10.1&10.2: Vectors in plane and space
3-dimensional Coordinate system
Consists of three mutually perpendicular
coordinate lines X-axis, Y-axis, and Z-axis
intersecting at the origin. The orientation of
the axes follows the right-handed rule.
Coordinate
13.1: Double Integrals
Let = (, ) be nonnegative and defined on a closed rectangle
= [, ] [, ] = cfw_(, )| ,
( , )
We divide the rectangle by using a partition = cfw_ : = 1, , of subrectangles,
with area of and choose a sample point ( , ) in each . Th
9.4: Polar Coordinates
* A coordinate system is just a way to represent a point in the plane by an ordered pair of
numbers called coordinates. Usually we use rectangular (or Cartesian) coordinates (, ),
which are directed distances from two perpendicular
10.3: The dot product
There are two types of multiplication of vectors.
Dot Product: The multiplication of two vectors produces a scalar
Cross Product: The multiplication of two vectors produces a vector
r
r
Given two vectors u = u1,u2 , u3 , v = v1, v
13.5: Triple integrals
We have used double integrals to integrate f ( x, y) over a 2-d region
To integrate f ( x, y, z ) over a 3-d region G we require triple integration
f ( x, y, z)dV
G
o Notation:
Evaluated as iterated integrals
Let G be the rectangu
6.3: Trigonometric techniques of integration
Integrals involving powers of trigonometric functions
(A) Integrals of the form
If m (or n) is an odd positive integer
2- use the identity 2 + 2 = 1
1- isolate a factor of sin (or cos )
3- make the substitut
12.1: Functions of several variables
Definition: A function f of two variables is a rule that assigns a unique real number
f ( x, y) to each point ( x, y) D R 2 . The set D is the domain of f and its range is
the set of values that f takes on, that is, f
12.6: The gradient and directional derivatives
* The directional derivative of the function
,
at the point " , " in the
direction of a unit vector #
$ = #% , # , denoted by &'$
" , " , is interpreted as:
r
Slope of surface z = f ( x, y) at ( x0 , y0 ) in
6.4: Integration of rational functions using partial fractions
To decompose a rational function () =
()
()
into partial fractions, proceed as follows:
(1) If degree () degree (), use long division to divide () by (), obtaining
() = a polynomial +
()
()
w
10.5: Lines and planes in space
To determine a line we require
a point on the line
a vector parallel to the line
The equation of a line L passing through P0 ( x0 , y0 , z0 ) and parallel to vector
r
v = a, b, c is given by any of the following forms
Para
13.3: Double integrals in polar coordinates
Note that:
= = ( + )( ) =
Area element dA
in rectangular coordinates: dA dxdy
in polar coordinates:
dA rdrd
1
Dr. Muhammad Islam Mustafa
How to convert?
Some double integrals are easier to evaluate in polar
6.6: Improper integrals
In defining a definite integral () we dealt with a function defined on a finite
interval [, ] and we assumed that does not have an infinite discontinuity.
In this section we extend the concept of a definite integral to the followin
6.2: Integration by parts
Let
= ( ) and
= ( ) be two functions of . Making use of the Product Rule for
differentiation
( ) ( ) = ( ) ( ) + ( ) ( ),
One can easily deduce the following formula for integration by parts
" )(*)+ (*),* = )(*)+(*) " +(*) (*),*
12.7: Extrema of functions of several variables
1
Dr. Muhammad Islam Mustafa
Q1: Locate and classify all critical points of the functions
(a) (, ) = 2 2
(b) (, ) = 3 2 3 6
(c) (, ) = 3 2 2 2 4 + 3 2
2
Dr. Muhammad Islam Mustafa
Calculus I
Q2: Find the
12.5: The Chain Rule
Recall for a function of single variable that if = () and = () are differentiable
functions, then the chain rule gives the derivative of with respect to as follows
=
We now extend the chain rule to functions of several variables.
