. Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential ------------------- . 5 x 11 5 x 11 1 9 x 2 6 x 8 x 2 x 4 2x 2 2 x 4 1 1 1 f 2 x 9( x 1) 3( x 2) 9( x 2) 2 4 2x 3 f3 x 2 x x 4x 5 ………………………………………………………………………. f1 x 1 60 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ If you succeeded, you know the basic rules for finding partial fractions and you may proceed to the next section. ------------------- 61 If, however, you encountered difficulties, it makes sense to return to the textbook and study the relevant pages 170–174, work through the relevant frames in the study guide and do some examples. You will be happier knowing the basics well when entering subsequent sections. Afterwards ------------------- 1 61 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . Laplace transforms 1 Objective: In the first section the definition of Laplace transforms will be explained and developed. It will prove useful, to copy all definitions and results into your notebook when going through this and subsequent sections. Then they will be readily accessible to you and you will not have to take recourse to the textbook for each and every single computation. READ 11.1 Introduction 11.2 The Laplace transform definition Textbook pages 321–322 When done . ------------------- 11.3 Solution of linear differential equations with constant coefficients 2 61 After having successfully gone through all preliminary steps, which might have been somewhat depletive in places, you now encounter applications and you will reap the benefits. Follow all the examples arduously and if necessary, consult the table on page 275. You will learn to transform a set of linear differential equations into a set of algebraic equations and after solving these to use the inverse transform, thus obtaining a solution of the original DEs. . READ 11.4 Solution of linear differential equations with constant coefficients Textbook pages 328–329 When done ------------------- 2 62 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . In the section you studied some new concepts have been introduced and defined. Write down at least three of them: 2 1. ………………………………………………….. 2. ……………………………………….. 3…………………………………………. ------------------- 3 . The tendency to overlook things we do not understand is natural and even necessary to survive. Nobody can understand everything. But if we even do not notice that we do not understand something, this may be dangerous. And may have consequences. Try to develop competence in noticing things you do not understand and develop a tendency to use indexes and encyclopedia. At least a few times per week. ------------------- 3 62 63 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G Flow density j 3 G Surface element vector A G G Flow of a vector field F through a surface A ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Try to write down on a separate sheet the definitions and meanings first using your memory but in case of difficulties by consulting your notes and if you do not succeed the textbook. ------------------- 4 . 17.5 Divergence of a vector field and Gauss’s theorem 63 The introduction of the concept of divergence is closely related to the last section on surface integrals. Study in the textbook 17.5 Divergence of a vector field 17.6 Gauss’s theorem Pages 475–479 Having concluded go to ------------------- 4 64 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G The flow density j is the quantity which passes through a unit area per time unit. The area is assumed to be perpendicular to the flow. 4 G The surface element vector is a vector A whose direction is perpendicular to the surface and whose magnitude is equal to the area A. Flow of a vector field through a surface element is given by the G G dot product of the two vectors F A . ------------------- 5 . G F represents a vector field. Complete the definition 64 G div F = ……………………… G div F is a …………………. ------------------- 5 65 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . We want the flow of a vector field through an even square surface whose magnitude is A. The surface belongs to the y-z plane. The flow hits the surface at an angle E G The flow vector is constant and given by j Ed S 5 2 ( j x , j y ,0 ) We decompose the task: G 1. We determine A G G 2. We calculate the flow j A The flow is: ………………… Further explanations wanted Solution found ------------------- 6 ------------------- 11 . G GFx GFy GFz div F Gx Gy Gz G div F is a scalar field. 65 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Calculate the divergence for the given vector field: G F ( x , y b, z 2 ) G div F … Determine the location of sources and sinks. ------------------- 6 66 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G We first determine the surface element vector A . Since the surface element belongs G to the y-z plane its direction is parallel to the x-axis. Thus, A also has the direction of the x-axis. 6 G G A (1,0,0) or A(1,0,0) Since the flow vector is to hit the surface element at an angle less G S we obtain A ……………… than 2 G Hint: remember the definition of the given j . ------------------- 7 . G div F (1 1 2 z ) (2 2 z ) 2(1 z ) 66 For the plane z 1 there are no sources or sinks. The space above this plane, z ! 