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is enough to guarantee the existence of the limit, just as the integral exists if g is continuous over before looking at how to compute a line integral, we need to examine the geometry captured by these integrals. suppose that for all points on a smooth planar curve imagine taking curve and projecting it “ up ” to the ... | openstax_calculus_volume_3_-_web | [
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number in the domain of r. therefore, ∞ ∞ ∞ | openstax_calculus_volume_3_-_web | [
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6. 13 find the value of where is the curve parameterized by note that in a scalar line integral, the integration is done with respect to arc length s, which can make a scalar line integral difficult to calculate. to make the calculations easier, we can translate to an integral with a variable of integration that is t. ... | openstax_calculus_volume_3_-_web | [
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can be viewed as a change in the t domain, scaled by the magnitude of vector example 6. 15 evaluating a line integral find the value of integral where is part of the helix parameterized by solution to compute a scalar line integral, we start by converting the variable of integration from arc length s to t. then, we can... | openstax_calculus_volume_3_-_web | [
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length of the wire. solution the length of the wire is given by where c is the curve with parameterization r. therefore, | openstax_calculus_volume_3_-_web | [
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6. 16 find the length of a wire with parameterization vector line integrals the second type of line integrals are vector line integrals, in which we integrate along a curve through a vector field. for example, let be a continuous vector field in ℝ that represents a force on a particle, and let c be a smooth curve in ℝ ... | openstax_calculus_volume_3_-_web | [
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6. 2 • line integrals 599 figure 6. 16 ( a ) an oriented curve between two points. ( b ) a closed oriented curve. let be a parameterization of c for such that the curve is traversed exactly once by the particle and the particle moves in the positive direction along c. divide the parameter interval into n subintervals o... | openstax_calculus_volume_3_-_web | [
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openstax. org orientation of the curve does matter. if we think of the line integral as computing work, then this makes sense : if you hike up a mountain, then the gravitational force of earth does negative work on you. if you walk down the mountain by the exact same path, then earth ’ s gravitational force does positi... | openstax_calculus_volume_3_-_web | [
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6. 2 • line integrals 601 figure 6. 18 this figure shows curve in vector field example 6. 19 reversing orientation find the value of integral where is the semicircle parameterized by and solution notice that this is the same problem as example 6. 18, except the orientation of the curve has been traversed. in this examp... | openstax_calculus_volume_3_-_web | [
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be precise, curve c is piecewise smooth if c can be written as a union of n smooth curves such that the endpoint of is the starting point of ( figure 6. 19 ). when curves satisfy the condition that the endpoint of is the starting point of we write their union as figure 6. 19 the union of is a piecewise smooth curve. ( ... | openstax_calculus_volume_3_-_web | [
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6. 2 • line integrals 603 the next theorem summarizes several key properties of vector line integrals. theorem 6. 5 properties of vector line integrals let f and g be continuous vector fields with domains that include the oriented smooth curve c. then i. ii. where k is a constant iii. iv. suppose instead that c is a pi... | openstax_calculus_volume_3_-_web | [
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parameterization of each side of the rectangle. here are four parameterizations ( note that they traverse c counterclockwise ) : therefore, notice that the value of this integral is positive, which should not be surprising. as we move along curve c1 from left to right, our movement flows in the general direction of the... | openstax_calculus_volume_3_-_web | [
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6. 19 calculate line integral where f is vector field and c is a triangle with vertices and oriented counterclockwise. applications of line integrals scalar line integrals have many applications. they can be used to calculate the length or mass of a wire, the surface area of a sheet of a given height, or the electric p... | openstax_calculus_volume_3_-_web | [
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. when we first defined vector line integrals, we used the concept of work to motivate the definition. therefore, it is not surprising that calculating the work done by a vector field representing a force is a standard use of vector line integrals. recall that if an object moves along curve c in force field f, then the... | openstax_calculus_volume_3_-_web | [
