EXAMPLE 6 Find the gradient vector field of f x, y x 2 y y 3. Plot the gradient vector field together with a contour map of f. How are they related?


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1 9 HAPTER 3 VETOR ALULU 4 _4 4 _4 FIGURE 5 EXAMPLE 6 Find the gradient vector field of f, 3. Plot the gradient vector field together with a contour map of f. How are the related? OLUTION The gradient vector field is given b f, f i f j i 3 j Figure 5 shows a contour map of f with the gradient vector field. Notice that the gradient vectors are perpendicular to the level curves, as we would epect from ection.6. Notice also that the gradient vectors are long where the level curves are close to each other and short where the are farther apart. That s because the length of the gradient vector is the value of the directional derivative of f and close level curves indicate a steep graph. A vector field F is called a conservative vector field if it is the gradient of some scalar function, that is, if there eists a function f such that F f. In this situation f is called a potential function for F. Not all vector fields are conservative, but such fields do arise frequentl in phsics. For eample, the gravitational field F in Eample 4 is conservative because if we define then f,, z mmg s z f,, z f i f j f z k mmg z 3 i F,, z mmg z 3 j mmgz z 3 k In ections 3.3 and 3.5 we will learn how to tell whether or not a given vector field is conservative. 3. Eercises ketch the vector field F b drawing a diagram like Figure 5 or Figure 9.. F, i j. F, i j 3. F, i j 4. F, i j 5. F, i j 6. F, i j s s 7. F,, z j 8. F,, z z j 9. F,, z j. F,, z j i 4 Match the vector fields F with the plots labeled I IV. Give reasons for our choices.. F,,. F, 3, 3 3. F, sin, sin 4. F, ln,
2 ETION 3. VETOR FIELD 93 I 5 II 6 use it to plot F, i 3 6 j _5 5 _5 _6 6 _6 A Eplain the appearance b finding the set of points, such that F,. r. Let F r r, where, and. Use a A to plot this vector field in various domains until ou can see what is happening. Describe the appearance of the plot and eplain it b finding the points where F. III 5 IV 5 4 Find the gradient vector field of f.. f, ln. f, e 3. f,, z s z 4. f,, z cosz _5 5 _ Find the gradient vector field f of f and sketch it. 5. f, 6. f, 4 _5 _5 5 8 Match the vector fields F on with the plots labeled I IV. Give reasons for our choices. 5. F,, z i j 3 k 6. F,, z i j z k 7. F,, z i j 3 k 8. F,, z i j z k 3 A 7 8 Plot the gradient vector field of f together with a contour map of f. Eplain how the are related to each other. 7. f, sin sin 8. f, sin 9 3 Match the functions f with the plots of their gradient vector fields (labeled I IV). Give reasons for our choices. 9. f, 3. f, 3. f, 3. f, s I II I 4 II 4 z _ z 4 4 _4 4 III IV _4 _4 z _ z _ III 4 _4 4 IV 4 _4 4 A 9. If ou have a A that plots vector fields (the command is fieldplot in Maple and PlotVectorField in Mathematica), _4 _4
3 94 HAPTER 3 VETOR ALULU 33. The flow lines (or streamlines) of a vector field are the paths followed b a particle whose velocit field is the given vector field. Thus, the vectors in a vector field are tangent to the flow lines. (a) Use a sketch of the vector field F, i j to draw some flow lines. From our sketches, can ou guess the equations of the flow lines? (b) If parametric equations of a flow line are t, t, eplain wh these functions satisf the differential equations ddt and ddt. Then solve the differential equations to find an equation of the flow line that passes through the point (, ). 34. (a) ketch the vector field F, i j and then sketch some flow lines. What shape do these flow lines appear to have? (b) If parametric equations of the flow lines are t, t, what differential equations do these functions satisf? Deduce that dd. (c) If a particle starts at the origin in the velocit field given b F, find an equation of the path it follows. 