ENGR 233 Study Guide - Final Guide: Curve, Divergence Theorem, Tangent Space
31 May 2018
School
Department
Course
Professor
Concordia University Department of Mathematics and Statistics
Course Number Sections
EMAT 233 P, Q
Examination Date Time Total Marks Pages
Final December 15, 2005 3 hours 100 2
Course Examiner Instructors
M. Bertola G. Dafni, A. McIntyre
Special Instructions: use of calculators and outside materials is NOT permitted.
Each problem is worth 10 marks.
Write your answers in the examination booklet. Write clearly and neatly and show all your work.
In order to receive full marks, you must justify your answers fully.
NOTE: You may use the answers to some problems in other problems. You may use symmetry
and geometrical considerations to simplify your computations, as long as these are justified.
1. (a) Identify the formula for the curvature of the curve traced by the motion of a particle
with position vector ~r(t), velocity ~v(t), and acceleration ~a(t):
κ=~v(t)·~a(t)
k~v(t)kor κ=k~v(t)×~a(t)k
k~v(t)kor κ=k~v(t)×~a(t)k
k~v(t)k3.
(b) Find the curvature of the curve Cgiven by the equation
~r(t) = acos tj+asin tk,0≤t≤2π.
(c) Give a simple description of the curve Cin part (a). What is its radius of curvature?
2. (a) Let f(x, y, z) be a scalar function with second-order partial derivatives. Show that
div(grad f) = ∂2
f
∂x2+∂2
f
∂y2+∂2
f
∂z2.
This can also be written as ~
∇ · ~
∇f=∇2
f, which is known as the Laplacian of f.
(b) Any scalar function ffor which ∇2
f= 0 is said to be harmonic. Verify that
f(x, y, z) = (x2+y2+z2)−1/2
is harmonic when (x, y, z)6= (0,0,0).
3. Find the equation of the tangent plane to the graph of the equation
z= 25 −x2−y2
at the point (3,−4,0).
Document Summary
Special instructions: use of calculators and outside materials is not permitted. Write clearly and neatly and show all your work. In order to receive full marks, you must justify your answers fully. Note: you may use the answers to some problems in other problems. You may use symmetry and geometrical considerations to simplify your computations, as long as these are justi ed. 1. (a) identify the formula for the curvature of the curve traced by the motion of a particle with position vector ~r(t), velocity ~v(t), and acceleration ~a(t): ~v(t) ~a(t) k~v(t)k or = k~v(t) ~a(t)k k~v(t)k or = k~v(t) ~a(t)k k~v(t)k3 (b) find the curvature of the curve c given by the equation. ~r(t) = a cos t j + a sin t k, 0 t 2 . (c) give a simple description of the curve c in part (a). 2. (a) let f (x, y, z) be a scalar function with second-order partial derivatives.
