### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2013

 FINAL EXAMINATION April 1998 TIME: 3 HOURS

CALCULATORS PERMITTED

 MARKS (20) 1. (a) Evaluate , where R is the region in between the curves y = x and y = x4, for . (b) Show that , where Da is the disc of radius a > 0, centred at the origin. (c) Evaluate where E is the region in bounded by the spheres of radii a and b (b > a > 0), centred at the origin. (10) 2. (a) Show that the integral satisfies , where S is the unit square . (b) Compute an approximate value for I by partitioning S by the lines and then using double sum with the integrand evaluated at the mid-points of the sub-regions defined by the partition. (20) 3. Consider the vector field defined on . (a) Compute div and curl. (b) Find a potential function for . (c) Evaluate the line integral where and is any smooth curve joining A to B. (d) Compute the flux of across the ellipsoid a2x2 + b2y2 + c2z2 = 1. (e) Show that where (10) 4. Use the change of coordinates in defined by , to show that where T is the triangular region in the (x,y) plane whose vertices are and (0,1). (15) 5. In this problem you may use the standard formulas for area of a disc and surface area of a sphere: and , respectively, where R is the radius. (a) Evaluate where is the circle of radius 3 centred at the origin, oriented clockwise. (b) Compute the value of where is the sphere x2 + y2 + z2 = a2, oriented by its outward normal and , where e is a constant and is the unit normal. Express e in terms of Q and a. (c) Suppose that two vector fields and , defined and smooth on , are related by curl . Show that the flux of J through a surface is equal to the circulation of around , where is the boundary of . [Assume that and are smooth and compatibly oriented.] (10) 6. (a) Determine which of the following sequences converges and obtain the limit in the convergent cases. (i) (ii) (b) Determine whether each of the following series is absolutely convergent , conditionally convergent or divergent. (i) (ii) (iii) (15) 7. (a) Find either the general (n ) term, or the terms of degree , in the Taylor series expansion about x = 0 (Maclaurin series) for (i) ; (ii) and determine the interval of convergence in each case. (b) Set x = 1/3 in (a) (ii) and compute the approximation to given by the sum of the first three non-zero terms of the Taylor series. Compute the error, if the exact value is . (c) Compute an approximation to by using the first three non-zero terms in the Taylor expansion for f(t) about t = 0. Use this result to approximate F(1/2). (100)