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- Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?az. Suppose F = (2xz + 3y²) a, + (4yz²) a;. (a) Calculate S[F·dS, where S is the shaded surface in Figure 1. (c) Based on your results for parts (a) and (b), what named theorem do you think is being satisfied here, if any? (b) Calculate SF· dl, where C is the A → B → C → D → A closed path in Figure 1. az C C (0,1,1) D (0,0,0) (A ay В ax Figure 1: Figure for Problem 1.2. Let S be the surface given by r(u, v) = (u, u² + v², v). Find the equation of the tangent plane at (2, 5, 1) in terms of (u, v) as well as in terms of (x, y, z).
- 4. Find an equation of the tangent plane to the surface x = u? + 1, y = v³ + 1, z = u + v at the point ( 5, 2, 3)If a parametric surface given by r₁(u, v) = f(u, v)i + g(u, v)j + h(u, v)k and -4 ≤ u ≤ 4,-2 ≤ v ≤ 2, has surface area equal to 8, what is the surface area of the parametric surface given by r2(u, v) = 5r₁(u, v) with −4 ≤ u ≤ 4, −2 ≤ v ≤ 2? 27Find the equation of the tangent plane to the surface z = at the point (2,-3, 10) O - 6y - z = 10 O * + 6y – z = –26 Or+ 4y-z = -20 X- 4y- z = 4