University of Waterloo Faculty of Mathematics. MATH 227: Calculus 3 for Honours Physics

Transcription

1 Universit of Waterloo Facult of Mathematics MATH 227: Calculus 3 for Honours Phsics Fall 2010 Lecture Notes E.R. Vrsca Department of Applied Mathematics c E.R. Vrsca

2 Lecture 1 (Relevant sections from Stewart, Calculus, Earl Transcendentals, Sith Edition: 14.1, 16.1) Introduction The purpose of this course is two-fold: 1. to develop further the calculus of functions of several variables, i.e., multivariable calculus, 2. to present the standard concepts and methods of differential and integral vector calculus. Even though Item 2 is technicall contained in Item 1, it warrants separate mention since vector calculus will soon be plaing an important role in our studies of Phsics. Indeed, from a historical perspective, vector calculus was developed in order to understand a number of phsical phenomena occurring in nature, e.g., gravitation, electromagnetism, fluid flow. Note that these phenomena require the concept of a field which eists in space and which can be used to eplain their behaviour. In our first Calculus course, ou were introduced to the idea of a real-valued function of a single real variable, i.e., f(). For each real number,, in some domain, the function f assigns a unique real number, which is denoted as f(). So the input to this function f is a real number, and its output is a real number. In the precusor to this course (hopefull!), e.g. MATH 128, ou were then introduced to the idea of a function of several variables, written as f(,) or f(,,z) or perhaps f(,,t). In general, the input to such a function f is an n-tuple: an ordered set of n real numbers, which will be written genericall as = ( 1, 2,, n ). Such an ordered n-tuple ma be considered as a point in R n. And the output of such functions was assumed to be a single real number. In all of the above eamples, the output of the function considered is a single real number, or scalar. As such, all of the above cases are eamples of scalar-valued functions. 2

3 Note: In books, a vector or n-tuple is usuall denoted as a boldfaced letter, e.g., = ( 1, 2,, n ). In lectures, I shall normall use arrow notation, e.g.,, since it is tedious to tr to write boldfaced characters on the blackboard. (The notation is also used b some lecturers.) Also, it will sometimes be convenient and I ve alread done it above to let, and z denote the independent variables, as opposed to 1, 2,. Mathematicall, the action of a scalar-valued function f is written as follows: f : R n R. Eamples: 1. n = 2: f(,) = n = 3: f(,,z) = z + z 5 3. Phsical quantities such as temperature, pressure and concentration are scalar quantities. For eample, the function T(,) = could represent the temperature on the surface of a hotplate at the point (,) for a suitabl defined region (,) R R 2. We shall often refer to these phsicall-relevant, scalar-valued functions as scalar fields, to differentiate them from vector fields that will also be an important part of this course. Unless otherwise indicated, the domain of the function f, to be denoted as D(f), will be defined as the set of all points R n such that f() is defined. And the range of f, to be denoted as R(f) is the set of all possible values of f() for D(f). As ou know well from first-ear calculus (and earlier), the graph of a function f : R R is a curve in R 2 defined b the association = f(). For each D(f), the point (,) = (,f()) lies on the graph of f. The graph of a function generall gives us a good idea of its action on points in its domain. It often allows us to identif hot/high regions as well as cool/low regions. 3

4 In MATH 128, ou saw that the graph of a scalar-valued function of two variables, f : R 2 R, could also be visualized: Graphicall, it is the surface in R 3 defined as z = f(,), sketched schematicall below. z z = f(, ). (a, b, f(a, b)) O. (a, b,0) Graph of f(, ) is represented b the surface z = f(, ) in R 3. Here, the input variables are and ou need both of them to identif a unique input to f. And for each input pair (,), f will produce an the output z = f(,). As such, the point (,,z) = (,,f(,)) is a point on the graph of f. We ll consider some eamples a little later in the course. The graph of a function of three variables f(,,z) would require four dimensions. As such, we must rel on other methods, e.g. level sets (to be revisited shortl), to get an idea of the action of such functions. We shall return to a more detailed analsis of scalar-valued functions in a few lectures. We now turn to vector-valued functions of several variables, written mathematicall as f : R n R m, where n and m are integers that are generall greater than one. Once again, the input of such a function is a point = ( 1, 2,, n ). But now the output is a point = ( 1, 2,, m ) in R m. Such functions can be written as m-tuples of scalar-valued functions, i.e., f() = f( 1, 2,, n ) = (f 1 ( 1, 2,, n ),,f m ( 1, 2,, n )) 4

