### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2013

 FINAL EXAMINATION April 1997 TIME: 3 HOURS

MARKS
(5)1. Show is a conservative vector field and find a potential function for .
(5)2. Evaluate: .
(5)3. Find the volume of the solid bounded above by the cone and below by z = x2 + y2.
(5)4. Evaluate: .

### You need not evaluate any integrals for the questions on the remainder of the exam.

(5)5. Set up an integral for the area of that part of the hyperbolic paraboloid z = y2 - x2 that lies between the cylinders x2 + y2 = 1 and x2 + y2 = 4.
(8)6. Let C be the closed curve consisting of the arc of y = x2 from (0,0) to (1,1) and then the straight line from (1,1) to (0,0).
(a) Set up as a single integral.
(b) Using Green's Theorem, set this up as a double integral.
(8)7. Set up the two integrals of Stokes Theorem for the vector field
and the surface consisting of that part of
z = 1 - x2 - y2 that lies above the xy plane.
(8)8. Set up the two integrals of the Divergence Theorem for the vector field and the unit ball .
(12)9. Find power series representations of the following (either present the general term or at least 5 non-zero terms):
(a)
(b)
(c)
(d)
(6)10. (a) Find the radius of convergence for .
(b) What is the sum of ?
(10)11. Use power series to solve the following differential equation. Give at least 5 non-zero terms or state the general term.

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