1 Introduction to the Characters of Calculus III

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1 1 Introduction to the Characters of Calculus III 1.1 The Sets R 3, R 2, and Subset Thereof Note: In the first part of these notes we will be working in a purely algebraic fashion with sets, subsets, and vectors in R 2 and R 3. We will in no way be graphing/plotting/drawing/visually representing these objects geometrically. From my perspective, graphically representing these objects will require a coordinate system. Coordinate systems are the subject of the second part of these notes. Only after coodinate systems have been introduced will we be able to geometrically represent sets/subsets and vectors. Please keep this distinction in mind when working through the first part of these notes, and endeavor to exercise your algebraic skills by resisting the urge to try to draw pictures. You algebraic skills will at least as useful, if not more useful, than your geometric intuition. Best, Professor Osborne Let s first define the sets R 3 and R 2 by In actual words, eq. (2) reads as R 2 = {(x,y) x,y R} (1) R 3 = {(x,y,z) x,y,z R} (2) the collection (indicated by the set symbols {}) of all three-tuples of numbers such that (indicated by the symbol ) x, y and z are each real numbers R is defined to be (indicated by the symbol =) R 3 (pronounced R three ). (Example) (1, 2, 4) R 3 b/c each of 1, 2, 4 R; (0, 0) R 2 b/c 0 R; (2, 0, 3), (π, ln(2),e 1 ) R 3 ; ( 1, 2, 3i) / R 3 b/c 1 = i / R Here is another set, this time a subset of R 2 S = {(x,y) x 2 + y 2 = 1} (3) In words, S is the sets of all two-tuples (x,y) such that x and y satisfy the algebraic equation x 2 + y 2 = 1. (Example) (1, 0) S b/c = 1; (1, 1) / S b/c = 2 1; (1/ 2, 1/ 2) S b/c squaring each of 1/ 2 and adding gives 1; (0, 3) / S (You Try) Find 2 others points S. Find 2 others points / S. (You Try) Given the set defined by (a) H is a subset of R x for what value of x? (b) Is (1, 0) H?; Is (0, 1) H?; Is (2, 3) H H = {(x,y) x 2 y 2 = 1} (4) (c) Find 2 other points H. Find 2 other points / H. (You Try) Given the set defined by G = {(x,y,z) xy } + z = 0 (5) In words: G is the set of all points which satisfy the algebraic expression x y + z = 0. 1

2 (a) G is a subset of R x for what value of x? (b) Is (1, 1, 1) G?; Is (2, 0, 2) G?; Is (1, 2, 1/2) G (c) Find 2 other points G. Find 2 other points / G. Conclussion: The sets R 3, R 2 and subsets therof are major characters in Multivariable Calculus (a.k.a Calculus III). The next major players are vectors. Homework: 1. Define your own two subsets for each of R 2 and R 3 and list two point in and out of your sets. 2. Define a subset of R 4. List two points in the set and out of the set. 1.2 Further Introductions Vectors Recall the sets R 2 = {(x,y) x,y R} R 3 = {(x,y,z) x,y,z R} Let us now choose two generic points in, R 3 called x (little x) and X (big X) denoted by x = (x 1,x 2,x 3 ) and X = (X 1,X 2,X 3 ). (Note) We can make the points x and X specific by choosing specific values for x 1,x 2,x 3 and X 1,X 2,X 3, say x 1 = 1,x 2 = 2,x 3 = 3,X 1 = 4,X 2 = 5,X 3 = 6. We would then write x = (x 1,x 2,x 3 ) = (1, 2, 3) and X = (X 1,X 2,X 3 ) = (4, 5, 6) Let us now define a new entity called the vector from x to X [denoted by xx] and defined by xx = (X 1 x 1,X 2 x 2,X 3 x 3 ) (6) (Example) For x = (x 1,x 2,x 3 ) = (1, 2, 3) and X = (X 1,X 2,X 3 ) = (4, 5, 6) then xx = (4 1, 5 2, 6 3) = (3, 3, 3) (You Try) Given the points P = (1, 2, 3) and Q = (2, 5, 3), find the vectors PQ and QP. (Notatial Note:) A question that can be asked at this point is, (Q:) What is the relationship of points in R 3 to vectors? Let X = (X 1,X 2,X 3 ) be a point in R 3 and let us denote the special point (0, 0, 0) by the symbol O (for Origin). It follows that the vector from O to X is OX = (X 1 0,X 2 0,X 3 0) = (X 1,X 2,X 3 ). (A:) That is, points X R 3 can be viewed as vectors in from O = (0, 0, 0) to that point X. In other words, since both the point and vector are represented by (X 1,X 2,X 3 ) it look like there is no distinction between points and vectors when using the origin 2