3. Line Integrals In this section we define an integral that is similar to a single integral ecept that instead of integrating over an interval a, b, we integrate over a curve. uch integrals are called line integrals, although curve integrals would be better terminolog. The were invented in the earl 9th centur to solve problems involving fluid flow, forces, electricit, and magnetism. We start with a plane curve given b the parametric equations t t a t b P P P a FIGURE P i P* i (* i, * i ) P i t i t* i t i P n b t or, equivalentl, b the vector equation rt t i t j, and we assume that is a smooth curve. [This means that r is continuous and rt. ee ection..] If we divide the parameter interval a, b into n subintervals t i, t i of equal width and we let i t i and i t i, then the corresponding points P i i, i divide into n subarcs with lengths s, s,..., s n. (ee Figure.) We choose an point P i * i *, i * in the ith subarc. (This corresponds to a point t* i in t i, t i.) Now if f is an function of two variables whose domain includes the curve, we evaluate f at the point i *, i *, multipl b the length s i of the subarc, and form the sum n f i *, i * s i i which is similar to a Riemann sum. Then we take the limit of these sums and make the following definition b analog with a single integral. Definition If f is defined on a smooth curve given b Equations, then the line integral of f along is if this limit eists. f, ds lim n l n f i *, i * s i i In ection 6.3 we found that the length of is b L dt d dt d dt a
4 ETION 3. LINE INTEGRAL 933 Figure 3 shows the twisted cubic in Eample 8 and some tpical vectors acting at three points on. EXAMPLE 8 Evaluate F dr, where F,, z i z j z k and is the twisted cubic given b t t z t 3 t.5 F{r()} z (,, ).5 F{r(3/4)} F{r(/)} FIGURE 3 OLUTION We have Thus F dr Frt t 3 i t 5 j t 4 k t 3 5t 6 dt t 4 rt t i t j t 3 k rt i t j 3t k Frt rt dt 4 5t Finall, we note the connection between line integrals of vector fields and line integrals of scalar fields. uppose the vector field F on 3 is given in component form b the equation F P i Q j R k. We use Definition 3 to compute its line integral along : F dr b a b P i Q j R k t i t j zt k dt a b Pt, t, ztt Qt, t, ztt Rt, t, ztzt dt a Frt rt dt But this last integral is precisel the line integral in (). Therefore, we have F dr P d Q d R dz where F P i Q j R k For eample, the integral d z d dz in Eample 6 could be epressed as F dr where F,, z i z j k 3. Eercises Evaluate the line integral, where is the given curve.. ds, : t, t, t. ds, : t 4, t 3, t 3. 4 ds, is the right half of the circle 6 4. sin d, is the arc of the curve 4 from, to, 5. d d, consists of line segments from, to, and from, to 3,
5 934 HAPTER 3 VETOR ALULU 6. s d s d, consists of the shortest arc of the circle from, to, and the line segment from, to 4, ds, : 4 sin t, 4 cos t, z 3t, t 8. z ds, is the line segment from (, 6, ) to (4,, 5) 9. e z ds, is the line segment from (,, ) to (,, 3). z d dz, : st, t, z t, t. z d z d dz, consists of line segments from,, to,,, from,, to,, 3, and from,, 3 to,, 4. z d z d dz, consists of line segments from,, to,,, from,, to, 3,, and from, 3, to, 3, 3. Let F be the vector field shown in the figure. (a) If is the vertical line segment from 3, 3 to 3, 3, determine whether F dr is positive, negative, or zero. (b) If is the counterclockwiseoriented circle with radius 3 and center the origin, determine whether F dr is positive, negative, or zero. 3 _3 3 _3 4. The figure shows a vector field F and two curves and. Are the line integrals of F over and positive, negative, or zero? Eplain. A 5 8 Evaluate the line integral F dr, where is given b the vector function rt. 5. F, 3 i s j, rt t i t 3 j, t 6. F,, z z i z j k, rt t i t j t 3 k, t 7. F,, z sin i cos j z k, rt t 3 i t j t k, t 9 Use a graph of the vector field F and the curve to guess whether the line integral of F over is positive, negative, or zero. Then evaluate the line integral. 9. F, i j, is the arc of the circle 4 traversed counterclockwise from (, ) to,. F,, s i s j is the parabola from, to (, ). (a) Evaluate the line integral F dr, where F, e i j and is given b rt t i t 3 j, t. ; (b) Illustrate part (a) b using a graphing calculator or computer to graph and the vectors from the vector field corresponding to t, s, and (as in Figure 3).. (a) Evaluate the line integral F dr, where F,, z i z j k and is given b rt t i 3t j t k, t. ; (b) Illustrate part (a) b using a computer to graph and the vectors from the vector field corresponding to t and (as in Figure 3). A 3. Find the eact value of 3 5 ds, where is the part of the astroid cos 3 t, sin 3 t in the first quadrant. A 8. F,, z i j z k, rt sin t i cos t j t k, t 4. (a) Find the work done b the force field F, i j on a particle that moves once around the circle 4 oriented in the counterclockwise direction. (b) Use a computer algebra sstem to graph the force field and circle on the same screen. Use the graph to eplain our answer to part (a). 5. A thin wire is bent into the shape of a semicircle 4,. If the linear densit is a constant k, find the mass and center of mass of the wire. 6. Find the mass and center of mass of a thin wire in the shape of a quartercircle r,,, if the densit function is,.
