Math 2002 Winter 2017 Written Assignment 3 Due Friday March 24th , 2017 Name (Print): BOO 1. Are the following expression defined? If so, state whether the results is a scalar or a vector. Assume F is a sufficiently differentiable vector field and f is a sufficiently differentiable scalar valued function. (a) (1 point) ( f ) (b) (1 point) (f ) (c) (1 point) F (d) (1 point) ( f ) (e) (1 point) ( f ) (f) (1 point) (f ) (g) (1 point) ( F) (h) (1 point) ( F) 2. Prove the following properties of the divergence and curl. Assume F and G are differentiable vector fields and c is a real number. (a) (3 points) (F + G) = F + G (b) (3 points) (cF) = c( F) 3. (5 points) A sphere of radius R is sliced parallel to the equatorial plane at distance R h from the equatorial plane (see figure). Find the general formula for the surface area of the resulting spherical cap with thickness h. Math 2002 Written Assignment 3 - Page 2 of 2 Due Friday March 24th , 2017 4. True or False? Explain your answer (otherwise you won't get points). (a) (2 points) A paddle wheel with its axis in direction h0, 1, 1i would not spin when put in the vector field F = h1, 1, 2i hx, y, zi. (b) (2 points) Stokes' Theorem relates flux of a vector field F across a surface to the values of F on the boundary of the surface. (c) (2 points) A vector field of the form F = ha + f (x), b + g(y), c + h(z)i, where a, b, c are constants, has zero circulation on a closed curve. (d) (2 points) If a vector field has zero circulation on all simple closed smooth curves C in a region D, then F is conservative on D