1. (a) Let f = f(x,y) be a smooth function. Define the matrix Show that A...

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1. (a) Let f = f(x,y) be a smooth function. Define the matrix Show that A = 0 1 -1 ] Vf AVf = 0 and V (AV) = 0. (b) Let F be a smooth vector field and let be a smooth function. Show that V. (oF) = (V F) + (Vo) F. Hint: Recall that if (x, y, z) is a scalar function, and F = (F1, F2, F3) is a vector field, then OF is the vector field F = (0F1, F2, F3). (c) Let f = f(x, y, z) be a smooth function. Show that () = 0. (d) Let F = F(x, y, z) be a smooth vector field. Show that ( F) = 0. 1. (a) Let f = f(x,y) be a smooth function. Define the matrix Show that A = 0 1 -1 ] Vf AVf = 0 and V (AV) = 0. (b) Let F be a smooth vector field and let be a smooth function. Show that V. (oF) = (V F) + (Vo) F. Hint: Recall that if (x, y, z) is a scalar function, and F = (F1, F2, F3) is a vector field, then OF is the vector field F = (0F1, F2, F3). (c) Let f = f(x, y, z) be a smooth function. Show that () = 0. (d) Let F = F(x, y, z) be a smooth vector field. Show that ( F) = 0.