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Math 311-^
July 1, 2005: slide #
Math 311-^
July 1, 2005: slide #
Reminder^ The comprehensive final exam is 1:00–3:
PM, Tuesday, July 5, in this room.Please bring paper (or a bluebook) to the exam.The exam covers everything on the syllabus.There are 10 questions on the exam.
Math 311-^
July 1, 2005: slide #
Review problems: vector calculus^ •^ Evaluate the line integral
∫~ F^ ·^ d ~ x^ when^ C^ ~ F ( x , y , z ) = ( yz , xz , xy ) and^ C^ is the curve describedparametrically by g ( t ) = (cos( t ), 1^ +^ sin( t ),^ t ),^0 ≤^ t^ ≤
2 π.
-^ Evaluate the surface integral
∫∫^2 zdσ^ when^ S^ is the surface^ S^ √^2 described by z = 1 −^ x − (^2) y ,^ z^ ≥^0.
-^ Evaluate the volume integral
∫∫∫^2 xdx dy dz^ when^ B^ is the^ B^ (^2 2) ball described by x +^ y +^ z
2 ≤^1.
∫∫ • Evaluate the flux integral ~ F^ ·^ ~ n dσ^ when S (^) (^222) ~ F ( x , y , z ) = ( y , z , x ) and S is the open surface described^2 22 2 parametrically by g ( u , v ) = ( u ,^ v , 1^ −^ u −^ v ),^ u +^ v <^
with normal vector oriented upward.
Math 311-^
July 1, 2005: slide #
Review problems: vector calculus^ •^ Consider the unit cube in
(^3) R determined by the threevectors (1, 0, 0), (0, 1, 0), and^ (0, 0, 1). Orient the surface withthe outward-pointing normal vector. Let^ S^ be the union of thefive faces of the cube on which^ z^ >^0. Evaluate the integral ∫∫^222 ~~curl F · ~ n dσ when F ( x , y ,^ z ) = ( y ,^ z ,^ x ). S t^2 t^3 t • Find a point ( a , b , c ) on the curve^ f^ ( t ) = ( e ,^ e ,^ e )^ and a^2 2 point ( d , e , f ) on the surface g ( u ,^ v ) = ( u ,^ v ,^ u +^ v )^ suchthat the tangent line to the curve at^ ( a ,^ b ,^ c )^ does notintersect the tangent plane to the surface at^ ( d ,^ e ,^ f^ ). • Construct an invertible coordinate transformation( ) ( ) x u^2 = g in some region of^ R such that the area yvu + v^ element transforms via dx dy^ =^ edu dv.
Math 311-^
July 1, 2005: slide #
Review problems: linear algebra^ ^ −^34 −^16 36 ^ •^ If^ A^ =^ −^18 −^6 18 ^ −^47 −^21
, find a^3 ×^2 matrix^ B^ such that ( ) (^2 0) AB = B. 0 3 ( )^ (^ )^ (^ ) (^111) • Suppose ~ u = , ~ u =^ ,^ ~ v =^ , and^ ~ v = 1 2 1 2 012
(^ )^2.^5
Find a matrix^ B^ such that if^ a
~ u +^ a ~ u =^ b ~ v +^ b ~ v , then 11 22 11 22 (^ )^ (^ )^ a^ b^11 =^ B^.^ a^ b^22 • True or false? If^ V and^ V are two subspaces of^1
(^3) R , both of dimension^2 , then there exists a linear transformation^3 3^ f^ :^ R →^ R such that^ f^ ( V ) =^1
V.^2
Math 311-^
July 1, 2005: slide #
Review problems: linear algebra^ •^ Construct a (non-standard) inner product on
(^2) R for which the vectors^ (1, 0)^ and^ (1, 1)^ become orthogonal. • Give an example of a linear transformation
(^3 2) f : R →^ R such that the vectors^ (1, 0)^ and^ (1, 1
)^ form a basis for the image and the vector^ (2, 4, 2)^ is a basis for the null space. • In the vector space^ P^ of polynomials, express
x^ as a linear combination of^ p ( x ) =^1 +^ x^1
(^22) + x ,^ p ( x ) =^1 +^2 x^ +^3 x ,^2 (^2) and p ( x ) = 2 − x + 4 x. (^3) • The trace of a square matrix is the sum of the elements onthe main diagonal: for example, ( )1 2 3 trace = 1 + 5 + 9 =^15 4 5 67 8 9
. On the vector space of^3 ×
matrices, is the trace a^ linear
function?