1 , consists of sinks. The space beneath this plane, z d 1 , consists of sources ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ All correct, go Another exercise wanted 7 ------------------- 69 ------------------- 67 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G A A(1,0,0) 7 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ No more problem More explication wanted ------------------- 9 ------------------- 8 . G We have no sink or sources if div F 0 . With the preceding example this was the case for G divF 21 z 0 . The equation 1 z 0 or z 1 represents a plane G which is parallel to the x-y-plane. For it div F is negative. This means the space above the plane z 1 consists of sinks. G Beneath the plane z 1 divF is positive and thus the space consists of sources. 67 Calculate the divergence for the following vector field: G F xG 2 div F 1, y, z 5 : ………………… Find the spaces where we have sources or sinks and where the field is free of sources and sinks. ------------------- 8 68 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . The surface element belongs to the y-z plane. Thus, its surface element vector is directed in the x-direction. But there are two possibilities. It may point in the positive or in the negative x-direction. We have to decide which of these directions is the right one. In our exercise is given the condition that the angle between both given vectors is to be less than 90° or 8 S . 2 G G In our exercise the x-component of j is negative. Thus, A has to point in the negative direction of the x-axis. From this G we obtain: A = (–1,0,0,) ------------------- 9 . G divF 2 x 1 1 2 x 1 For the plane x 1 the field is free of sinks and sources. The space to the left of this plane, defined by x 1 , consists of sinks. The space to the right of this plane, defined by x ! 1 , consists of sources. 68 Go on to ------------------- 9 69 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G G We determined the surface element vector A to be A A(1,0,0) 9 G Now we can determine the flow I of j through the surface: G j ( jx , j y , 0 ) G G I = j A ………………………… Solution found Further help wanted ------------------- 11 ------------------- 10 . Calculate the divergence for the vector field G F x , G divF 3 y 3 , 3z 69 ……………….. Where do you find sinks and sources? ……………………………………………………………………………………………………….... ……………………………………………………………………………………………………….... ……………………………………………………………………………………………………….... ------------------- 10 70 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G We remember the dot product of two vectors a is defined by G G a b a x bx a y b y a z bz G (a x , a y , , a z ) and b (bx , b y , bz ) 10 Now you should be able to calculate using the results obtained before: G G j A = …………………………………………………….. ------------------- 11 . G div F 3x 2 3 y 2 3 3 x 2 y 2 1 Sources and sinks vanish for x 2 y 2 1 . This is a circle with radius 1 which does not depend on the value of z. 70 Thus it is a cylinder. G The space inside the cylinder consists of sinks since div F is negative. G The space outside the cylinder consists of sources since div F is positive. ------------------- 11 71 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . I jx A 11 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Give the surface element vector for the four given surfaces the magnitude of which is always A ------------------- 12 . In the textbook the Nabla operator has been introduced. It is a new notation which seems to be merely formal. But it will prove quite useful because it is a short-hand notation that simplifies notations once you have familiarity with it. G §G G G · ¨¨ , ¸ , G G G x y z ¸¹ © G …………………. ------------------- 12 71 72 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . In each case we have two answers. The difference is the sign. This time a vector field that determines the direction is not given. G 1. A A(0,0,1) . G 2. A A 2 A(0,0,1) G (1,0,1) ) or A G 3. A A G 4. A A 3 2 G or A (1,1,1) or (1,0,1) or 12 A 2 G A G A (1,0,1) A ( 1,1,1) 3 A 2 (1,0,1) All correct Errors or explanation wanted ------------------- 16 ------------------- 13 . G §G G G · ¨¨ Gx , Gy , Gz ¸¸ © ¹ 72 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Using the Nabla operator we obtain a short-hand notation to represent the gradient of a scalar field f x, y, z and G the divergence of a vector field F x, y, z . We start with the calculation of the gradient: G grad f x, y, z f x, y, z =…………………………….. ------------------- 13 73 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G The surface element vector A is perpendicular to the surface. Thus, we look at first for a vector perpendicular to the surface, the magnitude of which does not matter for the time being. We give the first two solutions then obtain the next two solutions. a has not been defined yet. a will be defined later. 