6 ETION 3. LINE INTEGRAL (a) Write the formulas similar to Equations 4 for the center of mass,, z of a thin wire with densit function,, z in the shape of a space curve. (b) Find the center of mass of a wire in the shape of the heli sin t, cos t, z 3t, t, if the densit is a constant k. 8. Find the mass and center of mass of a wire in the shape of the heli t, cos t, z sin t, t, if the densit at an point is equal to the square of the distance from the origin. 9. If a wire with linear densit, lies along a plane curve, its moments of inertia about the  and aes are defined as revolutions, how much work is done b the man against gravit in climbing to the top? 36. uppose there is a hole in the can of paint in Eercise 35 and 9 lb of paint leak steadil out of the can during the man s ascent. How much work is done? 37. An object moves along the curve shown in the figure from (, ) to (9, 8). The lengths of the vectors in the force field F are measured in newtons b the scales on the aes. Estimate the work done b F on the object. (meters) I, ds I, ds Find the moments of inertia for the wire in Eample If a wire with linear densit,, z lies along a space curve, its moments of inertia about the , , and zaes are defined as I z,, z ds I z,, z ds I z,, z ds Find the moments of inertia for the wire in Eercise Find the work done b the force field F, i j in moving an object along an arch of the ccloid rt t sin t i cos t j, t. 3. Find the work done b the force field F, sin i j on a particle that moves along the parabola from, to, Find the work done b the force field F,, z z i j z k on a particle that moves along the curve rt t i t 3 j t 4 k, t. 34. The force eerted b an electric charge at the origin on a charged particle at a point,, z with position vector r,, z is Fr Kr r 3 where K is a constant. (ee Eample 5 in ection 3..) Find the work done as the particle moves along a straight line from,, to,, A 6lb man carries a 5lb can of paint up a helical staircase that encircles a silo with a radius of ft. If the silo is 9 ft high and the man makes eactl three complete 38. Eperiments show that a stead current I in a long wire produces a magnetic field B that is tangent to an circle that lies in the plane perpendicular to the wire and whose center is the ais of the wire (as in the figure). Ampère s Law relates the electric current to its magnetic effects and states that where I is the net current that passes through an surface bounded b a closed curve and is a constant called the permeabilit of free space. B taking to be a circle with radius r, show that the magnitude B B of the magnetic field at a distance r from the center of the wire is B B dr I B I r I (meters)
7 ETION 3.3 THE FUNDAMENTAL THEOREM FOR LINE INTEGRAL 943 Therefore 5 W m vb m va where v r is the velocit. The quantit m vt, that is, half the mass times the square of the speed, is called the kinetic energ of the object. Therefore, we can rewrite Equation 5 as 6 W KB KA which sas that the work done b the force field along is equal to the change in kinetic energ at the endpoints of. Now let s further assume that F is a conservative force field; that is, we can write F f. In phsics, the potential energ of an object at the point,, z is defined as P,, z f,, z, so we have F P. Then b Theorem we have W F dr P dr Prb Pra PA PB omparing this equation with Equation 6, we see that PA KA PB KB which sas that if an object moves from one point A to another point B under the influence of a conservative force field, then the sum of its potential energ and its kinetic energ remains constant. This is called the Law of onservation of Energ and it is the reason the vector field is called conservative. 3.3 Eercises. The figure shows a curve and a contour map of a function f whose gradient is continuous. Find f dr A table of values of a function f with continuous gradient is given. Find f dr, where has parametric equations t, t 3 t, t Determine whether or not F is a conservative vector field. If it is, find a function f such that F f F, 6 5 i 5 4 j F, 3 4 i 4 3 j F, e i e j F, e i e j F, cos cos i sin sin j F, ln i j 9. F, e sin i e cos j
8 944 HAPTER 3 VETOR ALULU. F, e 4 3 i e 4 j. F, i j; P,, Q4,. The figure shows the vector field F,, and three curves that start at (, ) and end at (3, ). (a) Eplain wh F dr has the same value for all three curves. (b) What is this common value? 3. Is the vector field shown in the figure conservative? Eplain. 3 8 (a) Find a function f such that F f and (b) use part (a) to evaluate F dr along the given curve.. F, i j, is the upper semicircle that starts at (, ) and ends at (, ) 3. F, 3 4 i 4 3 j, : rt st i t 3 j, 4. F, e i e j, : rt te t i t j, t 5. F,, z z i z j z k, is the line segment from,, to 4, 6, 3 6. F,, z z i j 3z k, : t, t, z t, t 7. F,, z cos z i cos z j sin z k, : rt t i sin t j t k, t 8. F,, z e i e j z e z k, : rt t i t j t 3 k, t 9 how that the line integral is independent of path and evaluate the integral. 9. sin d cos 3 d, is an path from, to 5,. 3 3 d d, is an path from, to 3, Find the work done b the force field F in moving an object from P to Q.. F, 3 i 3 j; P,, Q, 3 t A 4 5 From a plot of F guess whether it is conservative. Then determine whether our guess is correct F, sin i cos j F, 6. Let F f, where f, sin. Find curves and that are not closed and satisf the equation. (a) F dr 7. how that if the vector field F P i Q j R k is conservative and P, Q, R have continuous firstorder partial derivatives, then P Q i j s (b) P z R F dr Q z R 8. Use Eercise 7 to show that the line integral d d z dz is not independent of path. 9 3 Determine whether or not the given set is (a) open, (b) connected, and (c) simplconnected. 9.,, 3., 3., 4 3., or Let F, i j. (a) how that P Q. (b) how that F dr is not independent of path. [Hint: ompute F dr and F dr, where and are the upper and lower halves of the circle from, to,.] Does this contradict Theorem 6?