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6. 2 • line integrals 607 solution let c denote the given path. we need to find the value of to do this, we use equation 6. 9 : figure 6. 22 the curve and vector field for example 6. 23. flux and circulation we close this section by discussing two key concepts related to line integrals : flux across a plane curve and c... | openstax_calculus_volume_3_-_web | [
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a curve that represents a membrane, then the flux of f across c is the quantity of fluid flowing across c per unit time, or the rate of flow. more formally, let c be a plane curve parameterized by let be the vector that is normal to c at the endpoint of and points to the right as we traverse c in the positive direction... | openstax_calculus_volume_3_-_web | [
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6. 21 calculate the flux of across the line segment from to where the curve is oriented from left to right. let be a two - dimensional vector field. recall that integral is sometimes written as analogously, flux is sometimes written in the notation because the unit normal vector n is perpendicular to the unit tangent t... | openstax_calculus_volume_3_-_web | [
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and the vector from f form an angle of less than 90°, and therefore the corresponding dot product is positive. in example 6. 25, what if we had oriented the unit circle clockwise? we denote the unit circle oriented clockwise by then notice that the circulation is negative in this case. the reason for this is that the o... | openstax_calculus_volume_3_-_web | [
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6. 23 calculate the work done by field on a particle that traverses the unit circle. assume the particle begins its movement at section 6. 2 exercises 39. true or false? line integral is equal to a definite integral if c is a smooth curve defined on and if function is continuous on some region that contains curve c. 40... | openstax_calculus_volume_3_-_web | [
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circle of radius 2 centered at ( 3, 4 ) with linear mass density for the following exercises, evaluate the line integrals. 55. evaluate where and c is the part of the graph of from to 56. evaluate where is the helix 57. evaluate over the line segment from to 58. let c be the line segment from point ( 0, 1, 1 ) to point... | openstax_calculus_volume_3_-_web | [
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6. 2 • line integrals 613 67. 68. let f be vector field compute the work of integral where c is the path 69. compute the work done by force along path where 70. evaluate where and c is the segment of the unit circle going counterclockwise from to ( 0, 1 ). 71. force acts on a particle that travels from the origin to po... | openstax_calculus_volume_3_-_web | [
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integral where is curve from ( 1, 0 ) toward 82. [ t ] evaluate line integral where is the right half of circle 83. [ t ] evaluate where and c : 84. evaluate where and c is any path from to ( 5, 1 ). 85. find the line integral of over path c defined by from point ( 0, 0, 0 ) to point ( 2, 4, 8 ). 86. find the line inte... | openstax_calculus_volume_3_-_web | [
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6. 2 • line integrals 615 96. find the work done when an object moves in force field along the path given by 97. if an inverse force field f is given by where k is a constant, find the work done by f as its point of application moves along the x - axis from 98. david and sandra plan to evaluate line integral along a pa... | openstax_calculus_volume_3_-_web | [
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6. 3 conservative vector fields learning objectives 6. 3. 1 describe simple and closed curves ; define connected and simply connected regions. 6. 3. 2 explain how to find a potential function for a conservative vector field. 6. 3. 3 use the fundamental theorem for line integrals to evaluate a line integral in a vector ... | openstax_calculus_volume_3_-_web | [
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determining whether a curve is simple and closed is the curve with parameterization a simple closed curve? solution note that therefore, the curve is closed. the curve is not simple, however. to see this, note that and therefore the curve crosses itself at the origin ( figure 6. 26 ). figure 6. 26 a curve that is close... | openstax_calculus_volume_3_-_web | [
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6. 24 is the curve given by parameterization a simple closed curve? many of the theorems in this chapter relate an integral over a region to an integral over the boundary of the region, where the region ’ s boundary is a simple closed curve or a union of simple closed curves. to develop these theorems, we need two geom... | openstax_calculus_volume_3_-_web | [