1. G A a(0,0,1) 2 G .A a(1,0,0) 3. G A a(..............) 4, G A a(.................) 13 ------------------- 14 . G § Gf G Gf G Gf G · gradf x, y, z f x, y, z ¨¨ ex e y ez ¸¸ Gy Gz © Gx ¹ 73 G is a vector. In this case we compute the product of a vector with a scalar since the function f x, y, z is a scalar. The result is a vector since the product of a vector with a scalar is a vector. Now we calculate the divergence of a vector field G div F G G F x, y , z GFx G GFy G GFz G ez ex ey Gx Gy Gz G Again is a vector. But this time we compute G the dot product of the vector Nabla with the vector F . The result is a scalar. We remember well from vector algebra the dot product of two vectors results in a scalar. G Calculate M x, y, z for M ( x 2 y 2 z 2 ) : G grad M M …………………………….. G G G Calculate F for F x 2 , y 2 , z 2 : G G G div F F =………………………………… ------------------- 14 74 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G 1. A a(0,0,1) G 2. A a (1,0,1) G 3. A a(1,1,1) G 4. A a (1,0,1) 14 G Now we will determine a. A must have the magnitude A. To obtain this we have to choose a. For the first surface it is evident that: G A A(0,0,1) In this case a = A For the second exercise we have G A A2 A (1,0,1) We may prove it: A 2 (1 1) 2 2 G Now you may try to calculate a for the third exercise A .......(1,1,1) ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ------------------- 15 . grad M G M 2 x,2 y,2 z 2 xeGx 2 yeGy 2 zeGz 74 (The vector grad M may be written down in short-hand or extensively) G div F G G F 2 x 2 y 2 z ------------------- 15 75 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G . A A 3 (1,1,1) Prove: G ( A) 2 G We give the systematic solution. Given: A A2 (1 1 1) 3 15 A2 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ a(a x , a y , a z ). G G If A is to be of magnitude A this results in ( A) 2 A2 A 2 this results in a 2 (a x2 a y2 a z2 ) Thus, finally we obtain a A a a y2 a z2 2 x In our case no direction of the vector field is given. Thus, the sign of the surface element vector may be inverted. ------------------- 16 . In the textbook page 470 the electrical field of a sphere is discussed. Given a sphere with homogeneous charge distribution U , the total charge Q , and radius R. Then outside the surface of the sphere the electrical field is given by G E ( x, y , z ) Q 4SH 0 x, y , z x y2 z2 2 75 3 G Now calculate the divergence E outside the surface of the sphere. G div E =………………………………………………… ------------------- 16 76 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . 17.2 Surface integral Study in the textbook 16 17.2 Surface integral Textbook pages 464–466 This done go to ------------------- 17 . G div E 0 76 In case of difficulties consult the textbook again. ……………………………………………………………………………….. Inside the sphere the electrical field is given by G E ( x, y , z ) Q ( x, y , z ) 4SH 0 R 3 Calculate the divergence inside the sphere: : G div E …………………… ------------------- 17 77 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . The following integral is named ………. I G 17 G ³ F dA The circle in the integral I G G ³ F dA significates the integral has to be taken over a ……………………….surface. ------------------- 18 . G div E 3Q 4SH 0 R 3 U H0 77 Q 3 3 SR V and U. V 4 ………………………………………………………………………………………… Write down from memory the Gauss theorem. If possible do not consult the textbook. You should memorize this theorem. In case of difficulties remember ……………………………………………….. ……………………………………………….. ……………………………………………….. ------------------- 18 78 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . Surface integral 18 Closed surface ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Give at least three examples for a closed surface …. 1.: ……………………………………………………….. 2: ……………………………………………… 3:………………………………………………..………………………………………………...…… …………………………………….. ------------------- 19 . ³ div V G F dV G ³ F dA 78 AV ………………………………………………………………………………………. In the following we will use Gauss’s theorem to calculate the given electrical field inside and outside of the surface of the sphere. First calculate the electrical field outside the surface of the sphere: G E ………………………………….................. Solution found Help and explanations wanted 19 ------------------- 82 ------------------- 79 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . Control your examples using the definitions given in the textbook 19 This control is necessary if there are no right answers given. In your further study and in practice this is the normal case. ------------------- 20 . In the textbook page 470 the electric field of a sphere is given with radius R and charge density U . Outside of the sphere the electrical field is: G E ( x, y , z ) Q 4SH 0 x, y , z x y2 z2 2 79 3 We can obtain this result using Gauss’s theorem. The center of the sphere is to coincide with the origin of the coordinate system. First we calculate the total charge Q of the sphere. Given radius R and charge density