9 ETION 3.4 GREEN THEOREM (a) uppose that F is an inverse square force field, that is, Fr for some constant c, where r i j z k. Find the work done b F in moving an object from a point P along a path to a point P in terms of the distances d and d from these points to the origin. (b) An eample of an inverse square field is the gravitational field F mmgr r 3 discussed in Eample 4 in ection 3.. Use part (a) to find the work done b the gravitational field when Earth moves from aphelion cr r 3 (at a maimum distance of.5 8 km from the un) to perihelion (at a minimum distance of.47 8 km). (Use the values m kg, M.99 3 kg, and G 6.67 Nm kg. (c) Another eample of an inverse square field is the electric field E qqr r 3 discussed in Eample 5 in ection 3.. uppose that an electron with a charge of.6 9 is located at the origin. A positive unit charge is positioned a distance m from the electron and moves to a position half that distance from the electron. Use part (a) to find the work done b the electric field. (Use the value ) 3.4 Green s Theorem D Green s Theorem gives the relationship between a line integral around a simple closed curve and a double integral over the plane region D bounded b. (ee Figure. We assume that D consists of all points inside as well as all points on.) In stating Green s Theorem we use the convention that the positive orientation of a simple closed curve refers to a single counterclockwise traversal of. Thus, if is given b the vector function rt, a t b, then the region D is alwas on the left as the point rt traverses. (ee Figure.) FIGURE D D FIGURE (a) Positive orientation (b) Negative orientation Green s Theorem Let be a positivel oriented, piecewisesmooth, simple closed curve in the plane and let D be the region bounded b. If P and Q have continuous partial derivatives on an open region that contains D, then Q P da P d Q d D NOTE The notation P d Q d or g P d Q d is sometimes used to indicate that the line integral is calculated using the positive orientation of the closed curve. Another notation for the positivel oriented boundar
10 ETION 3.4 GREEN THEOREM Eercises A 4 Evaluate the line integral b two methods: (a) directl and (b) using Green s Theorem.. d 3 d, is the rectangle with vertices (, ), (, ), (, 3), and (, 3). d d, is the circle with center the origin and radius 3. d 3 d, is the triangle with vertices (, ), (, ), and (, ) 4. d d, consists of the arc of the parabola from, to, 4 and the line segments from, 4 to, 4 and from, 4 to, 5 6 Verif Green s Theorem b using a computer algebra sstem to evaluate both the line integral and the double integral. 5. P, 4 5, Q, 7 6, is the circle 6. P, sin, Q, sin, consists of the arc of the parabola from (, ) to (, ) followed b the line segment from (, ) to (, ) 7 6 Use Green s Theorem to evaluate the line integral along the given positivel oriented curve. 7. e d e d, is the square with sides,,, and 8. d 4 3 d, is the triangle with vertices (, ), (, 3), and (, 3) ( e s ) d cos d 9., is the boundar of the region enclosed b the parabolas and. tan d 3 sin d, is the boundar of the region enclosed b the parabola and the line 4., is the circle 3 d 3 d 4., is the ellipse sin d cos d 3. d d, consists of the line segment from, to, and the top half of the circle d 3 3 d, is the boundar of the region between the circles and 9 5., where F, i F dr j, consists of the circle 4 from, to (s, s) and the line segments from (s, s) to, and from, to, 6., where F, 6 i 5 F dr j, is the ellipse 4 7. Use Green s Theorem to find the work done b the force F, i j in moving a particle from the origin along the ais to,, then along the line segment to,, and then back to the origin along the ais. 8. A particle starts at the point,, moves along the ais to,, and then along the semicircle s4 to the starting point. Use Green s Theorem to find the work done on this particle b the force field F,, Find the area of the given region using one of the formulas in Equations The region bounded b the hpoccloid with vector equation rt cos 3 t i sin 3 t j, t. The region bounded b the curve with vector equation rt cos t i sin 3 t j, t. (a) If is the line segment connecting the point, to the point,, show that (b) If the vertices of a polgon, in counterclockwise order, are,,,,..., n, n, show that the area of the polgon is A 3 3 A (c) Find the area of the pentagon with vertices,,,,, 3,,, and,.. Let D be a region bounded b a simple closed path in the plane. Use Green s Theorem to prove that the coordinates of the centroid, of D are A d where A is the area of D. d d n n n n n n 3. Use Eercise to find the centroid of the triangle with vertices,,,, and,. 4. Use Eercise to find the centroid of a semicircular region of radius a. 5. A plane lamina with constant densit, occupies a region in the plane bounded b a simple closed path. how that its moments of inertia about the aes are I 3 3 d A d I 3 3 d