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6. 3 • conservative vector fields 617 definition a region d is a connected region if, for any two points and there is a path from to with a trace contained entirely inside d. a region d is a simply connected region if d is connected for any simple closed curve c that lies inside d, and curve c can be shrunk continuousl... | openstax_calculus_volume_3_-_web | [
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6. 25 is the region in the below image connected? is the region simply connected? fundamental theorem for line integrals now that we understand some basic curves and regions, let ’ s generalize the fundamental theorem of calculus to line integrals. recall that the fundamental theorem of calculus says that if a function... | openstax_calculus_volume_3_-_web | [
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6. 3 • conservative vector fields 619 the following theorem says that, under certain conditions, what happened in the previous example holds for any gradient field. the same theorem holds for vector line integrals, which we call the fundamental theorem for line integrals. theorem 6. 7 the fundamental theorem for line i... | openstax_calculus_volume_3_-_web | [
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to calculate the integral. note that this calculation is much more straightforward than the calculation we did in ( a ). as long as we have a potential function, calculating a line integral using the fundamental theorem for line integrals is much easier than calculating without the theorem. example 6. 29 illustrates a ... | openstax_calculus_volume_3_-_web | [
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6. 3 • conservative vector fields 621 calculate integral where c is the lower half of the unit circle oriented counterclockwise. the fundamental theorem for line integrals has two important consequences. the first consequence is that if f is conservative and c is a closed curve, then the circulation of f along c is zer... | openstax_calculus_volume_3_-_web | [
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the terminal point since f is conservative, there is a potential function for f. by the fundamental theorem for line integrals, therefore, and f is independent of path. figure 6. 29 the vector field is conservative, and therefore independent of path. to visualize what independence of path means, imagine three hikers cl... | openstax_calculus_volume_3_-_web | [
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6. 3 • conservative vector fields 623 proof we prove the theorem for vector fields in ℝ the proof for vector fields in ℝ is similar. to show that is conservative, we must find a potential function for f. to that end, let x be a fixed point in d. for any point in d, let c be a path from x to define by ( note that this d... | openstax_calculus_volume_3_-_web | [
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. example 6. 30 showing that a vector field is not conservative use path independence to show that vector field is not conservative. solution we can indicate that f is not conservative by showing that f is not path independent. we do so by giving two different paths, and that both start at and end at and yet let be the... | openstax_calculus_volume_3_-_web | [
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6. 27 show that is not path independent by considering the line segment from to and the piece of the graph of that goes from to conservative vector fields and potential functions as we have learned, the fundamental theorem for line integrals says that if f is conservative, then calculating has two steps : first, find a... | openstax_calculus_volume_3_-_web | [
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6. 28 find a potential function for the logic of the previous example extends to finding the potential function for any conservative vector field in ℝ thus, we have the following problem - solving strategy for finding potential functions : problem - solving strategy problem - solving stragegy : finding a potential func... | openstax_calculus_volume_3_-_web | [
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6. 29 find a potential function for we can apply the process of finding a potential function to a gravitational force. recall that, if an object has unit mass and is located at the origin, then the gravitational force in ℝ that the object exerts on another object of unit mass at the point is given by vector field where... | openstax_calculus_volume_3_-_web | [
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6. 30 find a potential function for the three - dimensional gravitational force testing a vector field until now, we have worked with vector fields that we know are conservative, but if we are not told that a vector field is conservative, we need to be able to test whether it is conservative. recall that, if f is conse... | openstax_calculus_volume_3_-_web | [
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6. 3 • conservative vector fields 629 example 6. 34 determining whether a vector field is conservative determine whether vector field is conservative. solution note that the domain of f is all of ℝ and ℝ is simply connected. therefore, we can use cross - partial property of conservative fields to determine whether f is... | openstax_calculus_volume_3_-_web | [
