ENGR 233 Study Guide - Final Guide: Parallelepiped, Multiple Integral, Tangent Space
31 May 2018
School
Department
Course
Professor
Concordia University
EMAT 233 - Final Exam
Instructors: Dafni, Dryanov, Enolskii, Keviczky, Kisilevsky, Korotkin, Shnirelman
Course Examiner: M. Bertola
Date: May 2006.
Time allowed: 3 hours.
[10] Problem 1. Compute the curvature κ(t) of the curve Cdefined by
~r(t) = ti+t3
3j+t2
2k.
[10] Problem 2. Find points on the surface x2+ 3y2+ 4z2−2xy = 16 at which the tangent plane is
parallel to the yz–plane.
[10] Problem 3. Find the direction in which the function below increases most rapidly at the indicated
point. Find also the maximum rate of increase.
f(x, y) = e2xsin(2y), P ≡(0, π/8)
[10] Problem 4. The D’Alambert equation
∂2
∂t2U(t, x, y)−∂2
∂x2U(t, x, y)−∂2
∂y2U(t, x, y) = 0.
describes the propagation of small waves on an elastic membrane. Show that the function defined as
U(t, x, y) := cos(ct −ax −by), c =pa2+b2
is a solution of the wave equation for any value of the constants a, b (where cis given by the formula
written on the right in the equation above).
[10] Problem 5.
Compute the line integral
ZC
(x+ 2y)dx+ (2x−y)dy
where Cis the contour indicated
in figure starting at (−4,1) and
ending at (−3,−1).
y
x
1
Document Summary
Instructors: dafni, dryanov, enolskii, keviczky, kisilevsky, korotkin, shnirelman. Compute the curvature (t) of the curve c de ned by. Find points on the surface x2 + 3y2 + 4z 2 2xy = 16 at which the tangent plane is parallel to the yz plane. Find the direction in which the function below increases most rapidly at the indicated point. Find also the maximum rate of increase. f (x, y) = e2x sin(2y) , Y2 u (t, x, y) = 0. describes the propagation of small waves on an elastic membrane. C (x + 2y)dx + (2x y)dy where c is the contour indicated in gure starting at ( 4, 1) and ending at ( 3, 1). x. Using the appropriate theorem (which you must state), compute the ux of the curl for the vector eld across the upper hemisphere. ~f = yi xj + z cos(z 3 + ln(1 + x2))k.