11 95 HAPTER 3 VETOR ALULU 6. Use Eercise 5 to find the moment of inertia of a circular disk of radius a with constant densit about a diameter. (ompare with Eample 4 in ection.5.) 7. If F is the vector field of Eample 5, show that F dr for ever simple closed path that does not pass through or enclose the origin. 8. omplete the proof of the special case of Green s Theorem b proving Equation Use Green s Theorem to prove the change of variables formula for a double integral (Formula.9.9) for the case where f, : R d d, u, v du dv Here R is the region in the plane that corresponds to the region in the uvplane under the transformation given b tu, v, hu, v. [Hint: Note that the left side is AR and appl the first part of Equation 5. onvert the line integral over R to a line integral over and appl Green s Theorem in the uvplane.] 3.5 url and Divergence In this section we define two operations that can be performed on vector fields and that pla a basic role in the applications of vector calculus to fluid flow and electricit and magnetism. Each operation resembles differentiation, but one produces a vector field whereas the other produces a scalar field. url If F P i Q j R k is a vector field on 3 and the partial derivatives of P, Q, and R all eist, then the curl of F is the vector field on 3 defined b curl F R z Q i P z R j Q P k As an aid to our memor, let s rewrite Equation using operator notation. We introduce the vector differential operator ( del ) as It has meaning when it operates on a scalar function to produce the gradient of f : f i f j f k f z f i f j f z k If we think of as a vector with components,, and z, we can also consider the formal cross product of with the vector field F as follows: i j k F z P Q R R z Q i P z R j Q P k curl F i j k z
12 958 HAPTER 3 VETOR ALULU b Green s Theorem. But the integrand in this double integral is just the divergence of F. o we have a second vector form of Green s Theorem. 3 F n ds D div F, da This version sas that the line integral of the normal component of F along is equal to the double integral of the divergence of F over the region D enclosed b. 3.5 Eercises 6 Find (a) the curl and (b) the divergence of the vector field.. F,, z i z j z k. F,, z z i z j k 3. F,, z z i k The vector field F is shown in the plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its zcomponent is.) (a) Is div F positive, negative, or zero? Eplain. (b) Determine whether curl F. If not, in which direction does curl F point? 7. F,, z e j e z k F,, z e sin i e cos j z k F,, z z i z j z z k 9.. Let f be a scalar field and F a vector field. tate whether each epression is meaningful. If not, eplain wh. If so, state whether it is a scalar field or a vector field. (a) curl f (b) grad f (c) div F (d) curlgrad f (e) grad F (f) graddiv F (g) divgrad f (h) graddiv f (i) curlcurl F (j) divdiv F (k) grad f div F (l) divcurlgrad f 6 Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F f F,, z z i z j k F,, z i j z k F,, z i z j k F,, z z 3 i z 3 j 3 z k F,, z e i e z j e k F,, z ze z i e z j e z k Is there a vector field G on 3 such that curl G i z j z k? Eplain. 8. Is there a vector field G on such that curl G z i z j k? Eplain. 9. how that an vector field of the form F,, z f i t j hz k where f, t, h are differentiable functions, is irrotational. 3
13 ETION 3.5 URL AND DIVERGENE 959. how that an vector field of the form is incompressible. 7 Prove the identit, assuming that the appropriate partial derivatives eist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F G, and F G are defined b f F,, z f,, zf,, z. divf G div F div G. curlf G curl F curl G 3. div f F f div F F f 4. curl f F f curl F f F 5. divf G G curl F F curl G 6. divf t Let r i j z k and. 8. Verif each identit. (a) r 3 (c) r 3 r 9. Verif each identit. (a) r rr (b) r (c) r rr 3 (d) ln r rr 3. If F rr p, find div F. Is there a value of p for which div F? 3. Use Green s Theorem in the form of Equation 3 to prove Green s first identit: D (b) where D and satisf the hpotheses of Green s Theorem and the appropriate partial derivatives of f and t eist and are continuous. (The quantit t n D n t occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of t.) 3. Use Green s first identit (Eercise 3) to prove Green s second identit: D F,, z f, z i t, z j h, k F G,, z F,, z G,, z F G,, z F,, z G,, z curl curl F grad div F F r r f t da f t n ds D rr 4r f t da f t t f da f t tf n ds where D and satisf the hpotheses of Green s Theorem and the appropriate partial derivatives of f and t eist and are continuous. 33. This eercise demonstrates a connection between the curl vector and rotations. Let B be a rigid bod rotating about the zais. The rotation can be described b the vector w k, where is the angular speed of B, that is, the tangential speed of an point P in B divided b the distance d from the ais of rotation. Let r,, z be the position vector of P. (a) B considering the angle in the figure, show that the velocit field of B is given b v w r. (b) how that v i j. (c) how that curl v w. B 34. Mawell s equations relating the electric field E and magnetic field H as the var with time in a region containing no charge and no current can be stated as follows: div E w z d div H curl H E curl E H c t c t where c is the speed of light. Use these equations to prove the following: (a) E E c t (b) H H c t (c) E E [Hint: Use Eercise 7.] c t (d) H H c t P v