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6. 31 determine whether is conservative. when using cross - partial property of conservative fields, it is important to remember that a theorem is a tool, and like any tool, it can be applied only under the right conditions. in the case of cross - partial property of conservative fields, the theorem can be applied only... | openstax_calculus_volume_3_-_web | [
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integral. if we are asked to calculate an integral of the form then our first question should be : is f conservative? if the answer is yes, then we should find a potential function and use the fundamental theorem for line integrals to calculate the integral. if the answer is no, then the fundamental theorem for line in... | openstax_calculus_volume_3_-_web | [
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6. 32 calculate integral where and c is a semicircle with starting point and endpoint example 6. 37 work done on a particle let be a force field. suppose that a particle begins its motion at the origin and ends its movement at any point in a plane that is not on the x - axis or the y - axis. furthermore, the particle ’... | openstax_calculus_volume_3_-_web | [
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6. 33 let and suppose that a particle moves from point to along any smooth curve. is the work done by f on the particle positive, negative, or zero? 632 6 • vector calculus access for free at openstax. org section 6. 3 exercises 99. true or false? if vector field f is conservative on the open and connected region d, th... | openstax_calculus_volume_3_-_web | [
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6. 3 • conservative vector fields 633 for the following exercises, evaluate the line integrals using the fundamental theorem of line integrals. 112. where c is any path from ( 0, 0 ) to ( 2, 4 ) 113. where c is the line segment from ( 0, 0 ) to ( 4, 4 ) 114. [ t ] where c is any smooth curve from ( 1, 1 ) to 115. find ... | openstax_calculus_volume_3_-_web | [
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find line integral of vector field along curve c parameterized by for the following exercises, show that the following vector fields are conservative by using a computer. calculate for the given curve. 136. c is the curve consisting of line segments from to to 137. c is parameterized by 138. [ t ] c is curve 139. the m... | openstax_calculus_volume_3_-_web | [
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6. 4 green ’ s theorem learning objectives 6. 4. 1 apply the circulation form of green ’ s theorem. 6. 4. 2 apply the flux form of green ’ s theorem. 6. 4. 3 calculate circulation and flux on more general regions. in this section, we examine green ’ s theorem, which is an extension of the fundamental theorem of calculu... | openstax_calculus_volume_3_-_web | [
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based solely on information about the boundary of d. green ’ s theorem also says we can calculate a line integral over a simple closed curve c based solely on information about the region that c encloses. in particular, green ’ s theorem connects a double integral over region d to a line integral around the boundary of... | openstax_calculus_volume_3_-_web | [
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fields, f satisfies the cross - partial condition, so therefore, which confirms green ’ s theorem in the case of conservative vector fields. proof let ’ s now prove that the circulation form of green ’ s theorem is true when the region d is a rectangle. let d be the rectangle oriented counterclockwise. then, the bounda... | openstax_calculus_volume_3_-_web | [
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6. 4 • green ’ s theorem 637 figure 6. 34 rectangle d is oriented counterclockwise. then, by the fundamental theorem of calculus, therefore, but, 638 6 • vector calculus access for free at openstax. org therefore, and we have proved green ’ s theorem in the case of a rectangle. to prove green ’ s theorem over a general... | openstax_calculus_volume_3_-_web | [
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6. 4 • green ’ s theorem 639 figure 6. 35 the line integral over the boundary of the rectangle can be transformed into a double integral over the rectangle. analysis if we were to evaluate this line integral without using green ’ s theorem, we would need to parameterize each side of the rectangle, break the line integr... | openstax_calculus_volume_3_-_web | [
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the area enclosed by ellipse ( figure 6. 37 ). | openstax_calculus_volume_3_-_web | [
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6. 4 • green ’ s theorem 641 figure 6. 37 ellipse is denoted by c. solution let c denote the ellipse and let d be the region enclosed by c. recall that ellipse c can be parameterized by calculating the area of d is equivalent to computing double integral to calculate this integral without green ’ s theorem, we would ne... | openstax_calculus_volume_3_-_web | [
