Concept explainers
Curvature of Plane Parametric Curves The curvature of a plane parametric curve
47. Find the curvature of the curve
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CALCULUS,EARLY TRANSCENDENTALS-ACCESS
- Curves on surfaces Verify that the curve r(t) lies on the given surface. Give the name of the surface. r(t) = (t cos t) i + (t sin t) j + t k; x2 + y2 = z2arrow_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 = ⟨ -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_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_forward
- Flux 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_forwardDierentiable 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_forwardWalking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane. a. In each case, find z'(t). b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). z = x2 + 4y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2πarrow_forward
- PICTURE FOR SUPPORT Use the stoke’s theorem to evaluate the anti derivative where F = and S is the surface of the solid bounded by the paraboloid Z= 1-x^2-y^2 and Z=0, C is the bounded of S with counter clockwise orientation .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 cap of the sphere x2 + y2 + z2 = 16, for 2√2 ≤ 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 frustum of the cone z2 = x2 + y2, for 2 ≤ z ≤ 8arrow_forward
- Walking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane. a.In each case, find z’(t). b.Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). z = 4x2 - y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2πarrow_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(a) Show that at every point on the curve r(u) = (e^u cos u, e^u sin u, e^u ) , the angle between the unit tangent vector and the z-axis is the same. Show that this is also true for the principal normal vector. (b) Give a parametric representation of the level surface e^(xyz) = 1. (c) Find the equation of the tangent plane to the surface z = 3 e x−y at the point (4, 4, 3).arrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning