Math 32B, Lecture 2 Fall 2010 Practice Problems for Final (sections 17.617.9
1.
(0 points)
(a) Let C be a simple closed piecewisesmooth space curve which lies entirely in a plane, and suppose that the plane has upwardpointing unit normal vector given by n = ai + bj + ck. Show that the area of the portion of the plane enclosed by C is 1 2 C
(bz − cy) dx + (cx − az) dy + (ay − bx) dz.
(b) Let C be a simple closed smooth curve in the plane 2x + 2y + z = 2, oriented counterclockwise as viewed from above. Show the the integral 2ydx + 3zdy − xdz C
depends only on the area of the region enclosed by C and not on the position or shape of C.
(0 points) A torus of revolution is the surface obtained by rotating a circle C in the xzplane about the zaxis in space. Assume that the radius of C is r > 0 and the center of C is (R, 0, 0), where R > r. (a) Find a parameterization of the torus.
2.
(b)
Find the surface area of the torus.
(0 points) Let f (x, y, z) and g(x, y, z) be scalar functions with continuous second partial derivatives, and let S be a surface with boundary curve C (where C is positivelyoriented relative to S). Show that ( f× S
3.
g) · dS =
C
f g · dr.
4.
(0 points) Let F = P i + Qj + Rk be a vector ﬁeld whose components are continuously diﬀerentiable. We deﬁne the notation F · to mean P Prove that (F · G) = (G · ) F + (F · )G + F × ( × G) + G × ( × F) . ∂ ∂ ∂ +Q +R . ∂x ∂y ∂z
5.
(0 points) Let F(x, y, z) be a vector ﬁeld on R3 − (0, 0, 0) given by F(x, y, z) = 2yz (x2 + y 2 + z 2 )2 i+ −2xz (x2 + y 2 + z 2 )2 j.
Let S be the closed surface which is the boundary of the region enclosed by the hyperboloid x2 + y 2 − z 2 = 4 and the planes z = 5 and z = −5 with outward orientation. Compute the ﬂux integral F · dS. S
...de Contaduría Pública y Administración
International Business Program

Instructional Guide for the Final Project 
The Managerial Process 

Professor: Lic. Ana Esperanza Álvarez García 
January  June 2013 
Final Assignment
The Final Assignment is an integrated business where, the first semester final assignment will serve as a base for making this semester final assignment.
I. General...
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3. Final Sketch:
After we completed our experiments with materials, I came up with a more developed concept and sketch. I used the same basic concept, but I added a city on the bank. I also put pieces that I thought described my life in the river this time. I decided to try and use powder paste for the river, and gouache for the background. This...
...This pack of BUS 600 Week 6 Final Paper includes:
Social Networking with WalMart
Business  General Business
Final Paper
The final assignment for this course is a Final Paper. The purpose of the Final Paper is for you to culminate the learning achieved in the course by completing a final research project on organizational communication processes. The Final Project represents...
...EVALUATION
The theme for my final GCSE Art and Design assignment was ‘Ordinary and/or Extraordinary’. I was given the exam paper just prior to going on a family holiday to the desert in Southern California and I decided that as Borrego Springs (the place that we visited) is such an unusual and extraordinary place, I would use it as my starting point in developing my ideas and investigations based on the theme. Borrego Springs provided me with both an intriguing place to...
...ADVANCED SPEECH
FINAL SPEECH EVALUATION
NAME______________________________
TOPIC_________________________________________
TYPE OF SPEECH________________________________
5 4 3 2 1
Rate the speaker on each point: Excellent Good Average Fair Poor
INTRODUCTION
Gained attention and interest 5 4 3 2 1
Clear transition sentence 5 4 3 2 1
Previewed body of...
...the dealer’s final price?
e. What is the dealer’s total savings?
III. Equation
a. Rebate = Original Price – Dealer Price
b. % Rebate = (Rebate / Original Price) x 100%
IV. Solving and Solution
a. Rebate = Original Price – Dealer Price
= $17995.00 – $16495.00
= $1,500.00 rebate
b. % Rebate = (Rebate / Original Price) x 100%
= ($1500.00 / $174995.00) x 100%
= 8.34% rebate
c. 15day discount = Dealer Price x Discount
= $16495.00 x 1%
= $164.95 15day discount...
...MBA 610: Final Paper Description and Grading Guide
Overview
In a wellcrafted, comprehensive paper, use specific cases, statutes, course resources, and course concepts to demonstrate your competency with regard to all of the following three points:
1. Areas of the law that you feel most affect business decisions (Law and Business Decisions)
2. Major underlying principles of the law that you feel are most impactful (Impactful Law Principles)
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Theoretical Models of Supervision
Most supervisors adapt the same theoretical model that they use in their practice to the supervision hour.
The psychoanalytic supervisor would not only pay attention to the developmental stages of the client but also those of the supervisee. The supervisor would pay special attention to the defensive...