V makes with the w axis and is the w coordinate of the point where the vector intersects or would intersect the unit circle More importantly note that cos is the cosine of the angle that V makes with the y axis Thus 6 is complementary to 04 and cos sina and in fact this is what we see Again note that this is the y coordinate of the point of intersection of the vector with the unit circle In R3 these directional cosines and directional angles are not as transparent Again they are de ned by V i z cosa 7 VH j y cos i 2y222 lt H7 k z cos y7 lVH xz2y222 lt This is one way to express a vector in terms of other known quantities However it is not a practical expression to use We seek a better way to describe these vectors in MATH 6102 090 Spring 2007 73 VECTOR PRODUCT lN R3 99 terms of known vectors Let Vi7 239 17 771 be a collection of nonzero orthogonal vectors in R and let El 6 R be an arbitrary vector Then we can write aiVi7 Ql H Ms Z 1 where 11 ElViHVlH2 for 239 1 771 To see how we get this7 you need to recall from linear algebra that any vector in R can be written uniquely as a linear combination of n orthogonal vectors Ela1V1anVn Since the vectors Vi are orthogonal when you take the dot product of i with Vi we are left with only one term on the right hand side of the equation V I39Vi aivi 39Vl39 Solving this for 11 gives us the result 73 Vector Product in R3 We de ne a product of two vectors that results in a vector7 but we will de ne this only in R3 There are generalizations to higher dimensions7 but they are not for this course We de ne the cross product of two vectors V o1i ozj wk and W w1i wzj wgk is the vector V gtlt W 11ng 7 o3w2i 13101 7 o1w3j 1le 7 o2w3k This is more easily remembered as the determinant of a 3 gtlt 3 matrix i j k 11 12 13 V X W w1 w2 w3 We have the following properties Theorem 74 Let u V and W be vectors in R3 and let 04 E R The cross product satis es the following properties V gtlt W 7W gtlt V antlcommutatlvlty ugtltVWuxvugtltW39 uVgtltWuxwVgtltW39 VgtltV0 PhikFel m aVXWaVXWVXaW MATH 6102 090 Spring 2007 100 CHAPTER 7 R3 EUCLlDEAN THREE SPACE AND ABOVE PROOF The rst and the last two are most easily proven from the determinant de nition Recall that if you switch two rows in a matrix then the determinant of the resulting matrix is 71 times the determinant of the rst matrix This handles the rst case Also recall that if a matrix has two equal rows then the determinant must be zero because the row vectors are dependent Lastly if you multiply any row by a constant then the determinant of the resulting matrix is that constant times the determinant of the rst matrix To check the two distributivity properties just use brute force and compute the products No nesse is necessary igtlti0 igtltjk igtltk7j jgtlti7k jgtltj0 jgtltki kgtltij kgtltj7i kgtltk0 Consider the expression u V gtlt W This is called the scalar triple product of u V and W Again by brute force we can show that U1 U2 U3 uVgtltW11 p2 p3 W1 102 W3 Theorem 75 Let V and W be vectors in R3 Then 1 The cross product V gtlt W is a vector that is perpendicular to both V and W 2 The magnitude ofVgtltW is HvgtltwH waH sint9 where 6 is the angle between V and W PROOF To show the rst part we need to show that V V gtlt W 0 as well as W V gtlt W 0 Since this is the scalar triple product each of these is computed from the determinant Since there is a repeated row in each of the determinants both are zero and we are done The second part is not so slick From the de nition of the vector product and the length of a vector we have that HV gtlt wllz 11ng 7 o3w22 11le 7 o3w12 11le 7 o2w12 Strategically add zero in the form of ofwf ogwg ogwg 7 111271 7 ogwg 7 3103 to the right side of the equation 222 22 22 22 22 22 22 22 22 UV gtlt Wll 1W1 1w2 iws 2w1 2102 2103 3w1 3102 3w3 2 2 2 2 2 2 7 mm 11sz 1ng 2o1w1o2w2 2o1w2o3w3 2o2w2o3w3 MATH 6102 090 Spring 2007 73 VECTOR PRODUCT lN R3 101 This gives llV X Wll2 l g 3W 103 10 i 1102 2102 131032 HVHZHWHZ VW2 llVllleWll2 HVHZHWHZCOSW V 2 W Zsin20 H H Now7 taking the square root of both sides we easily get HV gtlt Note that V sin2 lsin l sin 07 since 0 S 6 S 7T7 so therefore sin0 2 0 Thus7 our formula holds I Since we have already de ned i gtltj k we have chosen our orientation of the vector product This particular orientation is called the right hand rule Theorem 76 Vectors V and W are parallel if and only lfV gtlt W 0 Theorem 77 Let V and W be nonzero nonparallel vectors in R3 Then HV gtlt wll is the area of the parallelogram spanned by V and W and NV X W HVHHWH sin9 MATH 6102 090 Spring 2007