14 97 HAPTER 3 VETOR ALULU 3.6 Eercises. Let be the cube with vertices,,. Approimate s 3z d b using a Riemann sum as in Definition, taking the patches ij to be the squares that are the faces of the cube and the points P ij * to be the centers of the squares.. A surface consists of the clinder, z, together with its top and bottom disks. uppose ou know that f is a continuous function with f,,, f,, 3,and f,, 4. Estimate the value of f,, z d b using a Riemann sum, taking the patches ij to be four quarterclinders and the top and bottom disks. 3. Let H be the hemisphere z 5, z, and suppose f is a continuous function with f 3, 4, 5 7, f 3, 4, 5 8, f 3, 4, 5 9,and f 3, 4, 5. B dividing H into four patches, estimate the value of H f,, z d. 4. uppose that f,, z t(s z ), where t is a function of one variable such that t 5. Evaluate, where is the sphere z f,, z d Evaluate the surface integral. 5. z d, is the surface with parametric equations uv, u v, z u v, u v 6. s d, is the helicoid with vector equation ru, v u cos v i u sin v j v k, u, v 7. z d, is the part of the plane z 3 that lies above the rectangle, 3, 8. d, is the triangular region with vertices (,, ), (,, ), and (,, ) 9. z d, is the part of the plane z that lies in the first octant. d, is the surface z 3 3 3,,. d, is the surface 4z,, z. z d, is the part of the paraboloid 4 z that lies in front of the plane 3. z d, is the part of the plane z 3 that lies inside the clinder 4. d, is the boundar of the region enclosed b the clinder z and the planes and 5. z z d, is the hemisphere z 4, z 6. z d, is the part of the sphere z that lies above the cone z s 7. z d, is the part of the clinder 9 between the planes z and z 8. z d, consists of the clinder in Eercise 7 together with its top and bottom disks 9 7 Evaluate the surface integral F d for the given vector field F and the oriented surface. In other words, find the flu of F across. For closed surfaces, use the positive (outward) orientation. 9. F,, z i z j z k, is the part of the paraboloid z 4 that lies above the square,, and has upward orientation. F,, z i j z k, is the helicoid of Eercise 6 with upward orientation. F,, z ze i ze j z k, is the part of the plane z in the first octant and has downward orientation. F,, z i j z 4 k, is the part of the cone z s beneath the plane z with downward orientation 3. F,, z i j z k, is the sphere z 9 4. F,, z i j 3z k, is the hemisphere z s6 with upward orientation 5. F,, z j z k, consists of the paraboloid z,, and the disk z, 6. F,, z i j 5 k, is the surface of Eercise 4 7. F,, z i j 3z k, is the cube with vertices,,
15 ETION 3.7 TOKE THEOREM 97 A A A 8. Let be the surface z,,. (a) Evaluate z d correct to four decimal places. (b) Find the eact value of z d. 9. Find the value of z d correct to four decimal places, where is the part of the paraboloid z 3 that lies above the plane. 3. Find the flu of F,, z sinz i j z e 5 k across the part of the clinder 4 z 4 that lies above the plane and between the planes and with upward orientation. Illustrate b using a computer algebra sstem to draw the clinder and the vector field on the same screen. 3. Find a formula for F d similar to Formula for the case where is given b h, z and n is the unit normal that points toward the left. 3. Find a formula for F d similar to Formula for the case where is given b k, z and n is the unit normal that points forward (that is, toward the viewer when the aes are drawn in the usual wa). 33. Find the center of mass of the hemisphere z a, z, if it has constant densit. 34. Find the mass of a thin funnel in the shape of a cone z s, z 4, if its densit function is,, z z. 35. (a) Give an integral epression for the moment of inertia I z about the zais of a thin sheet in the shape of a surface if the densit function is. (b) Find the moment of inertia about the zais of the funnel in Eercise The conical surface z, z a, has constant densit k. Find (a) the center of mass and (b) the moment of inertia about the zais. 37. A fluid with densit flows with velocit v i j z k. Find the rate of flow upward through the paraboloid z 9 4, A fluid has densit 5 and velocit field v i j z k. Find the rate of flow outward through the sphere z Use Gauss s Law to find the charge contained in the solid hemisphere z a, z, if the electric field is E,, z i j z k. 4. Use Gauss s Law to find the charge enclosed b the cube with vertices,, if the electric field is E,, z i j z k. 4. The temperature at the point,, z in a substance with conductivit K 6.5 is u,, z z. Find the rate of heat flow inward across the clindrical surface z 6, The temperature at a point in a ball with conductivit K is inversel proportional to the distance from the center of the ball. Find the rate of heat flow across a sphere of radius a with center at the center of the ball. 3.7 tokes Theorem z n n tokes Theorem can be regarded as a higherdimensional version of Green s Theorem. Whereas Green s Theorem relates a double integral over a plane region D to a line integral around its plane boundar curve, tokes Theorem relates a surface integral over a surface to a line integral around the boundar curve of (which is a space curve). Figure shows an oriented surface with unit normal vector n. The orientation of induces the positive orientation of the boundar curve shown in the figure. This means that if ou walk in the positive direction around with our head pointing in the direction of n, then the surface will alwas be on our left. FIGURE tokes Theorem Let be an oriented piecewisesmooth surface that is bounded b a simple, closed, piecewisesmooth boundar curve with positive orientation. Let F be a vector field whose components have continuous partial derivatives on an open region in 3 that contains. Then F dr curl F d