3 O = (0, 0, 0). The only identifying feature is that we add the word point or vector designation. We will soon see that pictorially we should treat points and vectors differently. (Example) If X = (X 1,X 2,X 3 ) = (4, 5, 6) then OX = (X 1,X 2,X 3 ) = (4, 5, 6). If A = (1, 2) and O = (0, 0) then OA = (1, 2). (Notation Note) From now on we denote OX as just X (Note:) Realize that we are working with vectors purely algebraically, i.e. without drawing or geometrically representing them as this will require coordinate systems (more on this shortly). (You Try) Given the various points A = (2, 0, 1), B = (3, 2, 0), C = (1, 0, 5), D = (2, 3, 1), E = (2, 1), F = (3, 4), O 3 = (0, 0, 0), O 2 = (0, 0) Find A, F, AC, CA, CB, BA, CC, EF, O3 B, BC, O3 C. (Observation and You Try:) It will be trut that CA = CB + BA if we define addition of vectors entry by entry (check this for yourself). (Definition:) Given two vectors in R 3, v = (v 1,v 2,v 3 ) and w = (w 1,w 2,w 3 ) then the vector addition of v and w denoted v + w is defined to be v + w = (v 1 + w 1,v 2 + w 2,v 3 + w 3 ). (7) Define multiplication of a vector v with a scalar (constant) c by c v = (c v 1,c v 2,c v 3 ) (8) (Note:) This vector addition and scalar multipliation extend to R n for any n, but for this class we will stay focused on R 2, R 3, and subsets thereof. (You Try) Use your answers from above to verify that O 3 B + BC = O3 C. Find A + B. Find F + E. Find 2 A. Find 2 E. 2 On Coordinate Systems 2.1 The Standard Coordinate System To this point we have defined points and vectors in R 3 and R 2 purely algebraically by eqs. (1), (2), and 6. We have also defined vector addition algebraically by eq. (7). We have in no way appealed to pictures or geometric representations of these objects because this requires the use of a coordinate system as we now discuss. If you would like to picture R 3 think of an infinite box. 3

4 z 05 x0 0y 10 z 05 x0 0y Figure 1: (left) Think of R 3 as an infinite box. (middle) When a point Q R 3 is given, say Q = ( 5, 10, 10), we imagine gridding up R 3 and then counting -5 units in the x direction, -10 in the y direction, and -10 in the z direction and then drawing a green dot. (right) Different than points, vectors can be represented by (directed) arrows. For example, for the two points O = (0, 0, 0) (the standard origin, the red dot) and P = ( 10, 10, 10) (the blue dot), the vector OP = P O = ( 10, 10, 10) is represented as the arrow from the red dot to the blue dot. A closer view of the right plot is shown in Figure 2. (Definition:) A coordinate system on R 2 or R 3 is defined by two pieces of information 1. A choice of origin, i.e. either a specific point O 2 R 2 or O 3 R A choice of 2 or 3 vectors 2 for R 2 and 3 for R 3. (Step 1 Choose an Origin) Let us start with the choice of origin by choosing two points in R 3, say P = ( 10, 10, 10) and O = (0, 0, 0) which we will call the origins of two coordinate systems. 1. (Origin 1) If P = ( 10, 10, 10) is the origin then by this we mean that points in R 3 will be converted to vectors using the defintion PQ where Q is any point in R 3. For example, if Q = ( 5, 10, 10) is a point in R 3 then PQ = Q P = (5, 20, 0). Also, P O = O P = (10, 10, 10). 2. (Origin 2) If O = (0, 0, 0) is the origin then by this we mean that points in R 3 will be converted to vectors using the definition OP where P is any point in R 3. For example, For example, if Q = ( 5, 10, 10) is a point in R 3 then OP = ( 5, 10, 10). 3. You can choose any point in R 3 as an origin. So essentially, choosing an origin for the coordinate system is choosing, out of all possible points, a reference point from which to draw vectors from. See Figures 1 and 2, which we can now draw because we have specificed origins. (You Try) Given an origin B = (1, 0, 0), Express the point C = (3, 0, 0) and D = 4