inside the sphere U : Q ………………………………… ------------------- 20 80 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . The direction of the surface element vectors for closed surfaces is defined unambiguous ambiguous 20 ------------------- 21 ------------------- 22 . Q ³U 4S UR 3 3 dv VKugel 80 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Secondly we calculate the electrical field outside the sphere using Gauss’s theorem. G ³ divF dV V G ³ F dA A (V ) From electrodynamics we know that the flow of the electrical field through a closed surface is proportional to the enclosed total charge Q. G Q ³ E dA Surfase H0 Calculate the flow through the surfaces outside the sphere with the given radius Routside : G Q ³ E dA ........................... H0 ------------------- 21 81 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . You are right. With closed surfaces the sign is defined unambiguously. 21 With closed surfaces the surface element vector always points to ………… ………………………………………………………….…………………… ------------------- 23 . G ³ E dA 2 E 4SRaußen Q 81 H0 From this result we obtain: Q E 2 H 0 4SRaußen The field has spherical symmetry. The field vector points to the outside and its direction is given by the unit vector: G r ( x, y , z ) 2 r x y2 z2 Thus we obtain: G E r R x2 y2 z2 G Q r =…………………………...... H 0 4SR 2 r ------------------- 22 82 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . Unfortunately you are wrong. 22 Look again into the textbook and read again the definition or better the convention of the direction of surface elements. The surface element vectors are a) perpendicular to the surface b) they point with closed surfaces always to………………………….. ------------------- 23 . G E ( x, y , z ) Q 4SH 0 x, y , z x y2 z2 2 82 3 Now we are to calculate the electrical field inside the sphere. Let us regard a surface of a sphere inside with radius Rinside . We are to calculate the enclosed charge. Use Gauss’s theorem again and use the total charge Q of the original sphere. G Einnen =…………………………………………. Further help and detailed calculation Solution happily found 23 ------------------- 83 ------------------- 86 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . The surface element vector for closed surfaces always point to the outer space. 23 This is a convention, but it is worth memorizing. ------------------- 24 . The surface of an inner sphere encloses a part of the total charge given by: 83 Qinnen =……………… ------------------- 24 84 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . 17.3 Special cases of surface integrals Study in the textbook 24 17.3.1 Flow of a homogeneous vector field through a cuboid Pages 466–468 ------------------- 25 . Qinnen 4S 3 Rinnen U 3 ³ UdV 84 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ G Now we apply Gauss’s theorem. Given: div E G ³ E dA surface ³ div U . H0 G E dV volume We calculate both integrals: a) G ³ E dA =……………… Surface b) ³ div G EdV =………………. Volume Inserting into Gauss’s theorem we obtain:……………=…………………. ------------------- 25 85 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . Decide whether the following vector fields are homogeneous or not 25 Vector field is homogeneous G 1. F G 2. F yes no ( x, y , z ) x2 y2 z2 (1,0, x) 3. G F ( y , x, z ) 4. G F (6,3,5) 5. G F (2,0,0) ------------------- 26 . G a) ³ E dA b) ³ divEdV 2 E 4SRinnen G 85 U 4S 3 Rinnen H0 3 Inserting into Gauss’s theorem we obtain G U 4S 3 2 E 4SRinnen Rinnen H0 3 G U Rinnen E H0 3 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ G K To obtain E we have to regard the direction of E : G E E G r =……………………… r G Remember: r Rinside ------------------- 26 86 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . Homogeneous vector fields are 4. 5. 26 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ All correct Not all answers correct ------------------- 29 ------------------- 27 . G Einnen U x, y , z H0 3 86 Finally, we substitute U by the total charge Q of the original sphere using the known equation Q U G Einnen 4S 3 R 3 ………………….. ------------------- 27 87 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . In this case we suggest you read again the section you just studied. Theses questions did not imply calculations. Thus, you have problems with the understanding of the definition of a homogeneous vector field. Read again to try and find the correct answers. Vector field 27 homogeneous not homogeneous G 1. F (1,2,3) x2 y2 z2 G 2. F 1 1 1 ( , ; ) x y z 3. G F (1;0;0) 4. G F (x;0;0) 5. G F (21;1; ) ------------------- 28 . G Einnen Q ( x, y , z ) 4SH 0 R 3 87 In case of remaining difficulties repeat this section of the study guide and consult the textbook again. ------------------- Go on to 28 88 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G 1. F (1,2,3) x y2 z2 G 2. F 1 1 1 ( , ; ) x y z not homogeneous 2 28 not homogeneous 3. G F (1;0;0) homogeneous 4. G F (x;0;0) not homogeneous 5. G F (21;1; ) homogeneous ------------------- 29 . 