16 976 HAPTER 3 VETOR ALULU 3.7 Eercises. A hemisphere H and a portion P of a paraboloid are shown. uppose F is a vector field on 3 whose components have continuous partial derivatives. Eplain wh H z 4 H curl F d P 6 Use tokes Theorem to evaluate curl F d.. F,, z z i z j k, is the part of the paraboloid z 9 that lies above the plane z 5, oriented upward 3. F,, z e z i e z j z e k, is the hemisphere z 4, z, oriented upward 4. F,, z tan z i z j z k, is the part of the hemisphere s9 z that lies inside the clinder z 4, oriented in the direction of the positive ais 5. F,, z z i j z k, consists of the top and the four sides (but not the bottom) of the cube with vertices,,, oriented outward [Hint: Use Equation 3.] 6. F,, z i e z j k, consists of the four sides of the pramid with vertices,,,,,,,,,,,, and,, that lie to the right of the zplane, oriented in the direction of the positive ais [Hint: Use Equation 3.] 7 Use tokes Theorem to evaluate F dr. In each case is oriented counterclockwise as viewed from above. 7. F,, z i z j z k, is the triangle with vertices (,, ), (,, ), and (,, ) 8. F,, z e i e j e z k, is the boundar of the part of the plane z in the first octant curl F d P z 4 9. F,, z z i 4 j 5 k, is the curve of intersection of the plane z 4 and the clinder 4. F,, z i j k, is the boundar of the part of the paraboloid z in the first octant. (a) Use tokes Theorem to evaluate F dr, where F,, z z i j z k and is the curve of intersection of the plane z and the clinder 9 oriented counterclockwise as viewed from above. ; (b) Graph both the plane and the clinder with domains chosen so that ou can see the curve and the surface that ou used in part (a). ; (c) Find parametric equations for and use them to graph.. (a) Use tokes Theorem to evaluate F dr, where F,, z i 3 3 j k and is the curve of intersection of the hperbolic paraboloid z and the clinder oriented counterclockwise as viewed from above. ; (b) Graph both the hperbolic paraboloid and the clinder with domains chosen so that ou can see the curve and the surface that ou used in part (a). ; (c) Find parametric equations for and use them to graph. 3 5 Verif that tokes Theorem is true for the given vector field F and surface. 3. F,, z i j z k, is the part of the paraboloid z that lies below the plane z, oriented upward 4. F,, z i j z k, is the part of the plane z that lies in the first octant, oriented upward 5. F,, z i z j k, is the hemisphere z,, oriented in the direction of the positive ais 6. Let F,, z a 3 3z, b 3, cz 3 Let be the curve in Eercise and consider all possible smooth surfaces whose boundar curve is. Find the values of a, b, and c for which F d is independent of the choice of.
17 WRITING PROJET THREE MEN AND TWO THEOREM alculate the work done b the force field F,, z z i j z z k when a particle moves under its influence around the edge of the part of the sphere z 4 that lies in the first octant, in a counterclockwise direction as viewed from above. 8. Evaluate sin d z cos d 3 dz, where is the curve rt sin t, cos t, sin t, t. [Hint: Observe that lies on the surface z.] 9. If is a sphere and F satisfies the hpotheses of tokes Theorem, show that. curl F d. uppose and satisf the hpotheses of tokes Theorem and f, t have continuous secondorder partial derivatives. Use Eercises and 4 in ection 3.5 to show the following. (a) f t dr f t d (b) f f dr (c) f t tf dr Writing Project Three Men and Two Theorems The photograph shows a stainedglass window at ambridge Universit in honor of George Green. Although two of the most important theorems in vector calculus are named after George Green and George tokes, a third man, William Thomson (also known as Lord Kelvin), plaed a large role in the formulation, dissemination, and application of both of these results. All three men were interested in how the two theorems could help to eplain and predict phsical phenomena in electricit and magnetism and fluid flow. The basic facts of the stor are given in the margin notes on pages 946 and 97. Write a report on the historical origins of Green s Theorem and tokes Theorem. Eplain the similarities and relationship between the theorems. Discuss the roles that Green, Thomson, and tokes plaed in discovering these theorems and making them widel known. how how both theorems arose from the investigation of electricit and magnetism and were later used to stud a variet of phsical problems. The dictionar edited b Gillispie [] is a good source for both biographical and scientific information. The book b Hutchinson [5] gives an account of tokes life and the book b Thompson [8] is a biograph of Lord Kelvin. The articles b GrattanGuinness [3] and Gra [4] and the book b annell [] give background on the etraordinar life and works of Green. Additional historical and mathematical information is found in the books b Katz [6] and Kline [7].. D. M. annell, George Green, Mathematician and Phsicist : The Background to his Life and Work (London: Athlone Press, 993).... Gillispie, ed., Dictionar of cientific Biograph (New York: cribner s, 974). ee the article on Green b P. J. Wallis in Volume XV and the articles on Thomson b Jed Buchwald and on tokes b E. M. Parkinson in Volume XIII. 3. I. GrattanGuinness, Wh did George Green write his essa of 88 on electricit and magnetism? Amer. Math. Monthl, Vol. (995), pp J. Gra, There was a joll miller. The New cientist, Vol. 39 (993), pp G. E. Hutchinson, The Enchanted Voage (New Haven: Yale Universit Press, 96). 6. Victor Katz, A Histor of Mathematics: An Introduction (New York: Harperollins, 993), pp Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oford Universit Press, 97), pp lvanus P. Thompson, The Life of Lord Kelvin (New York: helsea, 976).