5 10 z y x Figure 2: A Closer View of Figure 1 (right). Different than points, vectors can be represented by (directed) arrows. For O = (0, 0, 0) (red) and P = ( 10, 10, 10) (blue), the vector OP = P O = ( 10, 10, 10) is represented as the arrow from red to blue. (1, 0, 4) as vectors relative to this origin. Sketch a picture of all points and the two vectors. (Step 2 Choose the legs of the coordinate system) Now we move to the second piece of information needed to define a coordinate system; a set of 2 or 3 vectors. To illustrate we begin with R 3 with origin O = (0, 0, 0). From the three points e 1 = (1, 0, 0),e 2 = (0, 1, 0),e 3 = (0, 0, 1) we find the three vectors Oe 1 = e 1 = (1, 0, 0), Oe 2 = e 2 = (0, 1, 0), Oe 3 = e 3 = (0, 0, 1). The three vectors e 1, e 3, e 3 complete the specification of a special coordinate system for R 3 which we denote by (O, e 1, e 2, e 3 ) (9) and call the standard coordinate system on R 3 (See Figure 3). (Note:) Emphasis has been placed on the word a in a coordinate system because there are many, many, many coordinate systems one can choose for R 3 ; One can simply change the origin or one can change the set of three vectors. 5

6 Figure 3: The Standard Coordinate System. (left) The red point is O = (0, 0, 0), while the three black points are e 1 = (1, 0, 0),e 2 = (0, 1, 0),e 3 = (0, 0, 1). The three arrows (black) e 1 = e 1 O, e 2 = e 2 O, e 3 = e 3 O point in the x,y,z directions, respectively. (right) The vector (blue) X = (X 1,X 2,X 3 ) = (1, 1, 1) = 1 e 1 +1 e 2 +1 e 3 expressed in the standard coordinate system (O, e 1, e 2, e 3 ). 2.2 What Is a Coordinate Systems Good For? In short, a coordinate system allows one to specify the coordinates of a vector which we now explain. We first illustrate using the standard coordinate system from equation 9. (Example:) Choose any point A in R 3, say A = (1, 2, 3). Express A as a vector in the coordinate sysem (O, e 1, e 2, e 3 ). 1. First A = OA = (1, 2, 3). This is using that O is our choice of origin for the coordinate system. 2. Now that the point A is the vector A use the scalar multiplication and vector addition of eqs. (8) and (7) to write A as A = (1, 2, 3) = 1 (1, 0, 0) + 2 (0, 1, 0) + 3 (0, 0, 1) (using properties eq.(8)and eq.(7)of vectors) = 1 e e e 3 (10) These steps are so easy because of our choice of the vectors e 1, e 2, e 3. (Terminology:) Once you have the vector A written in the form of eq. (10) we say that the vector has been decomposed or broken down in the standard coordinate system. 3. (Definition:) The numbers 1, 2 and 3 in front of the e s from eq. (10) are called the coordinates of A relative to the coordinate system (O, e 1, e 2, e 3 ). 4. (Definition:) The coordinate vector of A is formed from the coordinates of A. For the example above, the coordinate vector is (1, 2, 3). 6

