Chapter(1
Vector'Analysis
1
1
Units of Chapter 1
• 1.1 Vector Algebra
• 1.2 Differential Calculus
• 1.3 Integral Calculus
• 1.4 Curvilinear Coordinates
• 1.5 The Dirac Delta Function
• 1.6 The Theory of Vector Fields
2
2
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Vector Analysis: Lecture Notes and Exercises, Lecture notes of Vector Analysis
These lecture notes provide a comprehensive introduction to vector analysis, covering fundamental concepts such as vector algebra, differential calculus, integral calculus, and curvilinear coordinates. The notes are enriched with exercises that allow students to apply the concepts learned and deepen their understanding. The content is well-structured and presented in a clear and concise manner, making it suitable for both university and high school students.
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2015/2016
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Chapter 1
Vector Analysis
1
1
Units of Chapter 1
- 1.1 Vector Algebra
- 1.2 Differential Calculus
- 1.3 Integral Calculus
- 1.4 Curvilinear Coordinates
- 1.5 The Dirac Delta Function
- 1.6 The Theory of Vector Fields
2
- We already know this: a vector is designated by its magnitude (i.e., its length )
and direction (i.e. orientation).
- Notation: Unless stated explicitly, a vector will be written in bold font (or in
regular font with an arrow on top) and the magnitude will be in ordinary font.
- The vectors A and B below on the left are equal because they have length
and orientation; their respective origins do not matter.
- Minus A (- A ) is a vector of the same magnitude but of opposite direction
(see picture on the right)
3
1.1.1 Vector Algebra: Vector Operations
Dr. Peters, PHYS 3150- Fall 2016
3
Basic Vector Operations: (Vector addition and three kinds of multiplication)
(i) Vector Addition. The easiest approach is the “tail-to-tip” method. Given two vectors
A and B , place the tail of B to the tip of A. The sum A + B is the vector joining the tail of
A to the tip of B.
Dr. Archie Peters, PHYS 311 Fall 2014
4
+
=
=
The following properties clearly follow from the pictures:
( )
A "! A "#!
A " B! A "# B!!
A "! " A!!
1.1.1 Vector Algebra: Vector Operations
Question (dot product): Suppose that the angle between vectors A and B is θ.
Determine the dot product of the following vectors with themselves: (i) C = A - B (ii)
D = A + B
7
Solution:
$ $ !"#!
&(%' & % & %
$ $ $
= $ " $ + $ # # =! + " " !"
$ = " $ " = $ " $ " $ + $
" " "!!!
A A# " !# "! " " "!! "!!
$ $ !"#!
&((%' & % & %
$ $ $
= # + # + # " # =! + " + !"
= + # + = # + # + # +
" " "!!!
A A# " !# "! " " "!! "!!
Follow-up question: Under what condition is the length of C greater than that of D?
Under what condition is the length of C smaller than that of D?
1.1.1 Vector Algebra: Vector Operations
Dr. Peters, PHYS 3150- Fall 2016
7
Basic Vector Operations: (Vector addition and three kinds of multiplication)
(iv) Cross product (or vector product) of two vectors. The cross product of vectors
A and B is defined by
8
1.1.1 Vector Algebra: Vector Operations
The following properties follow from the definition of the cross-product:
!,-'.+'()*& !"#$%
,2&$.$!1,)$&/0&.'$ &
3 T'$& *,!$6.,4)& 45 &
8$!8$),6"#(!&.4 &8#()$&64).(,),)-&/4.'& &()& %&
,2&(&"),.&<$6.4!&;<$6.4!& 45 &#$)-.'&9:&
3 ='$!$&
3 2,) &
"!
A #
"!
A " # = !"! "
( ) ( )
I+)!"#$!%&'()!!
67 !2+! !"#5! !"&!.""11&1-!(4&#! I 3 !!2#!."(/'01"-!
! # A"! # "! A "
Question (vector product): Is the cross product associative, that is, is it true that
( A × B) × C = A ×( B × C)?
Solution: No it is not associative. For example, suppose both A and B are unit vectors
along the x-axis and C is a unit vector along the y-axis. Then A × B is zero; B × C is
along the z-axis. ( A × B) × C will be zero but A ×( B × C) will be along the y-axis.
9
1.1.1 Vector Algebra: Vector Operations
9
Component form representation of vectors:
Let be unit vectors along the x , y , and z axis respectively. Then an
arbitrary vector A can be expanded in terms of these basis vectors:
10
1.1.2 Vector Algebra: Component Form
61)2)" ." ." ""52)"2)+')%34)0-"1)"/."-.","%#&'#()(*+"#$"!