17.6 Curl of a vector field and Stoke’s theorem 88 The concept of curl will be introduced. After completing section 17.7 it is suggested you have a break and a cup of coffee or tea. But do not forget to take notes of new definitions on a separate sheet. Study 17.7 Curl of a vector field. 17.8 Stoke’s theorem Textbook pages 480–485 Having studied ------------------- 29 89 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G Determine the flow of the field F ( x, y, z ) (1,4,3) through a cube whose faces are parallel to the axis of the coordinate system. 29 G The flow I of the vector field F is I =……………………….. Solution found One more hint wanted ------------------- 31 ------------------- 30 . G If a vector field F1 is curl free we have: …………=……….. G If a vector field F2 has curl we have gilt……...…..=……….. 89 ------------------- 30 90 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G Hint: F (1,4,3) is a homogeneous vector field. Thus, you can apply rule 17.7 given in the textbook. 30 G Give the flow of the vector field F (1;4;3) through a cube whose faces are not parallel to the axes of the coordinate system. I =……………………………….. ------------------- 31 . Field is curl free Field has curl G ³ FG ds 0 ³ F ds z 0 90 1 2 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ It is not that easy, but it is worthwhile to memorize the definition of curl or to reconstruct it using the G G G formula rot F = u F . G rot F = ……………… ------------------- 31 91 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . I=0 31 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ G For which of the given surfaces below does the flow of the homogeneous field F not vanish? The surfaces are closed. cuboid sphere ellipsoid (0,2,0) dumbbell-shaped ------------------- 32 . G rot F § GFz GFy · ¸ ¨ Gz ¸ ¨ Gy ¨ GF GF ¸ ¨ x z¸ Gx ¸ ¨ Gz ¨ GFy GF ¸ x ¸¸ ¨¨ G Gy ¹ x © 91 Write down as a determinant: G rot F ………………………… ------------------- 32 92 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . None of these. All are closed. A homogeneous vector field vanishes for all closed surfaces. 32 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ G Given the vector field F ( x, y, z ) . Does its flow through a sphere vanish if the sphere’s x y2 z2 center coincides with the origin of the coordinate system? 2 Answer found Help or more explanation wanted ------------------- 35 ------------------- 33 . G G G ex e y ez G rotF 92 G G G Gx Gy Gz Fx Fx Fz ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Expand the determinant and give: G rotF ………………………… ------------------- 33 93 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . The surface of the sphere is closed. G Given the vector field F ( x, y, z ) x y2 z2 2 33 . This vector field is not homogeneous. Sketch some vectors of the field along the axes of the coordinate system ------------------- 34 . G rot F § GFz GFy · ¸ ¨ Gz ¸ ¨ Gy ¨ GF GF ¸ ¨ x z¸ Gx ¸ ¨ Gz ¨ GFy GF ¸ x ¸¸ ¨¨ G Gy ¹ x © 93 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Regard a flow of water. The field of velocities may be given by: G v (1, ln z , 0) Calculate: G div v ……………………… G rot v ……………………… ------------------- 34 94 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . If vectors in all directions are sketched, the field 34 is represented as shown to the left. Imagine a sphere with its center at the origin of the coordinate system. The field passes through the surface always from the inner side to the outer side. Does the flow through the surface vanish? ------------------- 35 . G divv 0 G 1 rotv ( ,0,0) z 94 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Calculate the line integral for the rectangle with sides a and b for a complete circulation. G G Given F to be F ( x, y, z ) (5,0, z 2 ) G ³ F ds ……………………… ------------------- 35 95 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G No. The flow of F ( x, y, z ) does not vanish. x y2 z2 The field passes at all points of the surface from the inner space to the outer space. 2 35 G A vector field F has radial symmetry if 1………………………………………………… 2…………………………………………. In case of doubt look again at textbook chapter 13 (section “coordinate systems”). ------------------- 36 . The field of velocities is free of curl. Thus: G ³ F ds 95 0 ------------------- 36 96 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . A vector field has radial symmetry if 1. All vectors point in a radial direction and; 2. Its amount depends only on the radius r. 36 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Decide whether the following field possesses radial symmetry: G 2. Its amount depends only on r. F ( x, y, z ) ( x 2 y 2 z 2 )3 = G r r3 Solution found Further explanation or help wanted ------------------- 39 ------------------- 37 . 