5 In books, such functions are usuall denoted in boldface, i.e;, f(). Once again, on the blackboard, I shall use arrows, i.e. f( ). Eamples: 1. n = 1, m = 3: You have seen this case in first-ear calculus. The input is a single real number and the output is a point in R 3. This is an eample of the parametric representation of a curve in R 3. It was standard to let t be the parameter and ((t),(t),z(t)) the points on the curve. As an eample, f(t) = ((t),(t),z(t)) = (cos t,sin t,t), t R, defines a heli in R n = 2, m = 3: f(,) = (, 2 + 3, + 6). Here, f 1 (,) =, f 2 (,) = 2 + 3, f 3 (,) = n = 3, m = 2: f(,,z) = ( sin(z),e z ). Here, f 1 (,,z) = sin(z), f 2 (,,z) = e z. Apart from the first eample, does this definition of a general vector-valued function of several variables sill? Useless? Contrived? Perhaps. But consider the following situation. At each point = (,,z) in the earth s atmosphere, we would like to consider a number of phsical properties of the air, for eample: (1) temperature T, (2) pressure P and (3) velocit v, itself a vector quantit with three entries, as functions of. This would involve a function f : R 3 R 5, i.e., m = 3 and n = 5. In most of the applications to Phsics in this course, we shall be using m = n = 2 or 3, i.e., f : R n R n, n = 2,3. Usuall, the input to f will be a point = ( 1,, n ) in phsical space. And the output of f will most often be one of the following: 1. The velocit v() = (v 1 (),,v n ()) of a particle at a point R n, 2. The acceleration a() = (a 1 (),,a n ()) of a particle at a point R n, 3. The force f() = (f 1 (),,f n ()) eerted on a particle at a point R n. 5

6 Eamples: 1. n = 2 with f(,) = (f 1 (,),f 2 (,)) = ( 2 + 5, ). f defines a vector field for all (,) R 2. (Whatever this vector field might mean is not of concern at the moment.) 2. Some eamples relevant to Phsics are presented in the handouts that accompan this set. We shall often refer to these vector-valued functions as vector fields. 6

7 Lecture 2 (Relevant section from Stewart, Calculus, Earl Transcendentals, Sith Edition: 16.1) Visualization of vector fields We now face a problem: Practicall speaking, how do we graph a vector-valued function of several variables, f : R n R m? At each input point ( 1,, n ), we obtain an output point ( 1,, m ). On a piece of paper, we can deal with three dimensions at most, with the help of perspective. Even in a relativel small-dimensional case, e.g., n = m = 4, we re going to be in trouble. In most applications in Phsics, our vector fields will represent velocities or forces. It turns out that the visualization of these fields in terms of arrows is a good method of graphing them. And, indeed, it is often useful to be able to visualize these vector fields in this wa. For eample, a velocit vector field sketched in terms of arrows can give a reasonable idea of how a fluid is travelling in space. In R 2, the vector field f(,) = (f 1 (,),f 2 (,)) is graphicall represented b drawing an arrow representing the vector f 1 (,)i + f 2 (,)j that starts at the point (,). In this wa, we obtain the arrow sketched schematicall below. f(,) f 2 (,)j (,) f 1 (,)i pictorial arrow representation of the vector f(,) = f 1 (,)i + f 2 (,)j Just to repeat: The arrow, in particular its horizontal and vertical components, f 1 (,) and f 2 (,), respectivel, represents the value of f at (,), i.e., f(,). If ou now draw these arrows for a sufficient number of points (,), then ou can get an idea of the flow of the vector field. (Think of how iron filings are used to obtain the picture of the magnetic field surrounding a bar magnet.) Note: It is often advantageous to use a length scaling for the arrows that is different than the one used to label the coordinate aes. Otherwise, the arrows are either too tin or too large, in the latter 7

8 case overlapping each other to ield a rather confusing picture. Eamples: 1. f(,) = (1,0). Here f 1 (,) = 1 and f 2 (,) = 0. At each point (,) we simpl draw the arrow vector i + 0j = i. The result is sketched below The vector field f(, ) = i. Possible phsical interpretation: the velocit field of a thin fluid moving horizontall on the surface of a table at constant velocit in the forward direction. 2. f(,) = (,). Note that this can be written in compact notation as f() =, since = (,). At each point (,) we draw the arrow i + j. For eample, at the point (1,1), we draw the vector i + j. At (2, 1), we draw the vector 2i j. The result is sketched below The vector field f(, ) = i + j. This has the appearance of a fluid that emanates from the origin and flows in all directions, increasing speed as it moves awa from the origin. 8

9 I thank the student who, during our discussion in class, gave a possible phsical realization of this phenomenon: Imagine water coming out of a vertical pipe that is situated at the top of a circularl smmetric hill, perhaps in the shape of a cone. As the water flows down the hill, it increases in speed because of gravitational acceleration. 3. f(,) = (,0) for 0. At each point (,), 0, we draw the arrow i: The vector field f(, ) = i, 0. This has the appearance of a fluid that is moving horizontall, however the fluid particles are moving more quickl as we move awa from the -ais. This is not too far from the real motion of a fluid with friction particles at the wall, i.e., the -ais are stuck to it. 4. f(,) = (,). It s sometimes instructive to pick some eas points and then work around them. For eample: f(0,0) = (0,0), f(1,0) = (0,1), f(0,1) = ( 1,0), f( 1,0) = (0, 1): The vector field f(, ) = i + j. This has the semblance of some kind of rotational field. We ll return to this eample later. One of the main points of this course is to mathematicall characterize the various behaviours of these vector fields. For eample, without looking at the graphical representation of a vector field, can we determine if it is rotational or not? 9

10 Some eamples of important vector fields in Phsics The velocit field of a rotating disc Consider a thin circular disc (e.g., a rotating turntable for es! vinl records, or a rotating CD) that is rotating counterclockwise with angular frequenc ω with respect to a stationar laborator frame, as sketched below. ω. The laborator coordinate sstem defined b the - and -aes is fied on the table. A point P on the disc traces out a circular trajector in time. If r(t) = ((t),(t)) denotes the position vector of P at a certain time, as measured from the center of the disc, then the velocit vector v(t) of P is tangent to this circular trajector and therefore must be perpendicular to r(t) and point in the direction of motion, as sketched below. v(t) P r(t) θ(t) O Since the trajector of P is circular, it is convenient to use a polar coordinate representation, i.e., (t) = r cos θ(t), (t) = r sin θ(t), (1) where r = OP is the (constant) radius of the circular trajector. And what about the angle θ(t)? Well, we are given that the disc is rotating at a constant angular speed of ω. B definition, this means that dθ dt = ω. (2) 10

11 In other words, θ(t) is increasing linearl in time. We can integrate the above equation to obtain θ(t) = ωt + θ 0, (3) where θ 0 = θ(0), i.e., the angle between the vector OP and the positive -ais at time t = 0. (In class, we set θ 0 = 0 for convenience, but let s simpl keep it general here.) Therefore, the coordinates of point P with respect to the stationar coordinate sstem of the table are given b (t) = r cos(ωt + θ 0 ), (t) = r sin(ωt + θ 0 ). (4) We leave it as a small eercise (in fact, it is assigned in Problem Set No. 1) to show that the velocit field of points on the rotating disc is given b v = ωi + ωj = ω [ i + j ]. (5) This is rotational vector field eamined at the end of the previous lecture, multiplied b the scalar ω. Note that v is perpendicular to r: r v = (,) (,) = + = 0. (6) It is convenient to define the angular velocit vector ω = ωk which points in the direction of the ais of rotation. Here, the so-called right-handed convention is being applied: If ou slightl curl the fingers of our right hand and let them point in the direction of motion of P, then our thumb will point in the direction of ω, as shown in the diagram below. z ω = ωˆk r v (side view) Also note that v = ω r, (7) a result that ou ma have seen in first-ear Phsics. We verif it below: i j k ω r = 0 0 ω = ωi + ωj + 0k = v (8) 0 11

