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 4 Line Integrals in the Plane 4A Plane Vector Fields 4A 1 Describe geometrically how the vector fields determined by each of the following vector functions looks Tell for each what the largest region in which F is continuously differentiable is a a i bj a b constants b x i y j 4A 2 Write down the gradient field Vw for each of the following a w ax by b w lnr c w f r 4A 3 Write down an explicit expression for each of the following fields a Each vector has the same direction and magnitude as i 2j b The vector at x y is directed radially in towards the origin with magnitude r2 c The vector at x y is tangent to the circle through x y with center at the origin clockwise direction magnitude l r 2 d Each vector is parallel to i j but the magnitude varies 4A 4 The electromagnetic force field of a long straight wire along the z axis carrying a uniform current is a two dimensional field tangent to horizontal circles centered along the z axis in the direction given by the right hand rule thumb pointed in positive z direction and with magnitude k r Write an expression for this field 4B Line Integrals in the Plane 4B 1 For each of the fields F and corresponding curve C or curves Ci evaluate L F dr Use any convenient parametrization of C unless one is specified Begin by writing the r integral in the differential form b M dx N dy a F x2 y i 2x j C1 and C2 both run from 1 O to 1 O C1 the x axis C2 the parabola y 1 x2 b F xy i x2j C F y i x j d F y i C the quarter of the unit circle running from 0 l to 1 O C the triangle with vertices at 0 O 0 I 1 0 oriented clockwise C is the ellipse x 2 cost y sin t oriented counterclockwise x j C is the curve x t2 y t3 running from 1 l to 4 8 F x y i xy j C is the broken line running from 0 O to 0 2 to 1 2 e F 6y i f 4B 2 For the following fields F and curves C evaluate Jc F dr without any formal calculation appealing instead to the geometry of F and C yj C is the counterclockwise circle center at 0 0 radius a b F y i x j C is the counterclockwise circle center a t O O radius a a F x i E 18 02 EXERCISES 2 4B 3 Let F i j How would you place a directed line segment C of length one so that the value of Jc F dr would be a a maximum b a minimum c zero d what would the maximum and minimum values of the integral be 4C Gradient Fields and Exact Differentials 4C 1 Let f x y x3 y3 and C be y 2 x between 1 1 and 1 I directed upwards a Calculate F V f b Calculate the integral Jc F d r three different ways i directly ii by using path independence t o replace C by a simpler path iii by using the Fundamental Theorem for line integrals 4C 2 Let f x y s e x y and C be the path y l x from 1 l t o 0 m a Calculate F Vf b Calculate the integral Jc F d r i directly ii by using the Fundamental Theorem for line integrals 4C 3 Let f x y sin x cosy a Calculate F V f b What is the maximum value J F d r can have over all possible paths C in the c plane Give a path C for which this maximum value is attained 4C 4 The Fundamental Theorem for line integrals should really be called the First Fundamental Theorem There is an analogue for line integrals of the Second Fundamental Theorem also where you first integrate then differentiate it provides the justification for Method 1 in this section It runs If M dx N dy is path independent and f x y F d r then Vf M i N j The conclusion says f x M f y N prove the second of these Hint use the Second Fundamental Theorem of Calculus 4C 5 For each of the following tell for what value of the constants the field will be a gradient field and for this value find the corresponding mathematical potential function b F e X y x a i x j a F y2 25 i axy j 4C 6 Decide which of the following differentials is exact For each one that is exact express it in the form df a y d x x d y b y 2 x y d x x 2 y x d y ydx xdy c x s i n y d x ysinxdy X Y 4 LINE INTEGRALS IN T H E PLANE 4D Green s Theorem 4D 1 For each of the following fields F and closed positively oriented curves C evaluF d r both directly as a line integral and also by applying Green s theorem and L ate calculating a double integral a F 2yi xj b F x 2 i j C x2 y2 1 C rectangle joining 0 0 2 0 O l 2 l c F xyi y2j 4D 2 Show that c C y x2andy x O x l 4x3ydx x4dy 0 for all closed curves C 4D 3 Find the area inside the hypocycloid x2I3 y2I3 1 by using Green s theorem This curve can be parametrized by x cos3 8 y sin3 8 between suitable limits on 8 4D 4 Show that the value of y3dx x3dy around any positively oriented simple closed curve C is always positive 4D 5 Show that the value of b xy2dx x2y 2x dy around any square C in the xy plane depends only on the size of the square and not upon its position 4D 6 Show that b x2y dx xy2 dy 0 around any simple closed curve C 4D 7 Show that the value of jc y y 3 dx 2xy dy around any equilateral triangle C depends only on the size of the triangle and not upon its position in the xy plane 4E Two dimensional Flux 4E 1 Let F y i x j Recalling the interpretation of this field example 4 V1 5 or just by remembering how it looks geometrically evaluate with little or no calculation the flux integral Jc F n ds where a C is a circle of radius a centered a t O O directed counterclockwise b C is the line segment running from 1 0 to 1 O c C is …
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