17.7 Potential of a vector field Study 96 17.9 Potential of a vector field Textbook pages 485–487 ------------------- 37 97 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G Firstly sketch some of the vectors of the field F G r r3 37 ------------------- 38 . In the textbook we treated the example of the gravitational field of a sphere, the mass of 97 which is M. The case most familiar to us is the earth. At the surface of the earth, however, we simplify the situation by calculating with a homogeneous field of gravitation. In this simplification the x-y plane is parallel to the earth’s surface. The z-axis points upwards and the origin of the coordinate system coincides with the earth’s surface. This approximation holds for a lot of calculations. In this case the gravitational force acting on a mass m is given: G F ………………………………. ------------------- 38 98 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . Your sketch may be similar to the sketch to the left. The vectors point outwards. Because of r 3 in the denominator the amount of the vectors decreases with the distance from the origin. The vectors at other positions point in a radial direction as well. Since ( x, y, z ) = r we obtain: F ( x, y, z ) ( x y 2 z 2 )3 2 = r r3 38 1 r2 F only depends on r. Thus, this vector field has …………………………………… ------------------- 39 . G F 0, 0, m g 98 G In the case of a gravitational field the force F is the product of the mass m of the body in question G and the gravitational field vector Fg of the gravitational field. This field vector is in this case G G Fg g ez 0, 0, g In the following we discuss fields and their field vectors. This holds as well for electrical fields and field vectors. In the case of a static electrical field the force acting on a charge Q is the product of Q G with the electrical field vector E : G F G QE G The field is completely represented by the electrical field vector E . G Check gravitational field Fg . Is it free of curl? G rot Fg …………………. G Fg is…………………….. ------------------- 39 99 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G The vector field F has radial symmetry because G ( x, y, z ) F points in a radial direction and its amount depends only on r. 2 ( x y 2 z 2 )3 39 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ------------------- 40 . rot 0,0, g 0 99 G Fg is curl-free, it is a conservative field. G Calculate the potential of F G m Fg . Remember: The following convention holds in physics. For a given force field G Fg the potential M is the work done against the force field M x, y, z ………………….. ------------------- 40 100 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . 17.3.1 Flow of a spherically symmetrical field through a sphere 40 In this section we show the calculation of surface integrals through a surface of a sphere. In physics this is an important special case. Read in the textbook 17.3.2 Flow of a spherically symmetrical field through a sphere Textbook pages 468–469 ------------------- 41 . M x, y , z m g z C 100 The potential is thus determined except the integration constant C: Determine the potential given above for three different situations: 1) M1 0 for the ground floor with z0 0 M1 ………………. 2) M 2 M2 3) M 3 M3 0 for the underground of a high building with z0 10 ……………….. 0 for the roof of a high building with z0 90 ……………….. ------------------- 41 101 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G eGr G G Calculate the surface integral ³ F dA of the field F r2 G G r for a surface of a sphere with radius R. with er r G Wanted is the flow of F through the surface of the sphere. G 41 G ³ F dA = ……………………………………….. ------------------- 42 . M1 0,0, mgz M2 0,0, mg z 10 M3 0,0, mg z 90 101 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ G Give the gravitational field Fg for these three cases: G Fg1 ………………… G Fg 2 ………………… G Fg 3 ………………… ------------------- 42 102 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G G dA ³ F dA = ³ r 4SR 2 2 1 R2 4S 42 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ G Given a field of a force with radial symmetry: F (r ) a G er r3 with G G r er r G Compute the flow of the field of the force F (r ) through the surface of a sphere which has a distance R from the origin of the field. The center of the field is defined by r = 0. Solution found Explanation or help wanted ------------------- 44 ------------------- 43 . M1 M 2 M3 0,0, mgz 102 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ In the textbook you studied the gravitational field of a mass M. This mass is assumed to be distributed homogeneously in the inner space of a sphere with radius R. Outside of the sphere the gravitational field is: G x, y , z Fg x, y, z J M 3 x2 y2 z2 G Simplify using r x, y , z G Fg x, y, z ………………….. ------------------- 43 103 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G F (r ) The field of the force is given by Wanted: the flow given by G a G er r3 with G G r er r 43 G ³ F dA G First hint: The field has radial symmetry of the following form: F f (r ) er G G Second hint: ³ F dA is calculated for the general case in the section of the textbook you just studied. Repeat your study and try again: G G ³ F dA = ………………………………………………………………….. 