Graphing a Parametric Surface In Exercises 13—16, use a computer algebra system to graph the surface represented by| the
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Multivariable Calculus
- Verifying InequalitiesIn Exercises 53-64, verify a the Cauchy-Schwarz Inequality and b the triangle inequality for given vectors and inner products. Calculusf(x)=sinx, g(x)=cosx, f,g=0/4f(x)g(x)dxarrow_forwardCalculus In Exercises 65-68, show that f and g are orthogonal in the inner product space C[a,b]with the inner product f,g=abf(x)g(x)dx. C[/2,/2], f(x)=cosx, g(x)=sinxarrow_forwardParametric Representation. In Exercises 7-10, find a parametric representation of the solution set of the linear equation. x+y+z=1arrow_forward
- Verifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S, and closed curves C. Assume C has counterclockwise orientation and S has a consistent orientation. F = ⟨x, y, z⟩; S is the paraboloid z = 8 - x2 - y2, for0 ≤ z ≤ 8, and C is the circle x2 + y2 = 8 in the xy-plane.arrow_forwardVerifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S, and closed curves C. Assume C has counterclockwise orientation and S has a consistent orientation. F = ⟨y, -x, 10⟩; S is the upper half of the sphere x2 + y2 + z2 = 1 and C is the circle x2 + y2 = 1 in the xy-plane.arrow_forwardVerifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S, and closed curves C. Assume C has counterclockwise orientation and S has a consistent orientation. F = ⟨2z, -4x, 3y⟩; S is the cap of the sphere x2 + y2 + z2 = 169 above the plane z = 12 and C is the boundary of S.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_forwardVerifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S, and closed curves C. Assume C has counterclockwise orientation and S has a consistent orientation. F = ⟨y - z, z - x, x - y⟩; S is the cap of the sphere x2 + y2 + z2 = 16 above the plane z = √7 and C is the boundary of S.arrow_forwardStokes’ Theorem on closed surfaces Prove that if F satisfies theconditions of Stokes’ Theorem, then ∫∫S (∇ x F) ⋅ n dS = 0,where S is a smooth surface that encloses a region.arrow_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 = ⟨x, y, z⟩ across the slanted surface of the cone z2 = x2 + y2,for 0 ≤ z ≤ 1; normal vectors point upward.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_forwardDescribe and sketch the surface given by the function. g(x, y) = xarrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning