2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
1
1. Areas in Polar Coordinates
This section shows how to calculate areas of plane regions, lengths of curves, and areas of surfaces
of revolution in polar coordinates.
1.1. Areas in the Plane
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
2
1.2. Plane Area between Two Curves
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
3
1.3. Length of a Polar Curve
(2)
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
4
1.4. Area of a Surface of Revolution
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
5
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
6
Vectors
1. Vector in Plane
A quantity such as force, displacement, or velocity is called a vector and is represented by a
directed line segment.
π΄ = ππ΄ββββ ββ = ππ + ππ (ππππ‘ππ)
|π΄| = βπ2 + π2 (πΏππππ‘β ππ π£πππ‘ππ)
πΌ + π½ =π
2
cos πΌ = sin π½
π, π are the fundamental unit vectors
ππππ‘ π£πππ‘ππ = οΏ½ββοΏ½ =π΄
|π΄|=
ππ + ππ
|π΄|
β΄ οΏ½ββοΏ½ = (cos πΌ) π + (cos π½)π
π€βπππ, cos πΌ =π
|π΄| ; cos π½ =
π
|π΄|
Vector Algebra Operations
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
7
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
8
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
9
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
10
Notes
1. ππππ‘ππ (ππ + ππ) ππ ππππππππππ’πππ π‘π π‘βπ π£πππ‘ππ (ππ β ππ).
2. ππππ‘ππ (ππ + ππ) ππ ππππππππππ’πππ π‘π π‘βπ ππππ π‘βππ‘ βππ ππ πππ’ππ‘πππ (ππ₯ + ππ¦ = π).
3. ππππ‘ππ (ππ β ππ) ππ π π£πππ‘ππ ππ π‘βπ ππππ π‘βππ‘ βππ ππ πππ’ππ‘πππ (ππ₯ + ππ¦ = π).
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
11
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
12
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
13
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
14
Note:
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
15
The Dot Product
Dot products are also called inner or scalar products because the product results in a scalar, not a
vector.
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
16
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
17
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
18
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
19
Writing a Vector as a Sum of Orthogonal Vectors
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
20
The Cross Product
The vector is orthogonal to both u and v because it is a scalar n.
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
21
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
22
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
23
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
24
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
25
Triple Scalar or Box Product
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
26
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
27
Equations of Lines and Planes in Space
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
28
The Distance from a Point to a Line in Space
There are two methods to find the distance from a point to a line in the space as follows:
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
29
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
30
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
31
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
32
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
33
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
34
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
35
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
36
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
37
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
38
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
39
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
40
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
41
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
42
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
43
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
44
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
45
PARTIAL DERIVATIVES
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
46
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
47
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
48
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
49
The Chain Rule
Chain Rule for Functions of Two Independent Variables
Chain Rule for Functions of Three Independent Variables
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
50
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
51
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
52
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
53
Directional Derivatives and Gradient Vectors
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
54
H.W:
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
55
Tangent Planes and Normal Lines
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
56
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
57
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
58
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
59
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
60
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
61
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
62
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
63
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
64
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
65
Multiple Integrals
1. Double Integrals y
x
R
dy
dx
dA
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
66
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
67
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
68
Finding Limits of Integration
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
69
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
70
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
71
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
72
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
73
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
74
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
75
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
76
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
77
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
78
Moments and Centres of Mass for Thin Flat Plates
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
79
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
80
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
81
Double Integrals in Polar Form
Finding Limits of Integration
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
82
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
83
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
84
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
85
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
86
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
87
2nd
Year Mathematics Dr. Rafea Dakhil
2017-2018 Lectures
88
1296