ENGR 233 Study Guide - Final Guide: Multiple Integral, Joule, Divergence Theorem
31 May 2018
School
Department
Course
Professor
Concordia University
Department of Mathematics and Statistics
Course Number Section
EMAT 233/2 Q
Examination Date Time Pages
Winter December 2002 3 hours 2
Instructors Course Examiner
Bertola, Keviczky C. David
Special instructions
Answer all questions: NO CALCULATORS ALLOWED
[10] Problem 1. Compute the curvature κ(t) of the curve Cdefined by
~r(t) = ti+t2
2j+t3
3k.
[10] Problem 2. Consider the surface defined by
cos(zx)ey= 1
(a) Find the equation of the tangent plane to the surface at the point (π, 0,0)
(b) Find the equation of the normal line to the surface at the point (π, 0,0)
[10] Problem 3. Consider the function
G(x, y) = x2−cos(yx)
with x=x(u, v) = u2+v2and y=y(u, v) = uv. Compute the partial derivatives ∂G
∂u and ∂G
∂v . The answer
must be expressed in terms of u, v only.
[10] Problem 4. Consider the equation (D’Alambert equation):
∂2
∂t2U(t, x)−∂2
∂x2U(t, x) = 0.
(a) Prove that U(t, x) = ex−tis a solution.
(b) Prove that if f(s) is a twice-differentiable function (i.e. the second derivative exists and it is continuous)
then the two functions
U(+)(t, x) = f(x+t), U(−)(t, x) = f(x−t)
solve D’Alambert equation for both choices of the signs.
[10] Problem 5.
Consider the vector field
~
F(x, y, z) = 2xy +yz +2x
1 + x2i+x2+xzj+xy +z2k.
(a) Show that curl( ~
F) = ~
0. Can you find a function Φ(x, y, z) such that ~
F(x, y, z) =
~
∇Φ(x, y, z)? If yes then find the most general such function.
(b) Compute the line integral
ZC
~
F·d~r
where the curve Cis the piecewise smooth curve starting at A≡(0,0,0) and ending
at B≡(0,0,1) consisting of the spiral ~r(t) = htcos(t), t sin(t), tifor t∈[0,2π] joined
to the segment starting from (2π, 0,2π) and arriving at B≡(0,0,1).
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Document Summary
Compute the curvature (t) of the curve c de ned by. ~r(t) = ti + j + k. t2. Consider the function with x = x(u, v) = u2 + v2 and y = y(u, v) = uv. Compute the partial derivatives g must be expressed in terms of u, v only. T2 (a) prove that u (t, x) = ex t is a solution. (b) prove that if f (s) is a twice-di erentiable function (i. e. the second derivative exists and it is continuous) then the two functions. X2 solve d"alambert equation for both choices of the signs. U(+)(t, x) = f (x + t) , ~f (x, y, z) = (cid:18)2xy + yz + 1 + x2(cid:19) i +(cid:0)x2 + xz(cid:1) j +(cid:0)xy + z2(cid:1) k . (a) show that curl( ~f ) = ~0. Can you nd a function (x, y, z) such that ~f (x, y, z) =
