Unit 6 Line and Surface Integrals In this unit, we consider line integrals and surface integrals and the relationships between them. We also discuss the three theorems Green s theorem, the divergence theorem and Stokes s theorem that summarize the connections between line, surface, double and triple integrals. Note: Unit 6 is based on hapter 17 of the textbook, Salas and Hille s alculus: Several Variables, 7th ed., revised by Garret J. Etgen (New York: Wiley, 1995). All assigned readings and exercises are from that textbook, unless otherwise indicated. Objectives Detailed objectives are given in each of the sections listed below. 1 Line Integrals 2 The Fundamental Theorem for Line Integrals 3 Work, Energy and the onservation of Mechanical Energy 4 Alternative Notation; Line Integers and Arc Length 5 Green s Theorem 6 Parametrized Surfaces and Surface Area 7 Surface Integrals 8 The Vector Differential Operator 9 The Divergence Theorem 10 Stokes s Theorem Mathematics 365 / Study Guide 47
Objective 1 a. integrate a two-dimensional vector field over a path. b. integrate a three-dimensional vector field over a path. c. calculate the work done by a two-dimensional force operating over a path. d. calculate the work done by a three-dimensional force operating over a path. e. demonstrate various conclusions relating to line integrals. Read Section 17.1, pages 1115-1122. omplete problems 1-10 and odd-numbered problems 11-33 on pages 1123-1124. scalar field vector field total work done by a force over a smooth curve: W = [ Fr ( ( u )) i r ( u )] du line integral of vector field h over curve : b hr ( ) i dr= [ hr ( ( u)) i r ( u)] du a effect of sense-preserving change of parameter on line integral piecewise smooth curve Before you proceed to Objective 2, make certain that you can meet each of the sub-objectives listed under Objective 1. b a 48 alculus Several Variables
Objective 2 a. use the fundamental theorem for line integrals to solve line integral problems. b. demonstrate various conclusions relating to the fundamental theorem for line integrals. Read Section 17.2, pages 1124-1127. omplete odd-numbered problems 1-21 and problems 22-25 on pages 1128-1129. gradient field fundamental theorem for line integrals relationship between fundamental theorem for line integrals and fundamental theorem of calculus fundamental theorem for line integrals: closed curves Before you proceed to Objective 3, make certain that you can meet each of the sub-objectives listed under Objective 2. Objective 3 a. use the work-energy formula and the law of conservation of mechanical energy to solve problems relating to energy. b. demonstrate various conclusions relating to force, energy and work. Read Section 17.3, pages 1129-1131. Mathematics 365 / Study Guide 49
omplete problems 1-9 on page 1132. 1 1 work-energy formula W = m[()] v β m[( v α)] 2 2 conservative field 2 2 potential energy functions for a conservative force field total mechanical energy (E) law of conservation of mechanical energy 1 2 2 mv + U = E difference in potential energy escape velocity equipotential surfaces of a conservative force field Before you proceed to Objective 4, make certain that you can meet each of the sub-objectives listed under Objective 3. Objective 4 a. use the arc length formula to evaluate a line integral along a given path, to determine whether a given vector field is a gradient field, and to find length, mass, centre of mass and moment of inertia. b. demonstrate various conclusions relating to the arc length formula. Read Section 17.4, pages 1132-1135. omplete problems 1-29 and odd-numbered problems 31-39 on pages 1135-1136. 50 alculus Several Variables
alternative notation for line integrals: Pxyzdx (,, ) + Qxyzdy (,, ) + Rxyzdz (,, ) = ( ) i hr dr definition of line integrals with respect to arc length arc length equations for length, mass, centre of mass (vector and scalar forms), moment of inertia Before you proceed to Objective 5, make certain that you can meet each of the sub-objectives listed under Objective 4. Objective 5 a. use Green s theorem to convert a line integral along a boundary of a Jordan region into a double integral, and to convert a double integral to a line integral along the boundary of a Jordan region b. use Green s theorem to evaluate line integrals, and to determine work, area and moment of inertia. c. demonstrate various conclusions relating to Green s theorem. Read Section 17.5, pages 1137-1145. omplete odd-numbered problems 1-35 on pages 1145-1146. Jordan curve Jordan region δq δp Green s theorem: ( x, y) ( x, y) dx dy = P( x, y) dx + Q( x, y) dy Ω δx δy Mathematics 365 / Study Guide 51