12 Note that v = ω r sin θ = ωr, (9) since θ = π/2. We see that for a fied value of ω the speed of the point P increases as we move outward from the center of the disc. This makes sense: The motion of an point P is periodic with period T = 2π/ω. But the distance that a point P must travel in one rotation is 2πr, which increases with the radius r of the trajector. The relation (7) applies to the more general situation of a rigid bod that rotates about an ais in three dimensions. The result is proved on Page 629 of the course tetbook b Adams and Esse. 12

13 Lecture 3 Some eamples of important vector fields in Phsics (cont d) Gravitational force fields It is instructive to focus on a particular eample of a vector field of major importance in phsics: the gravitational field. Suppose that a mass M is located at the origin O in R 3. And suppose that there is another mass m situated at point P with coordinates = (,,z). (You can refer to the relevant figure in the set of illustrations of vector fields.) What do we know about gravitation? 1. First, we know that the force eerted b M on m points in the direction from m to M. In this case, it points in the direction of r, where r is the position vector of m. (For convenience, we have placed M at the origin.) Therefore, the gravitational force eerted b M on m will have the form F Mm = something r (10) where something will be a positive quantit that is not necessaril constant. 2. Secondl, we know that the magnitude of F Mm is proportional to the square of the distance between M and m. In other words, F Mm = K r 2 = K r2, (11) where K is the constant of proportionalit and, for convenience, we use the notation r = r. 3. The magnitude of F Mm is also proportional to each of M and m. We shall first consider the effect of distance on the force, for fied M and m. From Eq. (10), we have F Mm = something r (12) Comparing (12) and (11), it follows that Therefore, from (10), we have that something = K r3. (13) F Mm = K r3r. (14) 13

14 We now use Fact No. 3 to deduce that K = GMm, where G is a proportionalit constant. The final result is well known to ou: F Mm (r) = GMm r 3 r, (15) where G is the so-called gravitational constant. Note that this result can also be written as where ˆr = r denotes the unit vector pointing in the direction of r. r Question: What is the force F mm eerted b mass m on mass M? F Mm (r) = GMm r 2 ˆr, (16) Eq. (15) is a ver compact notation for the vector force field. If we epress it in Cartesian coordinates, and z, the result is rather complicated looking. First of all, acknowledging that r = i + j + zk, we have so that This epression will be useful later. r = r = z 2 (17) GMm F Mm (,,z) = ( z 2 [i + j + zk]. (18) ) 3/2 What does this vector field look like? As we reasoned above, at each point P with coordinates r = (,, z), the force eerted b mass M on mass m will point toward mass M, which is located at the origin O. Therefore all arrows points toward the origin. As we move awa from the origin/the mass M, however, the lengths of the arrows get smaller. Here is a rough sketch: z O The gravitational force field vector F = GMm r 3 r. 14

15 It is often convenient to define the gravitational field f that eists due to the presence of the mass M at the origin: It is the gravitational force per unit mass eerted b M, i.e. f = 1 m F Mm = GM r. (19) r3 The the force eerted on a mass m at point P with position r is given b F = mf. (20) Now consider the case where the mass M is not at the origin of our coordinate sstem but rather at a position r 0 = ( 0, 0,z 0 ). What is the force eerted b M on m in this situation? A little bit of phsical thinking should reveal that the gravitational force field that we obtained earlier which was centered about the origin would be shifted so that it is now centered about the point r 0 = ( 0, 0,z 0 ) where mass M is located. The net result is F Mm (r) = GMm r r 0 3[r r 0] (21) Phsicall, the force eerted b M on m has to be in the direction of the vector that points from m to M, which is the vector r 0 r = [r r 0 ]. And the strength of the vector field must be inversel proportional to the square of the distance between M and m. You can check that this is the case in (21). A sketch of this vector field is given in the handout from Lecture 1. Electrostatic force fields The electrostatic force on a test point charge q due to the presence of a point charge Q at the origin is given b F(r) = KQq r 3 r, (22) In SI Units, the constant K = 1, where ǫ 0 is known as the permittivit of the vacuum. The 4πǫ 0 electrostatic force is then written as F(r) = Qq 4πǫ 0 r3r, (23) We shall be using this notation throughout the course. Note that F is (1) repulsive when Q and q have the same sign and (2) attractive when the have opposite signs, as epected. The electric field due to the presence of charge Q at the origin is the electrostatic force per unit charge, i.e., E(r) = Q 4πǫ 0 r3r, (24) 15

