Instructors Solution Manual for ADVANCED CALCULUS
Gerald B. Folland
Contents
1 Setting the Stage 1.1 Euclidean Spaces and Vectors 1.2 Subsets of Euclidean Space . . 1.3 Limits and Continuity . . . . .
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
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
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
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 th
f : R3 R3 f (r, , ) = (r sin cos , r sin sin , r cos ). r z
(x, y, z ) =
r0 , 0 , 0 f (x0 , y0 , z0 ) = f (r0 , 0 , 0 )
f det(Df )(r0 , 0 , 0 ) = 0 2 det(Df )(r0 , 0 , 0 ) = r0 sin 0 = 0 r0 =
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 eq
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] Gi
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
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
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
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
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
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 in
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 . App
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
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 derivativ
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 funct
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)
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 tru
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 =
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
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 ab
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] Writ
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 =
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
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,
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 tha
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