18 ETION 3.8 THE DIVERGENE THEOREM 983 For the vector field in Figure 4, it appears that the vectors that end near P are shorter than the vectors that start near P. Thus, the net flow is outward near P, so div FP and P is a source. Near P, on the other hand, the incoming arrows are longer than the outgoing arrows. Here the net flow is inward, so div FP and P is a sink. We can use the formula for F to confirm this impression. ince F i j, we have div F, which is positive when. o the points above the line are sources and those below are sinks. P FIGURE 4 The vector field F= i+ j P 3.8 Eercises. A vector field F is shown. Use the interpretation of divergence derived in this section to determine whether div F is positive or negative at and at P. P _ P _ P. (a) Are the points and P sources or sinks for the vector field F shown in the figure? Give an eplanation based solel on the picture. (b) Given that F,,, use the definition of divergence to verif our answer to part (a). P P _ P _ 3 6 Verif that the Divergence Theorem is true for the vector field F on the region E. 3. F,, z 3 i j z k, E is the cube bounded b the planes,,,, z, and z 4. F,, z z i z j 3z k, E is the solid bounded b the paraboloid z and the plane z 5. F,, z i z j z k, E is the solid clinder, z 6. F,, z i j z k, E is the unit ball z 7 5 Use the Divergence Theorem to calculate the surface integral F d; that is, calculate the flu of F across. 7. F,, z e sin i e cos j z k, is the surface of the bo bounded b the planes,,,, z, and z 8. F,, z z 3 i z 3 j z 4 k, is the surface of the bo with vertices,, 3 9. F,, z 3 i e z j z 3 k, is the surface of the solid bounded b the clinder z and the planes and
19 984 HAPTER 3 VETOR ALULU A A. F,, z 3 i j z k, is the surface of the solid bounded b the hperboloid z and the planes z and z. F,, z sin z i cosz j cos z k, is the ellipsoid a b z c. F,, z 3 i z j 3 z k, is the surface of the solid bounded b the paraboloid z 4 and the plane 3. F,, z 3 i 3 j z 3 k, is the sphere z 4. F,, z 3 sin z i 3 z sin j 3z k, is the surface of the solid bounded b the hemispheres z s4, z s and the plane z 5. F,, z e tan z i s3 j sin k, is the surface of the solid that lies above the plane and below the surface z 4 4,, 6. Use a computer algebra sstem to plot the vector field F,, z sin cos i sin 3 cos 4 z j sin 5 z cos 6 k in the cube cut from the first octant b the planes,, and z. Then compute the flu across the surface of the cube. 7. Use the Divergence Theorem to evaluate F d, where F,, z z i ( 3 3 tan z) j z k and is the top half of the sphere z. [Hint: Note that is not a closed surface. First compute integrals over and, where is the disk, oriented downward, and.] 8. Let F,, z z tan i z 3 ln j z k. Find the flu of F across the part of the paraboloid z that lies above the plane z and is oriented upward. 9. Verif that div E for the electric field. Use the Divergence Theorem to evaluate E Q 3 z d where is the sphere z. 6 Prove each identit, assuming that and E satisf the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous secondorder partial derivatives.. a n d, where a is a constant vector. F d, where F,, z i j z k VE 3 curl F d D n f d E f dv f t n d E f t tf n d E 7. uppose and E satisf the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives. Prove that f t f t dv f n d E These surface and triple integrals of vector functions are vectors defined b integrating each component function. [Hint: tart b appling the Divergence Theorem to F fc, where c is an arbitrar constant vector.] 8. A solid occupies a region E with surface and is immersed in a liquid with constant densit. We set up a coordinate sstem so that the plane coincides with the surface of the liquid and positive values of z are measured downward into the liquid. Then the pressure at depth z is p tz, where t is the acceleration due to gravit (see ection 6.5). The total buoant force on the solid due to the pressure distribution is given b the surface integral F f t t f dv f dv pn d where n is the outer unit normal. Use the result of Eercise 7 to show that F Wk, where W is the weight of the liquid displaced b the solid. (Note that F is directed upward because z is directed downward.) The result is Archimedes principle: The buoant force on an object equals the weight of the displaced liquid.
20 ETION 3.9 UMMARY ummar The main results of this chapter are all higherdimensional versions of the Fundamental Theorem of alculus. To help ou remember them, we collect them together here (without hpotheses) so that ou can see more easil their essential similarit. Notice that in each case we have an integral of a derivative over a region on the left side, and the right side involves the values of the original function onl on the boundar of the region. Fundamental Theorem of alculus b F d Fb Fa a a b Fundamental Theorem for Line Integrals f dr f rb f ra r(a) r(b) Green s Theorem D Q P da P d Q d D n tokes Theorem curl F d F dr n Divergence Theorem E div F dv F d E n
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