44 ------------------- . G Fg JM G r r3 JM G 1 r r2 r 103 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ G Now we calculate the potential M ³ F dr for a path of integration in the direction of a radius. G r Then we have: dr dr . r r 1 r M JM ³ dr =……………………………. r r r0 2 ------------------- 44 104 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . ³ G G a G er dA r3 ³ f (r )e dA r 4SR 2 f ( R) 4SR 2 a R3 4S a R 44 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Have a short break ------------------- 45 . r M r JM ³ r0 ª1 r ª1 º JM « » ¬ r ¼r dr r2 104 0 1º M r JM « » ¬ r0 r ¼ We can scale the potential of the gravitational field of the mass M in a way that it vanishes for r o f , but the potential may be scaled as well to vanish for the surface for which holds r rsurface In the textbook we explained the first case. In the study guide we calculated the second case. Calculate the potential for the first case in order to vanish for infinity. M1 ………………….. ------------------- 45 105 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . 17.3.2 Application: The electrical field of a point charge. 45 Here we apply the new knowledge Study in the textbook 17.3.3 Application: The electrical field of a point charge Textbook page 470 ------------------- 46 . M1 JM 1 r 105 All correct Further explanation wanted 46 ------------------- 108 ------------------- 106 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . The calculations in the foregoing section are obvious. In this case using physical units facilitates the calculations. 46 Thus, we can proceed to the next section without further exercises. Go to ------------------- 47 . The potential of the gravitational field was represented by: r dr r2 r0 M r JM ³ 106 ª 1 1º » ¬ r0 r ¼ JM « The condition was M r f 0 1 0 or r 0 f . r0 A difficulty in understanding this may arise because r0 is the lower limit of integration and the lower limit can not be f while the upper limit of integration is finite. But this problem is solved; we invert the direction of integration. We integrate from r to r0 . In this case the bracket must be zero. This is obtained by letting r Thus we obtain dr r2 r0 M r JM ³ r0 JM ³ r dr r2 ………………… ------------------- 47 107 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . 17.4 General case of computing surface integrals 47 In the following section we compute the surface integral for a general case. This section is slightly formal and more difficult. It is worth studying this section if you are not in a hurry and did not have difficulty with the preceding subject matter. Decide according to your preferences. I prefer to skip section 17.4 for the time being and want to proceed ------------------- 55 I prefer to study section 17.4. Then study in the textbook 17.4 General case of computing surface integrals Textbook pages 470–474 ------------------- 48 . ª 1 1º » ¬ r0 r ¼ M r JM « 107 Now we let r0 grow to infinity to obtain the result of the foregoing frame. M1 JM 1 r Further problems may arise if we change signs. Do not underestimate the problems related to signs. It is always advisable to calculate meticulously. ------------------- 48 108 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . A rectangular plane A is defined by the points P0 (0,0,0) , P1 (4,0,0) , and P3 (4,0,0) . Thus it lies in the x-z plane. G An inhomogeneous vector field is given by F Compute 48 (0,2 x,0). G G ³ FdA = …….. Solution found Explanation or hints wanted ------------------- 54 ------------------- 49 . Now we will treat the second case and let the potential of the gravitational field vanish for the surface of the earth. ª 1 1º We calculated before: M r JM « » ¬ r0 r ¼ The height z above the surface is given by r Thus: M z 108 r0 z M z =………………………. ------------------- 49 109 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . In this case we have an inhomogeneous field. The field is quite simple, because it has G only one component in the y-direction. Remember the field was given by F (0,2x,0) G G To obtain I = ³ F dA 49 G G we determine F and dA A F ……….. G dA ……. I = ………….. Solutions found Further help and explanations ------------------- 52 ------------------- 50 . ª1 1 º » ¬ r0 r0 z ¼ M r JM « 109 For places near the surface we state z r0 . Applying this approximation we write: 1 1 1 | ........................... r0 z § z · r0 r0 ¨¨1 ¸¸ © r0 ¹ We insert this into the equation above and obtain: M z JM >............................@ ------------------- 50 110 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G G We have to determine F and dA . G Difficulties may arise in determining dA . Plane A lies in the x-z plane, as was pointed out in frame 48. Thus, the surface element vector points in the y-direction. 50 The amount of a differential surface element for this plane is given by dA = dx dz. The surface element vector in the y-direction with amount dxdz is given by G dA G G Remember: F is given to be F (..............) (.....................) 