7 7 7
!
! A # "
# "!
# "!
A A A
= A + A + A
(i) Scalar Triple product: Given three vectors A , B , and C , we denote the scalar
triple product by A ·( B × C ).
Using the vector components, we evaluate the scalar triple product via the 3x
determinant
Note the cyclical relationship :
Geometric interpretation of the scalar triple product : The scalar triple product is
the volume of the parallelepiped (see figure) with sides A , B , and C.
Dr. Archie Peters, PHYS 311 Fall 2014
13
1.1.3 Vector Algebra: Triple Products
13
(i) Vector Triple product: Given three vectors A , B , and C , we denote the vector
triple product by A ×( B × C ).
The vector triple product is simplified using the so-called BAC-CAB rule:
Notice that the cross product anti-commutativity yields:
The BAC-CAB rule can be used to reduce higher vector products to simple forms.
For example:
14
1.1.3 Vector Algebra: Triple Products
Location of an arbitrary point in 3D in Cartesian coordinates: This is denoted
by the triplet ( x , y , z ). The vector from the origin of the coordinate system to the
point ( x , y , z ) (see figure below) is called the position vector r :
15
1.1.3 Vector Algebra: Position, Displacement, and Separation Vectors
15
Useful definitions from the position vector
16
1.1.3 Vector Algebra: Position, Displacement, and Separation Vectors
i. Magnitude:
ii. Unit vector (length 1) parallel to r:
iii. Infinitesimal displacement vector :
Suppose that the system x , y , z forms a standard coordinate system (all three axis
perpendicular). In this coordinate system a vector A has components A
x
, A
y
, A
z
.
Now suppose that we rotate the x , y , z coordinate frame counterclockwise about
the x - axis through an angle of 𝜙 to obtain a new coordinate frame 𝑥#, 𝑦&, 𝑧̅ (see
the figure below). Obviously 𝑥 ̅ = 𝑥 because the rotation is about the x-axis.
The vector A remains unchanged (maintains it magnitude and direction) but it has
different components based on the coordinate system in which it is represented.
Note that x-component of A remains unchanged during this transformation: A
x
= 𝐴
̅ "
19
1.1.5 Vector Algebra: How Vectors Transform
19
Let A be the magnitude of A. In the old ( x , y , z ) system
20
1.1.5 Vector Algebra: How Vectors Transform
In the new ( 𝑥#, 𝑦&,
𝑧) system
In matrix form, the last set of equations may be expressed as:
For rotation about an arbitrary axis , the new components 𝐴
̅ "
, 𝐴
$
, 𝐴
̅ "
transform
relative to the old components A
x
, A
y
, A
z
as
21
1.1.5 Vector Algebra: How Vectors Transform
21
22
1.2.1 Differential Calculus: Ordinary derivatives
7%G4-5"S#.G/I1G G+I#-.!GGG!'$2'(3GG-%G/01G G+I#-.!GGG !"#$%&'()
<.GW.8-%.G";.GW-88.#.%"-I'9GG 9 G$8G GI!GG )
>.$?."#-5I''(9G G-!G";.G!'$4.=&#IW-.%"G"$G";.G"I%&.%"G"$G";.G5S#+.G / 1 )
AS44$!.G";I"G G-!GIG8S%5"-$%G$8G GI%WG-"!GW.#-+I"-+.G G .@-!")
25
1.2.3 Differential Calculus: The Gradient Operator !
The gradient operator has the formal appearance of a vector , acting in a
multiplicative fashion on the scalar function T :
We formally define gradient operator, which we shall simply refer to as “del” as:
!
!"#$%&!$'
25
26
1.2.3 Differential Calculus: The Gradient Operator
!
We know that a vector A can multiply is three ways:
56 2'3)/#34M1%),-$.-()%$ ./M),-($%++#$%&'()"!
86 2'3)/#34M1%),-$.-()%$ ./M),-&%)#$%&'()" 7
96 2'3)/#34M+(M3M$* "** 7
Similarly, the operator can act of scalar and vector functions in three ways: !
6!"54" 3 "10&-#("$'4&-O#4" "1O3"-/0"&(#.."+(#,'&-* ""%&'()"#$"!
8!"54" 3 "10&-#("$'4&-O#4" "1O3"-/0",#-"+(#,'&-* ""%,O10(704& 0 "#$"!
9!54" 3 ".&3)3("$'4&-O#4" *"" ""%7(3,O04-"#$"!
We have discussed the gradient of a scalar function. We will now discuss the
divergence and curl of a vector function.
27
1.2.4 Differential Calculus: The Divergence
The divergence of a vector function v is defined by:
( )
! !)G#+!(!"#$%&' !!!
