Obtain the circulation integral of the field F = (x+eªseny)î + (x + e* cosy) in the curve C defined by the right loop of the lemniscate p² = cos20 in the opposite direction to that of the needles of the clock.
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- Consider the conservative vector field given by:F (x, y) = (x - ycos (x), y - sin (x))A potential function that generates the vector field F corresponds to:Calculate the line integral in scalar field ∫C (x2y2-√x)dy is the arc of the curve y=√x of (1,1) to (4,2)A vector field is given by F = ((1 + xy)exy, x2exy). Compute the integral of the vector field over the curve r(t) = (cos(t), 2sin(t)), where t ranges from zero to π/2.
- Calculate the circulation of the field F around the closed curve C. Circulation means line integralF = x 3y 2 i + x 3y 2 j; curve C is the counterclockwise path around the rectangle with vertices at (0,0),(3,0).(3,2) and (0.2)Verify the claim made in the given problem below section d by showing that the net outward flux of F across C is positive.(Hint: If you use Green’s Theorem to evaluate the integral ∫C ƒ dy - g dx,convert to polar coordinates.) Divergence from a graph To gain some intuition about the divergence,consider the two-dimensional vector field F = ⟨ƒ, g⟩ = ⟨x2, y⟩ and a circle C of radius 2 centered at the origin (see figure).a. Without computing it, determine whether the two-dimensional divergence is positive or negative at the point Q(1, 1). Why?b. Confirm your conjecture in part (a) by computing the two-dimensional divergence at Q. c. Based on part (b), over what regions within the circle is the divergence positive and over what regions within the circle is the divergence negative?d. By inspection of the figure, on what part of the circle is the flux across the boundary outward? Is the net flux out…Find the outward flux of the field F= 8xyi + 8yzj + 8xzk across the surface of the cube cut from the first octant by the planes x=a, y=a, z=a
- 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.)Calculate the divergence of the scalar field F(x,y)=3x2+siny.