WebFinal answer. 11. Let S be outward oriented surface consisting of the top half of the sphere x2 +y2 +z2 = 16 and the disc x2 +y2 ≤ 16 at height z = 0. Let F = x2i+z2yj+zy2k be a vector field. Use Stokes theorem to compute ∬ S(∇× F)⋅NdS. WebStoke's theorem states that for a oriented, smooth surface Σ bounded simple, closed curve C with positive orientation that ∬ Σ ∇ × F ⋅ d Σ = ∫ C F ⋅ d r for a vector field F, where ∇ × F denotes the curl of F. Now the surface in question is the positive hemisphere of the unit sphere that is centered at the origin.
6.7 Stokes’ Theorem - Calculus Volume 3 OpenStax
WebStokes’ Theorem Example The following is an example of the time-saving power of Stokes’ Theorem. Ex: Let F~(x;y;z) = arctan(xyz)~i + (x+ xy+ ... the following oriented surfaces S. (a) Sis the unit sphere oriented by the outward pointing normal. (b) Sis the unit sphere oriented by the inward pointing normal. (c) Sis a torus with r= 1, R= 5 ... WebStokes’ Theorem allows us to compute a line integral over a closed curve in space. Stokes’ Theorem: ... Use the Divergence Theorem to evaluate ZZ S F · d S where F = h x + sin z, 2 y + cos x, 3 z + tan y i over the sphere x 2 + y 2 + z 2 = 4. Example 5: Let S be the surface of the solid bounded by the paraboloid z = 4-x 2-y 2 and the xy-plane. fritz rauth obituary
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WebRemember this form of Green's Theorem: where C is a simple closed positively-oriented curve that encloses a closed region, R, in the xy-plane. It measures circulation along the boundary curve, C. Stokes's Theorem generalizes this theorem to more interesting surfaces. Stokes's Theorem For F(x,y,z) = M(x,y,z)i+N(x,y,z)j+P(x,y,z)k, Web1 day ago · Use (a) parametrization; (b) Stokes' Theorem to compute ∮ C F ⋅ d r for the vector field F = (x 2 + z) i + (y 2 + 2 x) j + (z 2 − y) k and the curve C which is the intersection of the sphere x 2 + y 2 + z 2 = 1 with the cone z = x 2 + y 2 in the counterclockwise direction as viewed from above. The force of viscosity on a small sphere moving through a viscous fluid is given by: where: • Fd is the frictional force – known as Stokes' drag – acting on the interface between the fluid and the particle • μ is the dynamic viscosity (some authors use the symbol η) fritz rath