Verify Stokes' Theorem for fas F. dr, where F = -yr²i+ry2j + zk, and S is the part of the sphere x² + y² +2² = 4 that lies in the upper half-space z ≥ 0.
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- Consider the special shape pictured in the diagram below. It is a cylinder, centered on the origin with its axis oriented along z, and it has been partially hollowed to leave two cone-shaped cavities at the top and bottom of the cylinder. The radius of the object is a, its height is 2a, and the solid part of the object (the shaded region that is visible in the rightmost panel of the illustration above, which shows a drawing of the cross-section of the object) has a uniform volume charge density of po. Assume that the object is spinning counter clockwise about its cylinder axis at an angular frequency of w. Which of the following operations is part of the calculation of the magnitude of the current density that is associated with the motion of the rotating object as a function of r (select all that apply)?Let C be the region between the circles x² + y² = 1 and x² + y² = 4. Let 2 2 2 F(x, y) = (²3³-²3V², 30³² Compute the circulation across C. Flux = y³, x³ - 3x²Calculate the flux of vector field F = (xy°, x²y) across the circle of radius 1 centered at coordinates (0, –1).
- Calculate the flux of the given vector field by evaluating the line integral directly alongthe given curve for the below parts:(a) The vector field is ⃗ F = (x − y)⃗i + x⃗j. The curve is the circle x^2 + y^2 = 1in the xy-plane. Use the parameterization x = cos t and y = sin t.(b) The vector field is ⃗ F = (x − 1)⃗i + y⃗j. The curve is a circle of radius 3centered at (1, 1). The parametric form of this circle is⃗r = (1 + 3 cos t)⃗i + (1 + 3 sin t)⃗j, 0 ≤ t ≤ 2π(c) The vector field is ⃗F = x⃗i + y⃗j. The curve is the line segment from thepoint (0, 1) to the point (1, 3).This would be all of g. how do you get just the z component of the gravitational fieldDon't use chatgpt will upvote