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6. 35 find the area of the region enclosed by the curve with parameterization flux form of green ’ s theorem the circulation form of green ’ s theorem relates a double integral over region d to line integral where c is the boundary of d. the flux form of green ’ s theorem relates a double integral over region d to the ... | openstax_calculus_volume_3_-_web | [
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6. 4 • green ’ s theorem 643 proof recall that let and by the circulation form of green ’ s theorem, example 6. 41 applying green ’ s theorem for flux across a circle let c be a circle of radius r centered at the origin ( figure 6. 39 ) and let calculate the flux across c. figure 6. 39 curve c is a circle of radius r c... | openstax_calculus_volume_3_-_web | [
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6. 36 calculate the flux of across a unit circle oriented counterclockwise. example 6. 43 applying green ’ s theorem for water flow across a rectangle water flows from a spring located at the origin. the velocity of the water is modeled by vector field m / sec. find the amount of water per second that flows across the ... | openstax_calculus_volume_3_-_web | [
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the property for any point in the domain of g. ( stream functions play the same role for source - free fields that potential functions play for conservative fields. ) 4. example 6. 44 finding a stream function verify that rotation vector field is source free, and find a stream function for f. solution note that the dom... | openstax_calculus_volume_3_-_web | [
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6. 4 • green ’ s theorem 647 satisfies laplace ’ s equation laplace ’ s equation is foundational in the field of partial differential equations because it models such phenomena as gravitational and magnetic potentials in space, and the velocity potential of an ideal fluid. a function that satisfies laplace ’ s equation... | openstax_calculus_volume_3_-_web | [
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6. 38 is the function harmonic? green ’ s theorem on general regions green ’ s theorem, as stated, applies only to regions that are simply connected — that is, green ’ s theorem as stated so far cannot handle regions with holes. here, we extend green ’ s theorem so that it does work on regions with finitely many holes ... | openstax_calculus_volume_3_-_web | [
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given a positive orientation. the boundary of each simply connected region and is positively oriented. if f is a vector field defined on d, then green ’ s theorem says that therefore, green ’ s theorem still works on a region with holes. to see how this works in practice, consider annulus d in figure 6. 45 and suppose ... | openstax_calculus_volume_3_-_web | [
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6. 4 • green ’ s theorem 649 figure 6. 45 shows a path that traverses the boundary of d. notice that this path traverses the boundary of region returns to the starting point, and then traverses the boundary of region furthermore, as we walk along the path, the region is always on our left. notice that this traversal of... | openstax_calculus_volume_3_-_web | [
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connected because the only hole in the domain of f is at the origin. we showed in our discussion of cross - partials that f satisfies the cross - partial condition. if we restrict the domain of f just to c and the region it encloses, then f with this restricted domain is now defined on a simply connected domain. since ... | openstax_calculus_volume_3_-_web | [
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6. 39 calculate integral where d is the annulus given by the polar inequalities and student project measuring area from a boundary : the planimeter figure 6. 47 this magnetic resonance image of a patient ’ s brain shows a tumor, which is highlighted in red. ( credit : modification of work by christaras a, wikimedia com... | openstax_calculus_volume_3_-_web | [
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##s and the roller moves back and forth ( but does not rotate ). 652 6 • vector calculus access for free at openstax. org figure 6. 48 ( a ) a rolling planimeter. the pivot allows the tracer arm to rotate. the roller itself does not rotate ; it only moves back and forth. ( b ) an interior view of a rolling planimeter. ... | openstax_calculus_volume_3_-_web | [
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the tracer ( the length of the tracer arm ). figure 6. 49 mathematical analysis of the motion of the planimeter. 1. explain why the total distance through which the wheel rolls the small motion just described is 2. show that | openstax_calculus_volume_3_-_web | [
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6. 4 • green ’ s theorem 653 3. use step 2 to show that the total rolling distance of the wheel as the tracer traverses curve c is total wheel roll now that you have an equation for the total rolling distance of the wheel, connect this equation to green ’ s theorem to calculate area d enclosed by c. 4. show that 5. ass... | openstax_calculus_volume_3_-_web | [
