(1, 0, 1) (0, 0, 0) (1, 0, 0) (1, 1, 1) (a) Find a piecewise smooth parametrization of the path C shown in the figure. r(t) = (1, 0, 0) 0sts 1 (0, 0, 0) to (1, 0, 0) r(t) = (1,0, t) 0sts 1 (1, 0, 0) to (1, 0, 1) r(t) = (1, 1, 1) 0sts 1 (1, 0, 1) to (1, 1, 1) (b) Evaluate ² √ (8x + y² − 2) ds. 18.83 X
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- 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…Find the appropriate parametrization for the given piecewise smooth-curvein R^2, with the implied orientation. The curve C, which goes along the circle of radius 3, from the point(3,0) above the x-axis to the point (-3,0), and then in a straight line along the x-axis back to (3,0).Find a parametrization for the line segment beginning at P1 and ending at P2. P 1(-4, 0, 3) and P 2(0, 5, 0) a)x = -4t, y = -5t + 5, z = 3t, 0 ≤ t ≤ 1 b) x = 5t, y = -4t + 5, z = -2t, 0 ≤ t ≤ 1 c) x = 5t - 4, y = -4t, z = -2t + 3, 0 ≤ t ≤ 1 d) x = 4t - 4, y = 5t, z = -3t + 3, 0 ≤ t ≤ 1
- A point moves along the curve of intersection of the paraboloid z=x^2+5y^2 and the plane x=3. At what rate is z changing with y when the point is at (3,-1,14)?In V=R3 Let W1 be the xy-plane and let W2 be the z-zxis: W1={(x,y,0):x,y∈R} and W2={(0,0,z):z∈R} Show thatSuppose C is the curve from (0, 0) to (2, 0) to (2, 3) to (0, 3) to (0, 0). Find the work done by the vector fieldF(x, y) = <x^(3)−2y^(2), x + cos(√y)> on a particle moving along C.
- Use Green's theorem to calculate the work done by force F on a particle that is moving counterclockwise around closed path C. F(x, y)=(x^3/2 - 8y)i + (5x + 8 sqrt(y))j, C: boundary of a triangle with vertices (0,0), (5,0), and (0,5)Let F = <x-y+3z, x-2z, 3y-4x>, where C is the positively oriented, closed, triangular curve with vertices (3,0,0), (0,2,0), and (0,0,6), and S is part of the first octant bounded by C. Find the flux of F up through S.Compute the line integral∫C [2x3y2 dx + x4y dy]where C is the path that travels first from (1, 0) to (0, 1) along the partof the circle x2 + y2 = 1 that lies in the first quadrant and then from(0, 1) to (−1, 0) along the line segment that connects the two points.