symbols and elementary region counterclockwise integral over 1 clockwise integral over 2 Before you proceed to Objective 6, make certain that you can meet each of the sub-objectives listed under Objective 5. Objective 6 a. calculate fundamental vector products. b. provide a parametric representation for various surfaces expressed in xyzcoordinates. c. provide a formula in xyz-coordinates for various surfaces expressed in parametric form. d. use the appropriate formula to find the area of various surfaces. e. demonstrate various conclusions relating to fundamental vector products. Read Section 17.6, pages 1147-1158. omplete problems 1 and 3-21, and odd-numbered problems 23-35 on pages 1158-1159. parametrization of a surface (function, plane, sphere, cone, spiral ramp) fundamental vector product area of a parametrized surface continuously differentiable surface 52 alculus Several Variables
formulas for the area of a surface: area of S = N( u, v) du dv Ω area of 2 2 y S = [ f ( x, y)] + [ f ( x, y)] + 1 dx dy Ω x A = sec[ γ(x,y)] dx dy Ω Before you proceed to Objective 7, make certain that you can meet each of the sub-objectives listed under Objective 6. Objective 7 a. evaluate integrals over a surface. b. find the mass of a material surface. c. calculate the flux of a vector across a surface. Read Section 17.7, pages 1159-1168. omplete problems 1-12, 18-23, 25-27, 29, 31 and 32 on pages 1169-1170. material surface formula for the mass of a material surface: M = λ[ xuv (, ), yuv (, ), zuv (, )] Nuv (, ) dudv Ω formula for the surface integral of a scalar field continuous over a surface: H( xyz,, ) dσ= Hxuv [ (, ), yuv (, ), zuv (, )] Nuv (, ) dudv S Ω average value of a scalar field continuous over a surface G-weighted value of H on S Mathematics 365 / Study Guide 53
flux of a vector field across a surface in the direction of a unit normal closed piecewise-smooth function Before you proceed to Objective 8, make certain that you can meet each of the sub-objectives listed under Objective 7. Objective 8 a. find the divergence and curl of a vector field. b. calculate the Laplacian of a scalar field. c. demonstrate various conclusions relating to the vector differential operator. Read Section 17.8, pages 1170-1175. omplete problems 1-21, 23, 25 and 27-34 on pages 1175-1176. vector differential operator : δ δ δ = i + j + k δx δy δz gradient of f: f f f f δ δ δ = i + j + k f = δ + δ + δ δx δy δz i j k δx δy δz δv1 δv2 δv3 divergence of v: i v = + + δx δy δz curl of v: i j k δ δ δ δv3 δv2 δv1 δv3 δv2 δv1 v = = x y z y z i + j+ δ δ δ δ δ δz δx δx δy k v v v 1 2 3 54 alculus Several Variables
Laplacian operator: 2 2 2 2 δ f δ f δ f f = i ( f ) = + + 2 2 2 δx δy δz basic identities: the curl of a gradient is zero the divergence of a curl is zero product rule for divergence product rule for curl n i ( r r) = ( n + 3) r ( r n r)= 0 n Before you proceed to Objective 9, make certain that you can meet each of the sub-objectives listed under Objective 8. Objective 9 a. use the divergence theorem to calculate the flux out of a solid. b. demonstrate various conclusions relating to the divergence theorem. Read Section 17.9, pages 1176-1181. omplete odd-numbered problems 1-19 on pages 1181-1182. Green s theorem expressed in vector terms: ( i v) dx dy = ( v i n) ds divergence theorem (Gauss s theorem) as a higher-dimensional analogue of Green s theorem: ( i v) dx dy dz = ( v i n) dσ T divergence as outward flux per unit volume S Ω Mathematics 365 / Study Guide 55
positive divergence source sink solenoidal electric field Before you proceed to Objective 10, make certain that you can meet each of the sub-objectives listed under Objective 9. Objective 10 a. verify Stokes s theorem for particular examples of smooth surfaces with smooth bounding curves. b. solve problems using Stokes s theorem. c. demonstrate various conclusions relating to Stokes s theorem. Read Section 17.10, pages 1183-1189. omplete odd-numbered problems 1-11, and problems 12-20 on pages 1189-1190. Stokes s theorem: [( v) i n] dσ= v( r) i dr circulation per unit area irrotational S Before you complete the fourth and fifth tutor-marked assignments, and write the final examination, make certain that you can meet each of the subobjectives listed under Objective 10. 56 alculus Several Variables
Final Examination Before you begin the fourth tutor-marked assignment, contact the Office of the Registrar to request the final examination. Please see your Student Manual for further information. Assignments 3 and 4 omplete Tutor-marked Assignment 3 and Tutor-marked Assignment 4, both of which you will find in the Assignments for redit section of your Student Manual. Submit the completed assignments to your tutor for grading. Remember to include a Tutor-marked Exercise form, from the course package, with each assignment, and to keep a copy of your work, at least a rough draft, in case the original is lost in the mail. ourse Project omplete the ourse Project, which you will find in the Assignments for redit section of your Student Manual. Submit the completed assignment to your tutor for grading. Remember to include a Tutor-marked Exercise form, from the course package, with each assignment, and to keep a copy of your work, at least a rough draft, in case the original is lost in the mail. Mathematics 365 / Study Guide 57
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