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oriented in the counterclockwise direction 149. where c is the boundary of the region lying between the graphs of and oriented in the counterclockwise direction 150. where c is the boundary of the region lying between the graphs of and oriented in the counterclockwise direction 151. where c consists of line segment c1 ... | openstax_calculus_volume_3_-_web | [
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6. 4 • green ’ s theorem 655 158. calculate integral along triangle c with vertices ( 0, 0 ), ( 1, 0 ) and ( 1, 1 ), oriented counterclockwise, using green ’ s theorem. 159. evaluate integral where c is the curve that follows parabola then the line from ( 2, 4 ) to ( 2, 0 ), and finally the line from ( 2, 0 ) to ( 0, 0... | openstax_calculus_volume_3_-_web | [
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##ockwise. 173. evaluate where c is any piecewise, smooth simple closed curve enclosing the origin, traversed counterclockwise. | openstax_calculus_volume_3_-_web | [
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6. 4 • green ’ s theorem 657 for the following exercises, use green ’ s theorem to calculate the work done by force f on a particle that is moving counterclockwise around closed path c. 174. 175. c : boundary of a triangle with vertices ( 0, 0 ), ( 5, 0 ), and ( 0, 5 ) 176. evaluate where c is a unit circle oriented in... | openstax_calculus_volume_3_-_web | [
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circle oriented counterclockwise. 658 6 • vector calculus access for free at openstax. org 185. use green ’ s theorem to evaluate line integral where c is circle oriented in the counterclockwise direction. 186. use green ’ s theorem to evaluate line integral where c is ellipse and is oriented in the counterclockwise di... | openstax_calculus_volume_3_-_web | [
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6. 4 • green ’ s theorem 659 197. use green ’ s theorem to evaluate integral where and c is a unit circle oriented in the counterclockwise direction. 198. use green ’ s theorem in a plane to evaluate line integral where c is a closed curve of a region bounded by oriented in the counterclockwise direction. 199. calculat... | openstax_calculus_volume_3_-_web | [
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6. 5 divergence and curl learning objectives 6. 5. 1 determine divergence from the formula for a given vector field. 6. 5. 2 determine curl from the formula for a given vector field. 6. 5. 3 use the properties of curl and divergence to determine whether a vector field is conservative. in this section, we examine two im... | openstax_calculus_volume_3_-_web | [
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the divergence of f is defined by note the divergence of a vector field is not a vector field, but a scalar function. in terms of the gradient operator divergence can be written symbolically as the dot product note this is merely helpful notation, because the dot product of a vector of operators and a vector of functio... | openstax_calculus_volume_3_-_web | [
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6. 5 • divergence and curl 661 figure 6. 50 ( a ) vector field has zero divergence. ( b ) vector field also has zero divergence. by contrast, consider radial vector field in figure 6. 51. at any given point, more fluid is flowing in than is flowing out, and therefore the “ outgoingness ” of the field is negative. we ex... | openstax_calculus_volume_3_-_web | [
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when working with magnetic fields. a magnetic field is a vector field that models the influence of electric currents and magnetic materials. physicists use divergence in gauss ’ s law for magnetism, which states that if b is a magnetic field, then in other words, the divergence of a magnetic field is zero. example 6. 4... | openstax_calculus_volume_3_-_web | [
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6. 5 • divergence and curl 663 solution if f were magnetic, then its divergence would be zero. the divergence of f is and therefore f cannot model a magnetic field ( figure 6. 52 ). figure 6. 52 the divergence of vector field is one, so it cannot model a magnetic field. another application for divergence is detecting w... | openstax_calculus_volume_3_-_web | [
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6. 41 let be a rotational field where a and b are positive constants. is f source free? recall that the flux form of green ’ s theorem says that where c is a simple closed curve and d is the region enclosed by c. since green ’ s theorem is sometimes written as therefore, green ’ s theorem can be written in terms of div... | openstax_calculus_volume_3_-_web | [
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6. 42 for vector field find all points p such that the amount of fluid flowing in to p equals the amount of fluid flowing out of p. curl the second operation on a vector field that we examine is the curl, which measures the extent of rotation of the field about a point. suppose that f represents the velocity field of a... | openstax_calculus_volume_3_-_web | [
