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- Show that the curve = Vti + vt + (2t - 1) k is tangent to the surface x² + y2 -z = 1 when t = 1Show that the curve r(t) = ( t3 /4 - 2)i +(4/t-3)j+cos(t-2)k is tangent to the surface x3 + y3 + z3 - xyz = 0 at (0, -1, 1).Use the divergence theorem to solve following a) F=xi-yj bounded by the planes z=0 and z=1 and the cylinder x^2+y^2=a^2 b) F=xi+yj+(z^2 +1) with the same bounds as part a.
- Two surfaces S and S^(-) with a common point p have contact order ≥ 2 at p if there exist parametrization x(u,v) and x^(-)(u,v) in p of S and S^(-) respectively such that xu = x^(-)u, xv = x^(-)v, xuu = x^(-)uu, xuv = x^(-)uv, xvv = x^(-)vv at p. Prove the following: a. Let S and S^(-) have contact order greater than or equal to 2 at p; x:U -> S and x^(-): U -> S^(-) be arbitrary parametrizations in p of S and S^(-) respectively and f: V c R^(3) -> R be a differentiable function in a neighborhood V of p in R^(3). Then the partial derivatives of order smaller than or equal to 2 of f o x^(-): U -> R are zero in x bar^(-1)(p) iff the partial derivatives of order smaller than or equal to 2 of f o x: U -> R are zero in x^(-1) (p). b. Let S and S^(-) have contact of order smaller than or equal to 2 at p. Let z = f(x, y), z = f^(-) (x, y) be the equations in a neighborhood of p, of S and S^(-) respectively where the xy plane is the common tangent plane at p = (0, 0). Then the…1. Consider the surface defined by z = x2 + e9y ln(x-y). Let f(x,y) = x2 + e9y ln(x-y). ○ Compute ∇f at the point (1,0). ○ Compute the derivative of f(x,y) at the point (1,0) in the direction (3,-4). ○ Explain the geometric relationship between the answer found in part (a) and the surface defined above.Use the Divergent Theorem to find all values of b> −1 such that ∫∫SFdS = (45b/2) + (51/2), where F (x, y, z) = (sin (πx), zy^3, z^2 + 4x) and S is the parallelepiped with −1≤ x ≤b, 0≤ y ≤1 and 1≤ z ≤4.