1.
[5 marks] Suppose f : R 3 R is of class C 2 ( R 3 ) and g : R 2 R 3 is defined by 2 g ( x, y ) = ( x 3 y , x + 1, 2 y + 1) . Compute where = ( f g )( x, y ) in terms y2 of the derivatives of f . To have the same notations in all papers, please
2
consid
1. [4 marks] Find the shortest distance from the point P (1,1,4) to the line tangent to the curve r (t ) = (2t , 3 + e t , 6 e t ) at the point P0 (0, 2, 5) . The point P0 (0, 2, 5) corresponds to the parameter value t = 0 . So the tangent vector to the c
Remark: Questions are not necessarily in the order of a difficulty level. 1. [4 marks] Find the shortest distance from the point P ( 1, 1,4) to the line tangent to the curve r (t ) = (2t , 3 + e t , 6 e t ) at the point P0 (0, 2, 5) .
2
2.(a)[5 marks] Let
CHANGE OF VARIABLES THEOREM Here is the formulation of a strong version of Change of Variables Theorem which is more suitable in practice, and is more connected to topics that we have studied. Also the assumptions are easier to verify. .Note the meaning o
SOLUTIONS FOR TEST #1. 1(a) Suppose l is the tangent line to the curve C : g (t) = (t, t2 , t3 ) at (1, 1, 1), and is the plane tangent to the surface xy 2 + 2x2 yz + z 3 = 1 at (1, 0, 1). Find the equation of the line through (1, 1, 1) that is parallel t
MAT237Y Multivariable Calculus Summer 2009 Term Test Thursday, June 25 from 6:10pm to 8:00pm. Instructors: A. Hammerlindl and J. Uren 1. (a) [5 marks] Dene the directional derivative. (b) [5 marks] Give the denition of a convex set. 2. [10 marks] Is the s
1. Suppose that the function f : R 2 R is differentiable and g : R 3 R 2 is defined by g ( x, y, z ) = ( xz , yz ) . (u, v) (a) [4 marks] Let (u , v ) = g ( x, y , z ) . Find Dg and . ( x, z ) x u y u z u z 0 x zx (u, v) Dg = = , ( x, z ) = 0 y = yz x v y
1 [6 marks, 3 marks each part] Evaluate (a)
C
zds where C is parametrized by g(t) = (cos t, sin t, t),
0 t 2. g (t) = ( sin t, cos t, 1)
2 2
zds =
C
t
( sin t)2 + (cos t)2 + 12 dt = 2 22 8 = (2 )3/2 = 3/2 3 3
t 2 dt =
0
1
0
2 = 2 t3/2 3
2 0
(b)
C
ydx
MAT237Y Multivariable Calculus Summer 2009 Quiz #2 Tuesday, July 9 from 6:10pm to 7:00pm. Instructors: A. Hammerlindl and J. Uren 1. (a) [3 marks] State the denition of the Frchet derivative. e (b) [7 marks] Show that if f : Rn Rm and g : Rn Rm are Frchet
MAT237Y Multivariable Calculus Summer 2009 Quiz #2 SOLUTIONS Tuesday, July 9 from 6:10pm to 7:00pm. Instructors: A. Hammerlindl and J. Uren 1. (a) [3 marks] State the denition of the Frchet derivative. e Solution: For a point a, if there is a linear map s
1. [5 marks] Suppose f : R 3 R is of class C 2 ( R 3 ) and g : R 2 R 3 is defined by 2 where = ( f o g )( x, y ) in terms y2 of the derivatives of f . To have the same notations in all papers, please g( x, y ) = ( x 3 y, x 2 + 1, 2 y + 1) . Compute consid
1. (a) [7] Find the constant k 0 such that ( r k ) = 0 , where r = r , r = ( x, y , z ) 0 .
