ENGR 233 Study Guide - Final Guide: Conservative Force, Cylindrical Coordinate System, Tangent Space
31 May 2018
School
Department
Course
Professor
Concordia University
Course Number
ENGR 233
Examination Date Time Total Marks Pages
Final April 2007 3 hours 100 2
Course Coordinator Instructors
A. R. Sebak M. Bertola, R. Bhat, C. David, Haifaels, R. Stern
Special Instructions: use of calculators and outside materials is NOT permitted.
Each problem is worth 10 marks unless stated otherwise.
A one page formula sheet will be handed in during the final exam.
Problem 1. Let us denote by h(x, y, z) a scalar function and ~
F(x, y, z) a vector field. Identify
which of the following operations are not allowed and which ones make mathematical sense. Write
(Yes) when the expression is a valid one or (No) next to each letter in your booklet. In case the
expression is not valid.
Credit will NOT be given for correct answers unless reasons or justifications
(one sentence) are shown in the examination booklet.
(example: “The expression is not defined because the ..... of a scalar/vector function is not
defined.” or ”The expression is well defined because the .... of a scalar is defined.” ).
(a) grad (curl h) (b) grad (curl ~
F) (c) div (curl ~
F) (d) grad (div ~
F) (e) grad (div h)
(f) div (curl (grad h)) (g) curl (div (grad h)) (h) div (curl (grad ~
F))
(i) curl (div (grad ~
F)) (j) grad (div (grad h))
Problem 2. Find the equation of the tangent plane to the graph of the equation
z= 2 −x3+y2
at the point (3,−4,−9).
Problem 3. Set up, but do not evaluate the double integral of g(x, y) = e−x2+yover the region
bounded by the curves
y=p1−x2, y =p4−x2, y =x , x = 0
using polar coordinates. In particular you must write the integral as a suitable iterated integral
in the r, θ coordinates.
Problem 4. Find the directions in which the following function has the maximum and minimun
rates of change at the point (1,0,1). Find those rates.
F(x, y, z) = x3+y2x+z2x+yx2+y3+z2y
Problem 5. Using the divergence theorem, compute the flux of the vector field
~
F(x, y, z) = (2x−y2cos(z))i+ (y−ln(1 + x2))j+ ex2y3k
Document Summary
Instructors: bertola, r. bhat, c. david, haifaels, r. stern. Special instructions: use of calculators and outside materials is not permitted. Each problem is worth 10 marks unless stated otherwise. A one page formula sheet will be handed in during the nal exam. Let us denote by h(x, y, z) a scalar function and ~f (x, y, z) a vector eld. Identify which of the following operations are not allowed and which ones make mathematical sense. Write (yes) when the expression is a valid one or (no) next to each letter in your booklet. Find the equation of the tangent plane to the graph of the equation z = 2 x3 + y2 at the point (3, 4, 9). In particular you must write the integral as a suitable iterated integral in the r, coordinates. Find the directions in which the following function has the maximum and minimun rates of change at the point (1, 0, 1).
