Rutgers Business School
New Brunswick Undergraduate Program
Operations Management
33:623:386:90
Winter 2014
Text:
Dr. Martin Markowitz
(848) 445-3600
[email protected]
Spreadsheet Modeling and Decision Analysis
6th Edition/
Cliff T. Ragsdale
Multiple Integrals
Goal: Develop an integral for R-valued functions of n
variables analogous to the Denite integral from Calc. I
Given: D a nice subset of Rn , and f : D R
To Dene:
D
f dA or
D
f dV
dA or dV is meant to refer to an innitesimal piece
of are
Optimization of functions of n variables
As in Calculus of One Variable:
Denition A function f : Rn R has
a minimum at x on E provided f (x) f (y) for all
yE
a maximum at x on E provided f (x) f (y) for all
yE
a local minimum at x provided there is a n
Lines and Planes
Goal of course: Calculus of functions from Rm to Rn ,
m or n bigger than 1
This chapter: Functions from R to Rn (usually n = 3)
aka vector-valued functions,
or parametric functions
Recall: Dierentiation for real-valued functions is really
For all the fundamental notions of calculus of vectorvalued functions, we can either
(a) Give a basic denition, as in Calc.I; or
(b) Dene them in terms of components.
Continuity: A function F : R Rn (n > 1) is continuous at c if
lim F(t) = F(c)
tc
Derivat
Vectors (cont)
Denition: Dot Product (or scalar product):
Let u = u1 , u2 , , un , v = v1 , v2 , , vn
n
u v := u1 v1 + u2 v2 + + un vn =
ui vi
i=1
Remark: Odd kind of multiplication
Examples: (Class)
Important note:
uu= u
1
2
Algebraic properties of the d
Introduction to vectors
Notation: R = the real numbers
R2 = the set cfw_(a, b) : a, b R of ordered pairs of real
numbers
= the (Euclidean) plane, or Euclidean 2-space
(relative to a coordinate system)
R3 = the set cfw_(a, b, c) : a, b, c R of ordered trip
Vector Functions
Recall: A function F : R Rn (n > 1) is a vectorvalued function.
Write: F(t) = f1 (t), f2 (t), . . . , fn (t)
(fi (t) is the ith component of F)
Often think of F(t) as giving the coordinates of a point
in n-space, for example in R3 :
F(t)
Theorems of Pappus
Prof. D. Ross, Dept. of Math., U. of Hawaii at M noa
a
There are two Theorems of Pappus:
Theorem 1 First Theorem of Pappus: If a curve with arclength L is translated through
space in such a way that the curves centroid P travels a dista
Surface Integrals
Recall: to compute the arc length of a curve, or integrate
a function along the curve, we had to
1. parametrize the curve (possibly by x)
2. get a formula for the innitesimal piece ds of arc
in terms of the parameter
ds was obtained from
Graphing Equations in Polar Form
Given: A polar equation for a curve.
Goal: Graph it
Remarks: 1. Graph need not be perfect.
We would use computers for precise graphs.
2. Use graph (and process of graphing) to get an idea
of how curve is traced out as vari