We are given a vector field and a closed path. Vector Field: F(x,y) = (-x, y?) andClosed Path: r(1) = (2 cos(r), -4 sin (r)>Since we have our Path as a vector, we can use the General Formula to find Flux.Use the closed path to convert the vector field into a parametric vector field.(A) F(1) = (4 sin (r), 4 cos?(1))B F(1) = (-2cos (1), 4 sin(r) >© F(1) = (-2 cos (1), 16 sin²(1)>D F(1) = (-4sản (1), 2 cos(r)>Use the closed path to find the "y'(t)" and "x(t)".x'(1) = -2 cos (1)y'(r) = -4 sin (r)x'(1) = -2 sin (1)(B)y'(r) = -4 cos ()x'(1) = 2 cos ()y'(r) = -4 sin (1)D x'(1) = -2 sin (1)y'(1) = 4 cos (r)Now that you have found the parameterized vector field, "y'(t)", and "x'(t)" use them to find the dot product.F(1) - (v'(1), -x'()>(AF(1) - (r'(1). -x'(1)) = 4 sin (t) cos (t) - 16 sin (t) cos (t)(BF(1) - ('(1). -x'(1)) = 8 cos (1) + 32 sin (2)(c)r'(1). -x'(1)) = - 8 cos (1) sin (t)F(1)D F(1) - ('(1). -x'(1)) = -4 cos²(1) + 8 sin²(1)Now that we have found the value of our dot product, we can use it to find the flux of our situation.F(1) - ]dt and intervat: [0, 2×]Flux =O Fo - (v'(). -x'()>]dt = 0.56xFind the divergence of the vector field.ofдуAv.F = -1Bv.F = 5x + 2y2 - 8zO v.F =0O v.F = 4y - 5Set up the triple integral for our path.@ STI v.] de dy dzNow that you have found the Divergence and set up your triple integral, we can use them with the Divergence Theorem to find theFlux of our situation.- SSS [v .F] av@ SSS [v . F] av = 0® SS[ [v F] av = 15© J]/ [v F] a = - 125xO SS[[v F] av = 75

According to question,Given vector field,Fx, y⇀ = -x, y2,Closed path,rt⇀ = 2 cos t, -4 sin t

Answered: F(1) = (-4 sin(1), 2 cos(1) ) Use the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 78E

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Conservative fields Use Stokes’ Theorem to find the circulationof the vector field F = ∇(10 - x2 + y2 + z2) around anysmooth closed curve C with counterclockwise orientation.
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a) Calculate the line integral of the vector field F(x, y) = yi − 5xj from the point (0, 3) to the point (3, 0)(i) along the connecting line C1 between the points.(ii) along the arc C2 (shorter or quarter circle) of the circle centered at the origin.b) Does the vector field F have a potential?(The ratio of answers is π/2.)
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You are given the velocity field: v = y2i - x2j + z2k Evaluate the divergence and curl of this field at the point (2, 1, 1) and use your answers to select the 2 options that describe the behaviour of in the immediate vicinity of (2, 1, 1). A. A net outward flow B. No net inflow or outflow C. A net inward flow D. A flow corresponding to a clockwise rotation about the z -axis when viewed from the positive z -axisE. A flow corresponding to an anticlockwise rotation about the z -axis when viewed from the positive z -axisF. A flow corresponding to no rotation about the z -axis

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