10.4: The cross product
Recall that the determinant of
2
2 matrix is given by:
3
3 matrix is given by:
For two vectors
Q1: Given
,
,
and
!
2, 3,5 and
(a)
,
,
, the cross product is the vector
!
3,4,6. Find
(b)
(c)
Right hand rule
Q2: Find a vector perp
6.1: Review of formulas and techniques
Many integrals can be easily evaluated by using one of the following integration formulas
=
+1
+1
+ ,
1
1
= ln| +
sin = cos +
cos = sin +
sec 2 = tan +
sec tan = sec +
csc 2 = cot +
csc cot = csc +
12.3: Partial derivatives
The partial derivative of a function (, ) with respect to at the point
(0 , 0 ) is denoted by and is defined as
(0 , 0 ) = lim0
(0 +,0 )(0 ,0 )
It is the rate of change of in the -direction.
The partial derivative of (, ) with
7/8/2016
VECTORS AND THE GEOMETRY
OF SPACE
10.1 &10.2: VECTORS IN THE PLANE (Two dimensional)
AND IN SPACE (Three dimensional)
We denote the directed line segment extending from the
point P (the initial point) to the point Q (the terminal point)
by
We ref
6/14/2016
INTEGRATION TECHNIQUES
6.2: INTEGRATION BY PARTS
Let u = f (x) and v = g(x), are two differentiable functions, the
integration by parts rule given as follow:
To apply integration by parts, you need to make a careful
choice of u and dv so that th
7/18/2016
FUNCTIONS OF SEVERAL VARIABLES AND
PARTIAL DIFFERENTIATION
12.1: FUNCTIONS OF SEVERAL VARIABLES
Def: Let D be a set of all order pair of real number ( x, y ) . A
function of two variables is a rule that assigns a real
number f ( x, y ) to each o
7/18/2016
PARAMETRIC EQUATIONS AND
POLAR COORDINATES
9.4: POLAR COORDINATES
Def: An assignment of order pair of the form r , to a point in
a plane will be referred to as polar coordinates.
To define polar coordinates fix an origin O (called pole) and
init
12/13/2016
MULTIPLE INTEGRALS
13.1: DOUBLE INTEGRALS
Let f be a function of two variables that is defined on region R.
The double integral of f over R is defined by
Where dA dxdy or dA dydx.
Theorem: (Fubinis Theorem)
Suppose that f is integrable over the
3/13/2016
VECTORS AND THE GEOMETRY
OF SPACE
10.1 &10.2: VECTORS IN THE PLANE (Two dimensional)
AND IN SPACE (Three dimensional)
We denote the directed line segment extending from the
point P (the initial point) to the point Q (the terminal point)
by
We re
2/8/2016
INTEGRATION TECHNIQUES
6.2: INTEGRATION BY PARTS
Let u = f (x) and v = g(x), are two differentiable functions, the
integration by parts rule given as follow:
To apply integration by parts, you need to make a careful
choice of u and dv so that the
1
Date: 14 /04 /2016
Time: 16:00 17:15
Semester: Spring 2015/2016
University of Sharjah
College of Sciences
Department of Mathematics
Answer key Midterm Exam: Calculus II for Engineers (1440161)
Name:
Instructor:
I. D. #:
Section #:
Question
Possible
Mark
ASSIGNMENT
ASSIGNMENT
#2
#2
Specificationsand
andquantity
quantitysurveying
surveying
Specifications
Done by: Khalid jamal Muwahid
ID: U00015930
Date of submission: 27/03/2014
To : Dr. Salah Toubat
Exercise #1:Content
Companies last year revenue =
Estimat
University of Sharjah
Department of Mathematics
College of Sciences
Fall 2014/2015
-Course
Prerequisite
Instructor(s)
Coordiator
Office Hours
Linear Algebra (1440211)
Calculus I (1440131)
Dr. Mohammad Sababheh (Sec:51)
Dr. Ali Jaballah
Girls: TR 12:30-13: