Show that vector field F(x, y, z)=yzi+xzj + (xy + 2z)k, is conservative, then evaluate fF.dr, where C is any mooth space curve from (1, 2, 3) to (0,3,-4) (15 points)

2. Show that F is a conservative vector field and then use that fact to evaluate F. dr along the given curve: F(x, y, z) = (e)i + (xe³ + e²)j + (ye²)k and C is the line segment from (0, 2, 0) to (4, 0, 3).

Match each of the following planar vector fields with the corresponding plot. Note that the vectors are scaled to avoid overlap. F (x, y) = (√x² + y² (-3y 3 -3y F(x, y) = (√²+²√²+²) 3x 2 1 0 5 2 -3 3 2 - 0 -1 4 با 4 7 3 // -3 -2 Answer Bank -2 3y √x²+x² -1 0 0 F(x, y) = (- F(x, y) = (-3x 1 1 3x √x²+x² 2 2 /x²+x² 3 3 9 3x 22² +2² 3y /x² +1² 4 3 2 1 0 -2 -3 -4 4 3 2 N -3 -3 3 1906-28 24 ME BEIN 2143 MULTFIGHT 1220121 -1 0 ....... 3 2 3 4

Consider the curve C from (-2, 0, 1) to (6, 5, 2) and the conservative vector field F(x, y, z) = (Yz, xz + 2y, xy). Evaluate F. dr