51 ------------------- . Approximation: M z JM 1 § z· r0 ¨¨1 ¸¸ r 0 ¹ © | 1ª zº «1 » r0 ¬ r0 ¼ 110 z 2 r0 For the case of the earth with mass M and radius rE this corresponds to the potential we used in previous frames. There we used the expression M z 0,0 g z gz Both expressions are identical if we define g appropriately. Calculate and define: g …………………… ------------------- 51 111 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G dA (0, dx dz ,0) G F (0,2x,0) 51 G G We want to determine the flow I = ³ F dA ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ a A Now we can calculate the dot product in the integral using the results obtained: G G I = ³ F dA = ³ (0,2 x,0) (0, dxdz,0) A A ³ .. …………….…. A ------------------- 52 . g JM r0 111 2 ------------------- 52 112 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G G ³ FdA ³ 2 xdxdz A 52 A This integral must be calculated for plane A. Written correctly this is a double integral. Write this integral as a double integral and insert the limits given by our plane A: A³ 2 x dx dz ........... ........ ³ ³ 2 x dx dz x ...... z ..... ------------------- 53 . At the end of this chapter we recapitulate: 112 G Definition of divergence of a vector field F : G div F ……………………=……… ------------------- 53 113 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . 4 I 3 53 ³ 2 xdx dz ³ ³ 2 xdx dz x 0z 0 A You can calculate the double integral. It was explained in chapter 13 “Multiple integrals.” In case of difficulties you should repeat that chapter, at least section 13.2 “multiple integrals with constant limits.” I ³ 2 xdx dz A 4 3 ³ ³ 2 xdx dz =…………………………………… x 0z 0 ------------------- 54 . G div F wFx wFy wFz wx wy wz G G F 113 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ G Calculate the divergence of the given vector field F : G F § x3 y3 z 2 · ¨¨ , , ¸¸ 2¹ © 3 3 G div F =…………………. Distribution of sources and sinks: ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ------------------- 54 114 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . I G G ³ F dA 48 54 A Congratulations. You have mastered some difficult reasoning. ------------------- 55 . G div F x2 y2 z 114 No sinks or sources for: G div F 0 : Thus, we obtain for this condition 0 x 2 y 2 z or z x 2 y 2 . This represents a paraboloid of revolution around the z-axis. The inner space consists of sinks. The outer space consists of sources. ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Give Gauss’s theorem: ……………………………………………..=……………………………… ------------------- 55 115 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . Before we finish this section we should recapitulate. G In a vector field F we have a quadratic plane with an area of 2. The position is sketched below. G Give the surface element vector A . 55 G A =………. ------------------- 56 . ³ div G FdV V Volume integral G G ³ F dA 115 Surface integral. l ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ G Give the definition of the curl of a vector field F : G rot F G G u F ……………………………. ------------------- 56 116 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . G A (0, 2 , 2) G You may have factorized the root: A 2 (0,1,1) 56 Calculate for this plane A the flow of three vector fields: F1 (0,6,0) I1 ................ F2 (0,2,1) I2 ............. F3 (6,0,0) I3 .................... ------------------- 57 . G rot F § GFz GFy · ¸ ¨ Gz ¸ ¨ Gy ¨ GF GF ¸ ¨ x z¸ Gx ¸ ¨ Gz ¨ GFy GF ¸ x ¸¸ ¨¨ G x Gy ¹ © 116 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Given a flow of water G V z 2 G rot V , 0, 0 ……………….. ------------------- 57 117 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . I1 6 2 I2 3 2 I3 0 57 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ G Given a vector field with radial symmetry j . The origin coincides with the origin of the coordinate system. G The amount of j is constant. G Calculate the flow I of j through a sphere with radius R. The center of the sphere lies in the origin of the coordinate system. I =……………. ------------------- 58 . rot z 2 ,0,0 0,2 z,0 117 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Give Stoke’s theorem: ………………………..=………………………. ------------------- 58 118 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . I G G ³ j dA G 4SR 2 j 58 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Calculate the flow ) of a vector field G F (1,0,0) Through the sketched cuboid with the edges a = 6, b = 1, c = 3 ) =………………………………… ------------------- 59 . G ³ divFdV V Volume integral G G G ³ F dA ³ rotF dA A Surface integral Surface integral G ³ F ds C A Line integral The theorems of Gauss and Stokes are worth understanding and memorizing in order to avoid difficulties when applying them in physics. You have reached the end of chapter 17. You have now made considerable progress. 59 118 Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential . ) 0 59 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ G Given F (0.5, 0, 0.5) Calculate the flow for three planes: G A1 = (1, 1, 0) I 1 =………… A2 = (1, 0, 1) I 2 =………………. G A3 = (1, 1, 1,) I 3 =……………… ------------------- 60 Please continue on page 1 (bottom half) 60