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the leaf to rotate. if the curl is zero, then the leaf doesn ’ t rotate as it moves through the fluid. definition if is a vector field in ℝ and all exist, then the curl of f is defined by note that the curl of a vector field is a vector field, in contrast to divergence. the definition of curl can be difficult to rememb... | openstax_calculus_volume_3_-_web | [
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6. 5 • divergence and curl 667 example 6. 53 finding the curl of a two - dimensional vector field find the curl of solution notice that this vector field consists of vectors that are all parallel. in fact, each vector in the field is parallel to the x - axis. this fact might lead us to the conclusion that the field has... | openstax_calculus_volume_3_-_web | [
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by field 668 6 • vector calculus access for free at openstax. org example 6. 54 determining the spin of a gravitational field show that a gravitational field has no spin. solution to show that f has no spin, we calculate its curl. let and then, since the curl of the gravitational field is zero, the field has no spin. | openstax_calculus_volume_3_-_web | [
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6. 44 field models the flow of a fluid. show that if you drop a leaf into this fluid, as the leaf moves over time, the leaf does not rotate. using divergence and curl now that we understand the basic concepts of divergence and curl, we can discuss their properties and establish relationships between them and conservati... | openstax_calculus_volume_3_-_web | [
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6. 45 is it possible for to be the curl of a vector field? with the next two theorems, we show that if f is a conservative vector field then its curl is zero, and if the domain of f is simply connected then the converse is also true. this gives us another way to test whether a vector field is conservative. theorem 6. 1... | openstax_calculus_volume_3_-_web | [
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what is the divergence of a gradient? if is a function of two variables, then we abbreviate this “ double dot product ” as this operator is called the laplace operator, and in this notation laplace ’ s equation becomes therefore, a harmonic function is a function that becomes zero after taking the divergence of a gradi... | openstax_calculus_volume_3_-_web | [
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6. 5 • divergence and curl 671 section 6. 5 exercises for the following exercises, determine whether the statement is true or false. 206. if the coordinate functions of ℝ ℝ have continuous second partial derivatives, then equals zero. 207. 208. all vector fields of the form are conservative. 209. if then f is conservat... | openstax_calculus_volume_3_-_web | [
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( 0, 0, 3 ) 255. let for what value of a is f conservative? 256. given vector field on domain ℝ is f conservative? 257. given vector field on domain ℝ is f conservative? 258. find the work done by force field in moving an object from p ( 0, 1 ) to q ( 2, 0 ). is the force field conservative? 259. compute divergence 260... | openstax_calculus_volume_3_-_web | [
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6. 5 • divergence and curl 673 for the following exercises, consider a rigid body that is rotating about the x - axis counterclockwise with constant angular velocity if p is a point in the body located at the velocity at p is given by vector field 261. express f in terms of i, j, and k vectors. 262. find 263. find in t... | openstax_calculus_volume_3_-_web | [
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6. 6 surface integrals learning objectives 6. 6. 1 find the parametric representations of a cylinder, a cone, and a sphere. 6. 6. 2 describe the surface integral of a scalar - valued function over a parametric surface. 6. 6. 3 use a surface integral to calculate the area of a given surface. 6. 6. 4 explain the meaning ... | openstax_calculus_volume_3_-_web | [
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need to parameterize c. in a similar way, to calculate a surface integral over surface s, we need to parameterize s. that is, we need a working concept of a parameterized surface ( or a parametric surface ), in the same way that we already have a concept of a parameterized curve. a parameterized surface is given by a d... | openstax_calculus_volume_3_-_web | [
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6. 6 • surface integrals 675 visualize two families of curves that lie on s. in the first family of curves we hold u constant ; in the second family of curves we hold v constant. this allows us to build a “ skeleton ” of the surface, thereby getting an idea of its shape. first, suppose that u is a constant k. then the ... | openstax_calculus_volume_3_-_web | [
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6. 47 describe the surface with parameterization ∞ ∞ it follows from example 6. 58 that we can parameterize all cylinders of the form if s is a cylinder given by equation then a parameterization of s is ∞ ∞ we can also find different types of surfaces given their parameterization, or we can find a parameterization when... | openstax_calculus_volume_3_-_web | [
