Finding Work in a Conservative Force Field In Exercises 19-22, (a) show that
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- Flux of a vector field? Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.arrow_forwardCirculation on a half-annulus Consider the vector field F = ⟨y2, x2⟩on the half-annulus R = {(x, y): 1 ≤ x2 + y2 ≤ 9, y ≥ 0}, whose boundary is C. Find the circulation on C, assuming it has the orientation shown.arrow_forwardFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨y, x, x - y⟩ on ℝ3arrow_forward
- Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨ez, ez, ez (x - y)⟩ on ℝ3arrow_forwardFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨yz, xz, xy⟩ on ℝ3arrow_forwardFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨y + z, x + z, x + y⟩ on ℝ3arrow_forward
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