Use the definition on page 1123 to compute 5.(2.ry2– ya") dx + (x+z=12) dy+ (x2y – Bryz?) di Where C is the twisted cubi

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Use The Definition On Page 1123 To Compute 5 2 Ry2 Ya Dx X Z 12 Dy X2y Bryz Di Where C Is The Twisted Cubi 1 (33.38 KiB) Viewed 30 times

Use the definition on page 1123 to compute 5.(2.ry2– ya") dx + (x+z=12) dy+ (x2y – Bryz?) di Where C is the twisted cubic parametrized by r(t) = (t, t.1), 0<t<1. (Hint: if you're thrown off by the notation, recall that SP dx + Q dy + R dz is an alternate notation for Jof.dr. See Page 1124.) If we let F(x,y,z) = (2.ryz - y2)i + (ra: - 12) i + (ry - 3.ryz?) k, (the same vector field we integrated in 1), show that F is the gradient of f(x,y,z) = rạyz - ryz? Use the net change theorem (FTC analog) on page 1127 to compute the same path integral from part 1 What if instead of following the twisted cubic from (0,0,0) to (1,1,1) we followed the path 2t r(t) = ((sint, 1 - cost, 0<t</2. {t24), Does the value of SF dr change? Why or why not?