(b) [8] Evaluate (curlF) dx , where F( x, y, z ) = ( xz + y, yz, z ) and C is the segment of the curve of intersection of the plane y = x and the paraboloid z = x
HINTS TO ASSIGNMENT #4 1. The linear map T : R 2 R 2 is defined by T( x) = Ax where A is a 2 2 matrix and x is a column vector. The parametric equation of the line might be written as x = a + tv . Applying T we get the image T( x) = A(a + tv ) = Aa + tAv
ASSIGNMENT #4 1. Prove that the linear mapping T : R 2 R 2 maps straight lines into straight lines. Under what condition on T, parallelograms are mapped onto parallelograms? 2. Find a linear mapping T such that the image of the trapezoidal region with ver
HINTS TO ASSIGNMENT #3 1. (a) Use Th. 3.11 to write y = y ( x) or x = x ( y ) . For both
dy dx and not zero, dt dt differentiate y (t ) = y ( x (t ) using the chain rule. In cases one of the derivatives is zero investigate that at those points is either m
ASSIGNMENT # 2 1. Let f ( x, y ) = 5 x 3 y . (a) Show that the partial derivatives x (0,0) and y (0,0) exist, and evaluate them. (b) Prove that f is not differentiable at (0,0) . 2. Consider the function f ( x, y ) = 2 xy . In what directions at the point
HINTS TO ASSIGNMENT 1 (Since it is your first assignment mostly from the part of mathematics that you are not familiar with, the hints are exceptionally generous, often just a solution to the problem) 1. Consider the set S = cfw_( x, y ) : 0 < y 1 x, x 0
ASSIGNMENT #1 1. Give an example (in the set notation, not a picture) of a closed set S in R 2 such that S int S (that is the closure of S int is not equal to S ). 2. Let S be a set in R n . Is it true that every interior point of S is in S int ? Justify.
1. Let S be the portion of the cone z 2 = x 2 + y 2 with 0 z 2 and x 0 . (a) [2 marks] Parametrize the surface S using cylindrical coordinates.
G : D R 2 R 3 where G (r , ) = (r cos , r sin , r ), D = cfw_(r , ) : 0 r 2,
2
2
(b) [6 marks] Compute
i |
1.
Let S be the portion of the cone z 2 = x 2 + y 2 with 0 z 2 and x 0 . (a) [2 marks] Parametrize the surface S using cylindrical coordinates.
(b) [6 marks] Compute
(x
S
2
+ y 2 + x) dS
2
2.[8 marks] Evaluate
(curl F) n dS
S
, where F = yz i + xy k and
1. Let the surface S be parametrized by f (u , v ) = ( 2 u + v , u + e v , u 2 + v) and let P0 ( 2, 2,1) be a point in R 3 . (a) [2 marks] Show that the point P0 lies on the surface S. We should be able to find (u,v) such that f (u , v ) = ( 2 , 2 , 1) .
1. Let the surface S be parametrized by f (u , v) = (2 u + v , u + e v , u 2 + v) and let P0 (2, 2,1) be a point in R 3 . (a) [2 marks] Show that the point P0 lies on the surface S.
(b) [5 marks] Write the equation for the tangent plane to S at the point
1. Supposethatthefunction f : R 2 R isdifferentiableand g : R 3 R 2 isdefinedby g( x, y, z ) = ( xz, yz ) . (a) [4marks]Let (u , v ) = g ( x, y , z ) .FindDgand
(u , v ) . ( x, z )
(b) [6marks]If w = ( f g )( x, y ) = f ( xz , yz ) ,then x constant.Determ
1. (a) [7] Find the constant k 0 such that (r k ) = 0 , where r = r , r = ( x, y, z ) 0 .
r r (r k ) = (k r k 1 ) = k (r k 2 r ) = k[ r k 2 (3) + (k 2) r k 3 r ] = k (k + 1) r k 2 = 0 r r for k = 1 0 . Here the identity 5.28 from the text was used to shor
F : R2 R
F (x, y) = (x2 + y2 2)(y x2 )
S = F 1 (0) = cfw_(x, y) : S S F (x, y ) = 0 C 1 x = f (y)
S
F =0 . F =0 F = (2x(y x2 ) (x2 + y 2 2)2x, 2y (y x2 ) + (x2 + y 2 2).
y = x2 x2 + y2 = 2, (1, 1). (1, 1) (1, 1). S C 1 x = f (y)
= 2x(y x2 ) 2x
MAT237Y Multivariable Calculus Summer 2009 Quiz #1 Tuesday, May 26 from 6:10pm to 7:00pm. Instructors: A. Hammerlindl and J. Uren 1. (a) [5 marks] State the Triangle Inequality. (b) [5 marks] Show that the set U = cfw_(x, y ) R2 : 3 < (x, y ) < 7 is a nei
MAT237Y Multivariable Calculus Summer 2009 Solutions to the Quiz #1 Instructors: A. Hammerlindl and J. Uren 1. (a) [5 marks] State the Triangle Inequality. Solution: x + y x + y . Equivalently, x y x y . In terms of distances, the length of one side of a