4. Line Integrals in the Plane

# 4. Line Integrals in the Plane - 4 Line Integrals in the...

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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 + b j , a, b constants b) -xi - 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 + 2 j . 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 L F . dr. Use any convenient parametrization of C, unless one is specified. . Begin by writing the r integral in the differential form M dx + N dy. b 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 - x2 j ; 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 + xy j ; 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. F = xi + y j ; 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.

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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 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.
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4. Line Integrals in the Plane - 4 Line Integrals in the...

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