S 1.1 You are given three time dependent orthogonal unit vectors (orthonormal vectors) 1(t), 2(t),3(t) where...

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S 1.1 You are given three time dependent orthogonal unit vectors (orthonormal vectors) 1(t), 2(t),3(t) where 1(t) 1(t)=1, 1(t) 2(t)=0, e2(t) 2(t) = 1, 2(t)-3(t) = 0, 3(t) 3(t) = 1, 3(t) 1(t) = 0. (1) (a) Show that there exists functions W12, W13, W21, W23, W31, W32 such that de1 =W12e2+W13e3, dt de2 W211 +w233, dt de3 = W31e1+w322, dt (2) (b) Show that w12 = -W21, W13 = -W31, and w23 = -W32, so that out of the six possible functions, only three are unique. (c) Verify that these relations reproduce Eq. (1.3) and (1.4) in the special case of the polar coordinate system. S 1.1 You are given three time dependent orthogonal unit vectors (orthonormal vectors) 1(t), 2(t),3(t) where 1(t) 1(t)=1, 1(t) 2(t)=0, e2(t) 2(t) = 1, 2(t)-3(t) = 0, 3(t) 3(t) = 1, 3(t) 1(t) = 0. (1) (a) Show that there exists functions W12, W13, W21, W23, W31, W32 such that de1 =W12e2+W13e3, dt de2 W211 +w233, dt de3 = W31e1+w322, dt (2) (b) Show that w12 = -W21, W13 = -W31, and w23 = -W32, so that out of the six possible functions, only three are unique. (c) Verify that these relations reproduce Eq. (1.3) and (1.4) in the special case of the polar coordinate system.