16 so that the force eerted on a charge q at position r is given b F = qe. (25) In the case Q > 0, the arrows representing the electric field will point outward as sketched below. z O The electric field vector E = Q 4πǫr3r, for Q > 0. 16

17 Scalar-valued functions of several variables (Relevant section from Adams and Esse, Calculus, Several Variables, Seventh Edition: 12.1) Let us now return to the topic of scalar-valued functions of several variables, i.e f : R n R. (26) The input to such functions is a set of n real numbers, considered as a point ( 1, 2,, n ) R n, and the output is a scalar, i.e., a single real value. Let us consider some simple situations where such functions of several variables are necessar. 1. Let T(,,z) denote the temperature at a point (,,z) in our classroom. In fact, we know that the temperature changes from hour to hour, da to da, so that it would be better to the time dependence in addition: 2. Let T(,,z,t) denote the temperature at a point (,,z) at time t. 3. Let h(,) denote the height of the earth s surface above sea level at a point (,) that identifies that location uniquel for eample denotes the latitude and denotes the longitude of the point. In general, barring earthquakes or landslides, this function will remain constant over time. 4. Consider an idealized, square black-and-white photograph. At each point (, ) of the photograph, where 0, 1, the photo ehibits a shade of gre that ranges from 0 (black) to 1 (white). We let g(, ) denote the grescale function associated with the photograph. We use the term idealized because the grescale values of real photographs do not eist at points but are rather determined b the concentration of silver atoms over small regions. For digital images, and actuall take on discrete integer values, e.g., 1, 512, corresponding to the piels that make up the images. The grescale values g(, ) also assume discrete values. Most common are 8 bit per piel images, where the g(,) can assume the values {0,1,2,,255}. Note: For colour images, we need three numbers, e.g., the red, green and blue components, at a point (,) to define the colour at that point. We could use three image functions u i (,) or consider the image function to be a vector-valued function: u(,) = (r(,),g(,),b(,)), where r(, ), g(, ) and b(, ) denote, respectivel, the red, green and blue components of the piel at (,). 17

18 Representations of scalar-valued functions of several variables Graphs of scalar-valued functions Recall that the graph of a function f() of a single real variable is a curve in R 2. We plot the values (,) where = f(). The values represent the input of the function and the values represent the correspoinding outputs. What about the graphs of a function f(, ) of two variables? Clearl, we ll need three dimensions two for the (,) input variables and one for the output z = f(,). The graph of f(,) will generall be a surface in R 3 as we sketch schematicall below. z z = f(, ). (a, b, f(a, b)) O. (a, b,0) Eamples: 1. Perhaps one of the simplest functions to consider, the function z = f(,) = C, where C is a constant. This function is obviousl defined for all R. The graph of this function is the plane z = C, which is parallel to the -plane, passing through the point (0,0,C) on the z-ais. 2. The function z = f(,) = 1, which is defined for all (,) R 2. We can rearrange this definition to give + + z = 1, (27) which indicates that the points on the graph of f lie on a plane. (A normal of this plane is (1,1,1).) The portion of the plane that lies in the first quadrant,,z 0 is shown below. 3. The function z = f(,) = 2, which is defined for all (,) R 2. Here, f is obviousl independent of. The graph of f is obtained b simpl taking the parabola z = 2 at = 0 and translating it along the positive and negative -ais to produce the parabolic trough sketched below. 18

19 z 1 z = 1 1 O 1 z z = 2 O 4. The function z = f(,) = 2 + 2, defined for all (,) R 2. Note the following: f(,) 0, with f(0,0) = 0. For = 0, the parabola z = f(,0) = 2 represents the intersection of the graph/surface z = f(,) with the z-plane ( = 0). For = 0, the parabola z = f(0,) = 2 represents the intersection of the graph/surface z = f(,) with the z-plane ( = 0). For all points = C, i.e. points that lie on a circle, the function f(,) has the same value. This implies that the graph of f has circular smmetr, i.e., we can rotate the parabolas about the z-ais to produce the graph of f, a kind of parabolic bowl. Note that the graph of a function of three variables, i.e., w = f(,,z), would be a set in fourdimensional space. Obviousl, we are not able to sketch such sets in their entiret. The method of level sets or contours, to be discussed in the net section, will help in this case. 19

20 z z = O 20