Graphing a Parametric Surface
In Exercises 13-16, use a computer algebra system to graph the surface represented by the
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Calculus: Early Transcendental Functions
- 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 = ⟨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_forwardsketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter. Surface: x2+y2=4; z=x2 Parameter: x=2sin(t)arrow_forwardDisplay the values of the functions in Exercises 37–48 in two ways:(a) by sketching the surface z = ƒ(x, y) and (b) by drawing an assortment of level curves in the function’s domain. Label each level curve with its function value.arrow_forward
- 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_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_forwardSurface area of an ellipsoid Consider the ellipsoidx2/a2 + y2/b2 + z2/c2 = 1, where a, b, and c are positive real numbers.a. Show that the surface is described by the parametric equations r(u, ν) = ⟨a cos u sin ν, b sin u sin ν, c cos ν⟩ for 0 ≤ u ≤ 2π, 0 ≤ ν ≤ π.b. Write an integral for the surface area of the ellipsoid.arrow_forward
- VECTOR ANALYSIS Express V = x2 - 2y2 - z2 in spherical coordinates.arrow_forwardSketch the space curve represented by the intersection of the surfaces z = x2 + ,y2 x + y = 0 . Then represent the curve by a vectorvalued function using the parameter x = t .arrow_forwardConsider the scalar field (image provide). Use gradφ (image provide) to find a unit vector in the direction of gradφ. State how the direction of gradφ relates to surfaces of constant φ(x, y, z).arrow_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 = ⟨y - z, z - x, x - y⟩; S is the part of the plane z = 6 - ythat lies in the cylinder x2 + y2 = 16 and C is the boundary of S.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_forwardSketch the space curve represented by the intersection of the surfaces z = x2 + y2, y = 2. Then use the parameter x = t to find a vector-valued function for the space curvearrow_forward
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