7 (Q:) Why is the Standard Coordinate System (O, e 1, e 2, e 3 ) so nice? (A:) The coordinate vector of any vector A expressed in the standard coordinate system is the same as the vector itself. That is, for example, B = (2, 3, 4),C = ( 2, 1, 6) then B = (2, 3, 4) = 2 e e e 3. and C = ( 2, 1, 6) = 2 e 1 1 e e 3 so the coordinate vectors are (2, 3, 4) and ( 2, 1, 6). This is not always the case for other coordinate systems. As we see with the next section. 2.3 Non-Standard Coordinate Systems Let s begin with R 2 and choose two different coordinate systems #1 and #2 given by #1 (O, e 1, e 2 ) where O = (0, 0) and e 1 = (1, 0) and e 2 = (0, 1). #2 (A, v 1, v 2 ) where A = (2, 2) and v 1 = (1, 1) and v 2 = (1, 1). (Example:) Express the point C = ( 3, 3) R 2 as a vector in coordinate system #2. Further, what is the decomposition of this vector relative to coordinate system #2? To complete the example, we essentially answer both parts at the same time. First we express C as a vector in coordinate system #2 by doing C A then break it up into the relevant two parts AC = C A = ( 5, 5) v {}} 1 { = x 1 (1, 1) +x 2 (1, 1) }{{} v 2 = (x 1 + x 2,x 1 x 2 ) where x 1 and x 2 are the unknown coordinates. For two vectors to be equal (in this case ( 5, 5) and (x 1 + x 2,x 1 x 2 )) each entry of the vector must equal. That is, to find x 1,x 2, we must solve the system of equations x 1 + x 2 = 5 x 1 x 2 = 5. (11) The solution turns out to be x 1 = 5 and x 2 = 0 which you should check by AC = ( 5, 5) = 5 v v 2 (multiply out, does it equal ( 5, 5)?). (12) In conclussion, in this example the vector AC has been decomposed or broken down relative to coordinate system #2. The coordinate vector of AC relative to coordinate sysem #2 is ( 5, 0). 7

8 y x Figure 4: Two Coordinate Systems in R 2. The standard system is given by the black point and two black arrows, while a non standard system is given in by the red point and two red arrows. A single point P=(1,3) is shown in blue and expressed as a vector in each of the two coordinate systems (shown as blue arrows). It is only when a coordinate system is selected that we really know how to draw vectors/arrows. The point of this figure is that a single point P can have multiple vector descriptions depending on the coordinate systems selected. Homework: 1. Express the the points C = ( 1, 5) and D = (2, 4) as vectors in coord. sys. #2. What is the decomposition these vectors relative to #2. What are the coordinate vectors of these vectors relative to #2? 2. Given an origin A = (1, 3) and the points (2, 5) and (0, 1), define a coordinate system for R 2. Use this coordinate system to express the point C = ( 2, 5) as a vector. What are the coordiantes of this vector relative to the new coordinate system? 2.4 Choosing A Coordinate System Gives a Framework Within Which To Construct Plots (This Whole Subsection is One Example): Recall from Section 2.3, that we had two coordinate systems #1 (the standard coordinate system on R 2 ) given by O = (0, 0) and e 1 = (1, 0) and e 2 = (0, 1) and a non-standard coordinate system given by (A, v 1, v 2 ) where A = (2, 2) and v 1 = (1, 1) and v 2 = (1, 1). See the black and red vectors in Figure 4. 8

9 The entries of a coordinate vector can be interpreted as instructions about how to travel from the origin of the coordinate system in each of the directions to another point. For example, the point P = (1, 3) is expressed as a vector in the standard coordinate system as P = (1, 3) = 1 e e 3 and so the coordinates vector is (1, 3). The coordinate vector can be interpreted to be those instruction telling us how to move in the given coordinate system; starting at the origin O, go 1 step in the e 1 direction and then 3 steps in the e 2 direction and you will end at point P. This process is shown by the blue arrows in Figure 4. Considering now the second (red) coordinate system, the point P = (1, 3) is expressed as the vector (1 2, 3 2) = ( 1, 1) = 1 v v 1 and so the corrdinate vector is ( 1, 0). The coordinate vector can be interpreted to mean the following: if we start at the origin A and travel 1 steps (i.e. backwards 1 step) in the v 2 direction and then 0 steps in the v 1 direction then we will end at the point P. That is, the coordinate vector contains the information about how we are to move inside the given coordinate system. This process is shown in Figure 4. The point of this example is to show that points can have many ( many) different vector and coordinate vector descriptions. Because of its covenience, the standard coordinate system will be the one most often employed. That is, from now on unless otherwise stated, we will represent points and vectors relative to the standard coordinate system. That is, a given point corresponds to the vector X = (X 1,X 2,X 3 ) R 3 with coordinate vector X = (X 1,X 2,X 3 ) = X 1 e 1 + X 2 e 2 + X 3 e 3 (X 1,X 2,X 3 ). So, for example, the coordinate vector (2, 3, 4), unless otherwise stated, corresponds to the point X = (2, 3, 1) R 3 and the vector X = (2, 3, 1). It will be necessary from time to time throughout this class to change coordinate systems or to take the origin of the coordinate system to be other than O. When we get to this point we will be ready from our preparations here. Now test your understanding by doing the following problems. 9

10 y x Figure 5: Use This Figure For the Homework Problems. The red coordinate system is given by (A,v 1,v 2 ) = ((2, 2), (1, 1), (1, 1)). Homework: Use Figure 5 to answer the following questions. The points P 1 = (0, 5),P 2 = (0, 2),P 3 = (1, 3),P 4 = (3, 5),P 5 = (3, 2),P 6 = (4, 2) R 2 are given. Note that I said these vectors are coordinate vectors. Answer the following questions 1. Given the coordinate vectors (relative to coord. sys # 2), (1, 1), (2, 1), and ( 1, 1), what points correspond to these coordinate vectors? Check your answer algebraically. (For example:), given the coordinate vector (0,-1) this means we go 0 steps in v 1 and 1 step in v 2 so we end up at point P 3 = (1, 3). To check your work algebraically, notice that AP 3 = ( 1, 1) = 0 v 1 1 v 2 = 1 (1, 1) = ( 1, 1) (sweet, they re equal). 2. Try to use the figure to geometrically determine the coordinate vectors of the vectors AP 5 and AP 1. Do the algebra (similar to eq. (11)) to check your geometrically determined answer. 10

11 3 Now Pick Up Select Topics From Our Book (Briggs and Cochran). Let s take as a working assumtion that, unless otherwise stated, all points P R 2or3 will be represented in the standard coordinate system. (Notational Note:) Our book uses bold faced font to represent vectors, i.e. v = Ov is v. Also our book uses instead of e 1, e 2, e 3 the symbols i, j, k, respectively. You will also note that when describing the vector, our book uses > symbols. That is, instead of writing the vector X = (X 1,X 2,X 3 ) as we have done throughout these notes, our book writes X = X 1,X 2,X 3. In our class, I will use either depending on the convenience of the situation. Most often when I want to distinguish a point from a vector I will not rely on a notation, but come flat out and tell you in words that I want you to think of the object as a point or a vector. Now let s pick up some select topics from our book: 1. Magnitude of vectors in R 2 and R 3 and unit vectors (pgs. 684, 686)) 2. The dot product, its properties, and orthogonality (pg. 705) 3. Angle between vectors (pg. 704) 4. The cross product and its properties (pg. 714) 11