30#+0&G$-!$%&0G"G0&(&$#%!!#)!! 2 !!(!+0(1,G0!#)!.#/!+,-.!
+
- . . - +
Positive divergence Zero divergence
positive divergence
27
1.2.5 Differential Calculus: The Curl
Dr. Archie Peters, PHYS XXXX Fall
2014
The curl of a vector function v is defined by:
Geometric interpretation : The curl is a measure of how much the vector v “curls”
or rotates around the point in question.
Zero curl (irrotational)
non-zero curl (rotational)
1.3.1 Integral Calculus: Line, Surface, and Volume Integrals
Dr. Archie Peters, PHYS XXXX Fall
2014
(a) Line Integrals: A line integral has the following form
! !!
"
# $
!
!
45 " ""1."2$() 3 1'&2&'&("/.$0&"$." -+I"()&" "#$%%&'"!
"""
+ + #$%&"'!()* !"
,
(
!! #!! #" =!
" "
31
1.3.1 Integral Calculus: Line, Surface, and Volume Integrals
Dr. Archie Peters, PHYS XXXX Fall
2014
(b) Surface Integrals: A surface integral has the following form
#/N!2('-$! .($.(/0N'#,&$!-!+(!"#$%&'()!
!N"!&!
7 !S+($(! !N"!+(!4&5/N#0(!-%! !&/0!
8-N'(!+&*!!!
0N$('N-/!N" !.($.(/0N'#,&$!-!*+(!&$(&)!
!N"!&!2('-$!%#/'N-/!&/0! !N"!&/!N/%N/N("N4&,!.&+!-%!&$(&!S+-"(!
!!!!!<N/(5$&N-/!-2($!',-"(0!"#$%&'(!;7!()5)! 9&,,-/:
!!!<N/(5$&N-/!-2($!-.(/!"#$%&'(!!;7!()5)! .&.($:
!! !!
# #
# #
" "
" "
!
!
1.3.1 Integral Calculus: Line, Surface, and Volume Integrals
(c) Volume Integrals: A volume integral has the following form
(F!$*+I!"#$%C'
3F!$%!I2%C+0#"!2("#C!0((C1+F#$%!I/I$%.! I+F !!-.(C%!
8(C!%6#.2"%T!+F!$*%!5#C$%I+#F!0((C1+F#$%!I/I$%.! 4!
:%!9(C.!(9! !1%2%F1I!(F!$%!%.2"(/%1!0((C1+F#$%! I/I$%.
=*%C%! !+I!#!I0#"#C!9<F0$+(F!#F1! !+I!#F!+F9+F+$%I+.#"!;("<.%!%"%.%F$4!
!!-+F$%>C#$+(F!;("<.%!! '
?
!!
" "!
'
$
33
1.3.2 The Fundamental Theorem of Calculus
Fundamental theorem of calculus: the integral of a derivative over an interval
is given by the difference values of the function at the end points (boundaries).
$
"
!
b
1.3.4 The Fundamental Theorem of Divergences
This theorem is also Green’s theorem, Gauss’s integral theorem, or the divergence
theorem. It states that the volume integral of a derivative (that is, the divergence of a
vector function) is equal to the value of the function at the surface that bounds the
volume :
Interpretation. Suppose that v represent fluid flow. The divergence (spreading
out of vectors) can be likened to faucets pouring out liquid (left side of equation).
The right side of the equation is the flux of v, which is essentially the total amount of
fluid passing through the surface per unit time. Using this analogy the divergence
theorem is expressed as
37
1.3.5 The Fundamental Theorem of Curls (aka Stoke’s Theorem)
Stoke’s theorem states that
Deductions :
- depends only on the boundary line and not on the particular surface used.
- for any closed surface because the boundary collapses to a
single point (that is, d l =0)
1.3.6 Integration by parts
!!
!
"
!
"
!
"
39
1.4.1 Curvilinear Coordinates: Spherical Polar Coordinates
Dr. Archie Peters, PHYS XXXX Fall
2014
The spherical polar coordinates ( r , θ , ϕ ) of a point P with Cartesian coordinate is ( x , y ,
z ) is defined in the figure below.
r is the radial distance from the origin. θ is the polar angle (angle down from the z-axis).
ϕ is the azimuthal angle (the angle made with the positive x-axis by the projection of the
vector onto the xy plane). The relations between the Cartesian coordinates ( x , y , z ) and
the spherical polar coordinates are
Expressing ( r , θ , ϕ ) in terms ( x , y , z ) we obtain
( ( ( )( ( ( )(