MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms V11 Line Integrals in Space 1 Curves in space In order to generalize to three space our earlier work with line integrals in the plane we begin by recalling the relevant facts about parametrized space curves In 3 space a vector function of one variable is given as It is called continuous or differentiable or continuously differentiable if respectively x t y t and z t all have the corresponding property By placing the vector so that its tail is at the origin its head moves along a curve C as t varies This curve can be described therefore either by its position vector function I or by the three parametric equations The curves we will deal with will be finite connected and piecewise smooth this means that they have finite length they consist of one piece and they can be subdivided into a finite number of smaller pieces each of which is given as the position vector of a continuously differentiable function i e one whose derivative is continuous In addition the curves will be oriented or directed meaning that an arrow has been placed on them to indicate which direction is considered to be the positive one The curve is called closed if a point P moving on it always in the positive direction ultimately returns to its starting position as in the accompanying picture The derivative of r t is defined in terms of components by If the parameter t represents time we can think of drldt as the velocity vector v If we let s denote the arclength along C measured from some fixed starting point in the positive direction then in terms of s the magnitude and direction of v are given by vl Ixl ds dirv t t if dsldt if dsldt 0 0 Here t is the unit tangent vector pointing in the positive direction on C You can see from the picture that t is a unit vector since 2 V VECTOR INTEGRAL CALCLUS 2 Line integrals in space Let F M i assumed continuous Nj Pk be a vector field in space We define the line integral of the tangential component of F along an oriented curve C in space in the same way as for the plane We approximate C by an inscribed sequence of directed line segments A r k form the approximating sum then pass to the limit LF k 03 lim x r k A r k k The line integral is calculated just like the one in two dimensions if C is given by the position vector function r t one would write 6 as to 5 t 5 tl Using x y z components In particular if the parameter is the arclength s then 6 becomes since t drlds which shows that the line integral is the integral along C of the tangential component of F As in two dimensions this line integral represents the work done by the field F carrying a unit point mass or charge along the curve C Example 1 Find the work done by the electrostatic force field F y i carrying a positive unit point charge from 1 1 1 to 2 4 8 along a a line segment b the twisted cubic curve r t i t2j t3k zj xk in Solution a The line segment is given parametrically by Jd1 31t 11 dt 31 2 t2 b Here the curve is given by x t y t2 z t3 integral is llt 15 t 5 2 For this curve the line The different results for the two paths shows that for this field the line integral between two points depends on the path V11 LINE INTEGRALS IN SPACE 3 3 Gradient fields and path independence The two dimensional theory developed for line integrals in the plane generalizes easily to three space For the part where no new ideas are involved we will be brief just stating the results and in places sketching the proofs Definition Let F be a continuous vector field in a region D of space The line integral J F dr is called path independent if for any two points P and Q in the region D the value of Jc F dr along a directed curve C lying in D and running from P to Q depends only on the two endpoints and not on C An equivalent formulation is the proof of equivalence is the same as before 8 lQ F dr is path independent F dr 0 for every closed curve C in D Q P Definition Let f x y z be continuously differentiable in a region D The vector field is called the gradient field of f in D Any field of the form V f is called a gradient field T h e o r e m F i r s t fundamental t h e o r e m of calculus for line integrals I f f x y z is continuously differentiable in a region D then for any two points Pl P2 lying in D where the integral is taken along any curve C lying in D and running from PI to P2 In particular the line integral is path independent The proof is exactly the same as before use the chain rule to reduce it to the first fundamental theorem of calculus for functions of one variable There is also an analogue of the second fundamental theorem of calculus the one where we first integrate then differentiate Theorem Second fundamental t h e o r e m of calculus for line integrals Q F dr path independent in a region D and define Let F x y z be continuous and Jp X Y Z JI x y z F dr then O YO O Vf F inD Note that since the integral is path independent no C need be specified in 11 The theorem is proved in your book for line integrals in the plane The proof for line integrals in space is analogous Just as before these two theorems produce the three equivalent statements in D 12 F Vf Q PI F dr path independent Q F d r 0 for any closed C 4 V VECTOR INTEGRAL CALCLUS As in the two dimensional case if F is thought of as a force field then the gradient force fields are called conservative fields since the work done going around any closed path is zero i e energy is conserved If F V f then f is the called the mathematical potential function for F the physical potential function is defined to be f Example 2 Let f x y z x Y Z Calculate F V f and find isthehelixx cost y sint z t O t 7r Solution By differentiating F z i 1 0 O to 1 0 7r Therefore by lo 2yzj x y2 k No direct calculation of the line integral is needed notice Exercises Section 6D 1 F …
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