Math 263 Assignment 5 SOLUTIONS
1. The temperature at all points in the disc x2 + y 2 1 is T (x, y) = (x + y)ex
maximum and minimum temperatures on the disc.
2 y 2
. Find the
SOLUTION:
T (x, y) = (x+y)ex
2 y 2
Tx (x, y) = (12x2 2xy)ex
2 y 2
Ty (x, y) = (1
Math 263 Assignment 6 Solutions
Problem 1. Find the volume of the solid bounded by the surfaces z = 3x2 + 3y 2 and
z = 4 x2 y 2 .
Solution. The two paraboloids intersect when 3x2 + 3y 2 = 4 x2 y 2 or x2 + y 2 = 1.
Wrting down the given volume rst in Carte
Math 263 Assignment 2 Solutions
1) Find the angle at which the following curves intersect:
r2 (t) = (1 + t)i + t2 j + t3 k.
r1 (t) = cos t i + sin t j + t k,
Solution. We rst nd the point of intersection of the two curves,
by setting
(1)
cos t1 = 1 + t2 ,
Math 263 Assignment 7
SOLUTIONS
Problems to turn in:
(1) In each case sketch the region and then compute the volume of the solid region.
(a) The ice-cream cone region which is bounded above by the hemisphere z =
and below by the cone z = x2 + y 2 .
Soluti
Math 263 Assignment 4 Solutions
1) If z = f (x, y), where x = r cos and y = r sin , nd the quantities
z z
,
r
and
2z
.
r
Solution. By the chain rule,
z
= fx (r cos , r sin ) cos + fy (r cos , r sin ) sin ,
r
z
= fx (r cos , r sin )r sin + fy (r cos , r s
Math 263 Assignment 3 Solutions
1. (a) Draw a contour diagram for the function f (x, y) =
contours f (x, y) = 1, 2, 3 and 4.
(x 1)2 + (y 2)2 . Indicate the
(b) Calculate f (2, 3) and indicate this vector on your diagram.
(c) Consider z = f (x, y). Find th
Math 263 Assignment 9 - Solutions
1. Find the ux of F = (x2 + y 2 )k through the disk of radius 3 centred at the origin in the xy
plane and oriented upward.
Solution The unit normal vector to the surface is n = k. The ux is thus given by:
F .dS =
x2 + y 2
Math 263 Fall 2008, Test 2 Solutions
1. Let F(x, y, z) = (sin x, 2 cos x, 1 y 2 ).
(a) Calculate curl F.
(b) Calculate div F.
(c) Calculate div(curl F).
Solution:
(a) curl F = (2y)i (2 sin x)k.
(b) div F = cos x.
(c) div(curl F)=0. This is true for any ve
Math 263, Fall 2008
Midterm I Solutions
1. A rectangular box lies in the region x 0, y 0, z 0. One corner of the box is at
(0, 0, 0) and another corner lies on the sphere x2 + y 2 + z 2 = 3. Find the maximum
possible volume of the box.
(15 points)
Solutio
Math 263 Assignment 8 Solutions
Problem 1. Given that F(x, y, z) = (2xz + y 2 )i + 2xyj + (x2 + 3z 2 )k, nd a function f such
that F = f and use it to evaluate C F dr along the curve C : x = t2 , y = t + 1, z = 2t 1,
0 t 1.
Solution. Set
fx = 2xz + y 2 ,