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- Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?The base of the closed cubelike surface shown here is the unit square in the xy-plane. The four sides lie in the planes x = 0, x = 1, y = 0, and y = 1. The top is an arbitrary smooth surface whose identity is unknown. Let F = x i - 2y j + (z + 3)k, and suppose the outward flux of F through Side A is 1 and through Side B is -3. Can you conclude anything about the outward flux through the top? Give reasons for your answer.a. Find a parametrization for the hyperboloid of one sheet x2 + y2 - z2 = 1 in terms of the angle u associated with the circle x2 + y2 = r2 and the hyperbolic parameter u associated with the hyperbolic function r2 - z2 = 1. (Hint: cosh2 u - sinh2 u = 1.) b. Generalize the result in part (a) to the hyperboloid (x2/a2 ) + (y2/b2 ) - (z2/c2 ) = 1.
- find a parametrization of the surface. (There aremany correct ways to do these, so your answers may not be the sameas those in the back of the text.) The portion of the cylinder y2 + z2 = 9 between the planes x = 0 and x = 3How do you calculate the area of a parametrized surface in space? Of an implicitly defined surface F(x, y, z) = 0? Of the surface which is the graph of z = ƒ(x, y)? Give examples.Find a parametric representation for the surface. The part of the plane z = x + 2 that lies inside the cylinder x2 + y2 = 9. (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of s and/or ?.)
- Find a parametric representation for the surface. the part of the sphere x2 + y2 + z2 = 64 that lies above the cone z = sqrt(x^2 + y^2) (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of u and/or v.) where z > sqrt(x^2 + y^2)1. Evaluate the surface area:(a) The part of the paraboloid z = 1 − x2 − y2that lies above the plane z = −2.(b) The surface z =23(x3/2 + y3/2), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.(c) The part of the sphere x2 +y2 +z2 = 4 that lies within the cylinder x2 +y2 = 4x and above the xy-plane.(d) The area of the finite part of the paraboloid z = x2 + y2cut off by the plane z = 25.A smooth curve is normal to a surface ƒ(x, y, z) = c at a point of intersection if the curve’s velocity vector is a nonzero scalar multiple of ∇ƒ at the point. Show that the curve r(t) = sqrt(t) i + sqrt(t) j-1/4(t+3)k is normal to the surface x2 + y2 - z = 3 when t = 1.