Vector Analysis Level 1 Objective Problems 1) Which of the following second derivatives are zero? a) b) c) d) e) 2) Which of the following are correct? a) b) c) d) 3) The 3-dimensional coordinate system is rotated about the x-axis through an angle θ counter clockwise. Which of the following is a correct transformation from (x, y, z) to (x', y', z') (T in the superscript stands for transpose of the row vector)? a) b) c) d) Level 2 Objective Problems 1) Tick the correct option: a) b) and + c) + and d) None of the above. 2) Consider a generalized coordinate system (u, v, w) for which Which of the following are correct statements? a) In cylindrical coordinates, where b) In spherical coordinates, and c) For any scalar function T, . d) In spherical coordinates, 3) Which of the following are INCORRECT? a) b) A vector field E has the value inside a unit sphere centred at the origin and 0 outside. Then, everywhere. c) d) The graph of is the limit of triangles with base extending from to e) and vertex at . at the point . Subjective Problems 1) Prove that for a scalar field where the surface S has the boundary C. 2) Compute the divergence of the function Check the divergence theorem for this function, using your volume as the inverted hemispherical bowl of radius R, resting on the xy-plane and centred on the origin. [Use and ] 3) What is the charge density of a uniform, infinitesimally thin spherical shell of radius R and total charge Q, centred at the origin?