Verify Stokes's theorem for the vector field: B =r cos of+sin o a. By evaluating §, B dl over the semicircular contour shown below b. By evaluating ,(V x B) ds over the semicircular contour shown below
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- Consider the following function. Solve the divergence of this vector field.For vector field v(x, y) = (-xy, y), find all points P such that the amount of fluid flowing in to Pequals the amount of fluid flowing out of P. Select the correct answer below: O At all points P O At all points P, where y s0 O At all points P, where y = 1 O At all points P, where y = xCalculate the flux of vector field F = (xy°, x²y) across the circle of radius 1 centered at coordinates (0, –1).
- 2.6Suppose a unidirectional vector field E exists across an enclosed surface A as shown below. which among the following is/are true? 153 E The divergence volume integral value will only be equal to 0 if E is a uniform field. O The divergence volume integral value will be positive if the magnitude of E is increasing along the positive z-hat direction. The divergence volume integral value will always be equal to 0. Only the top and bottom cylindrical phases have non-zero divergence surface integral values.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).
- Consider one-surface hyperboloid is given by the equation r? y? 22 1. Parametrize the surface ř(v, 0) and find the normal vector.Compute the flux of the vector field F = 2zk through S, the upper hemisphere of radius 5 centered at the origin, oriented outward. flux =Calculate the flux of the vector field F(x, y, z) = 5i + 5j + zk through the closed circular cylinder of radius 4 centered about the z-axis for -6 ≤ z < 6, oriented away from the z-axis. Note: a closed cylinder has a top and a bottom. Flux = SS F.dĀ=