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6. 49 give a parameterization for the portion of cone lying in the first octant. we have discussed parameterizations of various surfaces, but two important types of surfaces need a separate discussion : spheres and graphs of two - variable functions. to parameterize a sphere, it is easiest to use spherical coordinates.... | openstax_calculus_volume_3_-_web | [
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just a point. for example, consider curve parameterization the image of this parameterization is simply point which is not a curve. notice also that the fact that the | openstax_calculus_volume_3_-_web | [
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6. 6 • surface integrals 679 derivative is zero indicates we are not actually looking at a curve. analogously, we would like a notion of regularity for surfaces so that a surface parameterization really does trace out a surface. to motivate the definition of regularity of a surface parameterization, consider parameteri... | openstax_calculus_volume_3_-_web | [
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in figure 6. 64 ( a ) can be parameterized by ( we can use technology to verify ). notice that vectors exist for any choice of u and v in the parameter domain, and the k component of this vector is zero only if or if or then the only choices for u that make the j component zero are or but, these choices of u do not mak... | openstax_calculus_volume_3_-_web | [
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6. 50 is the surface parameterization smooth? surface area of a parametric surface our goal is to define a surface integral, and as a first step we have examined how to parameterize a surface. the second step is to define the surface area of a parametric surface. the notation needed to develop this definition is used t... | openstax_calculus_volume_3_-_web | [
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0.01812882535159588,
0.0008246465004049242,
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0.034297727048397064,
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0.013711579144001007,
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6. 6 • surface integrals 681 to m and j ranges from 1 to n so that d is subdivided into mn rectangles. this division of d into subrectangles gives a corresponding division of surface s into pieces choose point in each piece point corresponds to point in the parameter domain. note that we can form a grid with lines that... | openstax_calculus_volume_3_-_web | [
0.0036148468498140574,
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0.01985052227973938,
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0.012472188100218773,
0.008109873160719872,
0.0006430540233850479,
0.004480946343392134,
0.08041422069072723,
0.04269522801041603,
0.01840384490787983,
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0... |
by and in other words, we scale the tangent vectors by the constants and to match the scale of the original division of rectangles in the parameter domain. therefore, the area of the parallelogram used to approximate the area of is varying point over all pieces and the previous approximation leads to the following defi... | openstax_calculus_volume_3_-_web | [
0.022061672061681747,
0.0137617327272892,
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0.002666444517672062,
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0.03807835280895233,
0.0607464425265789,
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-0.03297758847475052,
0.01... |
6. 6 • surface integrals 683 figure 6. 68 the right circular cone with radius r = kh and height h has parameterization with a parameterization in hand, we can calculate the surface area of the cone using equation 6. 18. the tangent vectors are and therefore, the magnitude of this vector is by equation 6. 18, the surfac... | openstax_calculus_volume_3_-_web | [
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0.01430367399007082,
0.0311740729957819,
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0.002810513600707054,
0.008230684325098991,
0.009040968492627144,
0.017813464626669884,
0.0027427750173956156,
0.04671110585331917,
0.05095212161540985,
0.0245286226272583,
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0.0024750... |
6. 6 • surface integrals 685 using the same techniques. let be a positive single - variable function on the domain and let s be the surface obtained by rotating about the x - axis ( figure 6. 69 ). let be the angle of rotation. then, s can be parameterized with parameters x and by figure 6. 69 we can parameterize a sur... | openstax_calculus_volume_3_-_web | [
0.009919770993292332,
0.01550334133207798,
0.022572612389922142,
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0.004216240253299475,
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0.05863582342863083,
0.061399124562740326,
0.028821812942624092,
-0.012378597632050514,
-0.... |
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