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Work In Exercises 25-28, use Green’s Theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C.
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Calculus: Early Transcendental Functions
- Use Green's Theorem to find the work done by the force F(x, y)= x(x + y)i + xy2j in moving a particle from the origin along the x-axis to (2, 0), then along the line segment to (0, 2), and then back to the origin along the y-axis.arrow_forwardConsiser the position-dependent force F = (5 z)i + (2 y² + 5)j+ (5 æy+ 3xz+ 5x + 3z)k, acting on a particle. a. Find the work done, W, by the force as the particle moves from the point (x, y, z) = (0,0,3) to the point (x, y, z) = (2, 1, 7) along the following paths: i. the path defined by the position vector r = 2 ti +t°j+ (3+4t)k, for 0 < t < 1; W = sin (a) f Ω a ii. the path defined by the straight line x = 2 y, z = 2 x + 3. W =arrow_forwardUse Green's Theorem to find the work done by the force field F(x, y) = Vgi + Væj on a particle that moves along the stated path. The particle moves counterclockwise one time around the closed curve given by the equations y = 0, x = 7, and y = 9 NOTE: Enter the exact answer. Warrow_forward
- Using Green's Theorem, find the outward flux of F across the closed curve C.F = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (6, 0), and (0, 6) a) 216 b) 72 c) 0 d) 36arrow_forwardConsider the vector-valued function r(t) = cos ti + sin tj + In(cos t)k. (a) Find the vectors T, N, and B of r at the point P = (1,0,0). (b) Find the tangential and normal components of the acceler- ation of r at the point P = (1,0,0).arrow_forwardParametrize the intersection of the surfaces y – z? = x – 8, y? + z? = 36 using t = y as the parameter (two vector functions are needed). (Use symbolic notation and fractions where needed.) x (t) = y(t) = z (t) = +arrow_forward
- An airplane took off from point (3, 0, 0) at an initial velocity, v(0)= 3j with an acceleration vector of a(t) = (-3 cos t)i + (-3 sin t)j + 2k. Determine the vector function, r(t), to represent the path of the airplane as a function of t.arrow_forwardAssume that an object is moving along a parametric curve and the three vector function. T (t), N(t), and B (t) all exist at a particular point on that curve. CIRCLE the ONE statement below that MUST BE TRUE: (a) B. T=1 (b) T x B = N (B is the binormal vector.) v (t) (c) N (t) = |v (t)| (d) N (t) always points in the direction of velocity v (t). (e) a (t) lies in the same plane as T (t) and N (t).arrow_forwardSketch the vector field. F(x, y) =〈y, 1〉arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage