MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable CalculusFall 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) -xi -yj 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/r2. 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 F .dr. Use any convenient parametrization of C, unless one is specified. . Begin by writing L 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: the quarter of the unit circle running from (0,l) to (1,O). C) F = y i -x j ; C: the triangle with vertices at (0, O), (0, I), (1, 0), oriented clockwise. d) F = y i ; C is the ellipse x = 2 cost, y = sin t, oriented counterclockwise. e) F = 6y i + x j ; C is the curve x = t2, y = t3, running from (1,l) to (4,8). f) F = (x +y) i + xyj ; C is the broken line running from (0,O) to (0,2) to (1,2). 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. a) F = xi +yj ; C is the counterclockwise circle, center at (0, 0), radius a. b) F = y i -x j ; C is the counterclockwise circle, center at (O,O), radius a.2 E. 18.02 EXERCISES 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 y2 = x, between (1, -1) and (1, I), directed upwards. a) Calculate F = V f. b) Calculate the integral Jc F .dr three different ways: (i) directly; (ii) by using path-independence to replace C by a simpler path (iii) by using the Fundamental Theorem for line integrals. 4C-2 Let f (x, y) = sexy, and C be the path y = l/x from (1,l) to (0, m). a) Calculate F = Vf b) Calculate the integral Jc F .dr (i) directly; (ii) by using the Fundamental Theorem for line integrals. 4C-3 Let f (x, y) = sin x cosy . a) Calculate F = Vf. b) What is the maximum value J a cF . dr can have over all possible paths C in the plane? Give path C for which this maximum value is attained. 4C-4* The Fundamental Theorem for line integrals should really be called the First Fun- damental 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 1in this section. It runs: If M dx + N dy is path-independent, and f (x, y) = F.dr, then Vf = Mi +Nj The conclusion says fx = M, fy = 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. a) F = (y2 + 25) i + axy j b) F=eX+y((x+a)i +xj) 4C-6 Decide which of the following differentials is exact. For each one that is exact, express it in the form df . a) ydx-xdy b) y(2x+y)dx+x(2y+x)dy ydx -xdyc)* xsinydx+ ysinxdy (X+Y)~4. LINE INTEGRALS IN THE PLANE 4D. Green's Theorem 4D-1 LFor each of the following fields F and closed positively oriented curves C, evalu-F . dr both directly, as a line integral, and also by applying Green's theorem and ate calculating a double integral. a)F: 2yi+xj, C: x2+y2=1 b) F : x2(i + j ) C: rectangle joining (0,0), (2,0), (O,l), (2,l) c)F: xyi+y2j, C:y=x2andy=x, O<x<l 4D-2 Show that 4x3ydx +x4dy = 0 for all closed curves C. !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 bxy2dx+ (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 jcy(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 at (O,O), directed counterclockwise. b) C is the line segment running from (-1,0) to (1,O) c) C is the line running from (0,O) to (1,O) . 4E-2 Let F be the constant vector field i + j. Where would you place a directed line segment C of length one in the plane so that the flux across C would be a) maximal b) minimal c) zero d) -1 e) what would the maximal and minimal values be? 4E-3 Let F = x2 i +xyj . Evaluate Jc F .n ds if C is given by r(t) = (t + 1) i +t2j , where 0 < t < 1; …
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