Curvature of Plane Parametric Curves The curvature of a plane parametric curve
48. Find the curvature of the curve
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CALCULUS,EARLY TRANSCENDENTALS-ACCESS
- Parametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The portion of the cylinder x2 + y2 = 9 in the first octant, for 0 ≤ z ≤ 3arrow_forwardFlux of the radial field Consider the radial vector field F = ⟨ƒ, g, h⟩ = ⟨x, y, z⟩. Is the upward flux of the field greater across the hemisphere x2 + y2 + z2 = 1, for z ≥ 0, or across the paraboloid z = 1 - x2 - y2, for z ≥ 0?Note that the two surfaces have the same base in the xy-plane and the same high point (0, 0, 1). Use the explicit description for the hemisphere and a parametric description for the paraboloid.arrow_forwardSurface integrals using a parametric description Evaluate the surface integral ∫∫S ƒ dS using a parametric description of the surface. ƒ(x, y, z) = y, where S is the cylinder x2 + y2 = 9, 0 ≤ z ≤ 3arrow_forward
- Surface integrals using a parametric description Evaluate the surface integral ∫∫S ƒ dS using a parametric description of the surface. ƒ(x, y, z) = x, where S is the cylinder x2 + z2 = 1, 0 ≤ y ≤ 3arrow_forwardParametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The cone z2 = 4(x2 + y2), for 0 ≤ z ≤ 4arrow_forwardParametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The cap of the sphere x2 + y2 + z2 = 16, for 2√2 ≤ z ≤ 4arrow_forward
- Surface integrals of vector fields Find the flux of the following vector field across the given surface with the specified orientation. You may use either an explicit or a parametric description of the surface. F = ⟨ -y, x, 1⟩ across the cylinder y = x2, for 0 ≤ x ≤ 1,0 ≤ z ≤ 4; normal vectors point in the general direction of thepositive y-axis.arrow_forwardSurface integrals of vector fields Find the flux of the following vector field across the given surface with the specified orientation. You may use either an explicit or a parametric description of the surface. F = ⟨e-y, 2z, xy⟩ across the curved sides of the surfaceS = {(x, y, z): z = cos y, | y | ≤ π, 0 ≤ x ≤ 4}; normal vectorspoint upward.arrow_forwardPath of steepest descent Consider each of the following surfaces and the point P on the surface. a. Find the gradient of ƒ. b. Let C’ be the path of steepest descent on the surface beginning at P, and let C be the projection of C’ on the xy-plane. Find an equation of C in the xy-plane. c. Find parametric equations for the path C’ on the surface. ƒ(x, y) = 4 - x2 - 2y2 (a paraboloid); P(1, 1, 1)arrow_forward
- Dierentiable curves with zero torsion lie in planes That a sufficiently di¡erentiable curve with zero torsion lies in a plane is a special case of the fact that a particle whose velocity remains perpendicular to a fixed vector C moves in a plane perpendicular to C. This, in turn, can be viewed as the following result. Suppose r(t) = ƒ(t)i + g(t)j + h(t)k is twice di¡erentiable for all t in an interval 3a, b4 , that r = 0 when t = a, and that v # k = 0 for all t in 3a, b4 . Show that h(t) = 0 for all t in 3a, b4 . (Hint: Start with a = d2r/dt2 and apply the initial conditions in reverse order.)arrow_forwardSurface integrals of vector fields Find the flux of the following vector field across the given surface with the specified orientation. You may use either an explicit or a parametric description of the surface. F = ⟨x, y, z⟩ across the slanted surface of the cone z2 = x2 + y2,for 0 ≤ z ≤ 1; normal vectors point upward.arrow_forwardParametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The frustum of the cone z2 = x2 + y2, for 2 ≤ z ≤ 8arrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning