electromagnetic mempelajari perambatan gelombang elektromagentik pada sistem telekomunikasi.

1. 1. VEKTOR ANALISIS
1. VEKTOR ANALISIS
ELEKTROMAGNETIK
ELEKTROMAGNETIK
TERAPAN
TERAPAN
By
By Dwi
Dwi Andi
Andi Nurmantris
Nurmantris

4. OUTLINE
OUTLINE
1. Vektor Analysis
a) Scalar dan vektor
b) Vektor Algebra
c) Coordinate system
d) Calculus of scalar and vektor
(gradient, divergence, curl)
e) Divergence Theorem
f) Stoke’sTheorem

5. MAXWELL’S EQUATION
MAXWELL’S EQUATION

6. MAXWELL’S EQUATION
MAXWELL’S EQUATION
DON'T
PANIC!
Thisis where we are heading
r
=
Ñ D
.
ò
ò =
v
s
dv
ds
.
D r
0
. =
Ñ B
0
=
ò
s
ds
.
B
t
B
E
¶
¶
-
=
´
Ñ ò ò ¶
¶
-
=
s
ds
.
t
B
dl
.
E
t
D
E
H
¶
¶
+
=
´
Ñ s ò
ò ¶
¶
+
=
s
l
ds
).
t
D
E
(
dl
.
H s

7. WHY MAXWELL’S EQUATION?
WHY MAXWELL’S EQUATION?
• How EM field produced?
• How EM wave produced?
• How does a capasitor store energy?
• How does an antenna radiate or receive signals?
• How do Electromagnetic fields propagate in space?
• What really happens when electromagnetic energy
travels from one end of waveguide to the others?

8. OUTLINE
OUTLINE
1.Vektor Analysis
a) Scalar and vector
b) Vektor Algebra
c) Coordinate system
d) Calculus of scalar and vektor

9. Scalar and Vector
Scalar and Vector
Skalar : Besaran yang hanya memiliki nilai
Vektor : Besaran yang memiliki nilai dan arah
Contoh : Temperatur, laju, jarak, dll
Contoh : Medan listrik, medan magnet, Gaya,
kecepatan, posisi, percepatan, dll
a. Scalar Discrete Quantities : can be described with a single numeric
value
b. Scalar Fields Quantities : that the magnitude must be described as
one Function of (typically) space and/or time
a. Vector Discrete Quantities : must be described with both a single
value magnitude and a direction
b. Vector Fields Quantities: quantities that both magnitude and/or
direction must be described as a function of (typically) space and/or
time

10. Scalar and Vector
Scalar and Vector
• Scalar Quantities
ü Berat
ü Tinggi
ü Temperatur
• Vector Quantities
ü Force
ü velocities
• Scalar Fields
ü Berat dalam fungsi
waktu
ü Surface Temperatur
saat ini pada area
tertentu
• Vector Fields
ü Surface wind
velocities
ContohPhysical Quantity dalam scalardan vektor

11. Vector Representation
Vector Representation
symbolically we can represent a vector
quantity as an arrow: q The lengthof the arrow is
proportionalto the magnitude of
the vector quantity.
q The orientation of the arrow
indicatesthe direction ofthe
vectorquantity.
q The variable names of a vector
quantity will alwaysbe either
boldfaceorhave an overbar
Dua buah vektor dikatakan sama jika kedua vektor tersebut
memiliki magnitude dan arah yang identik
A
r
B
C

12. Vector Representation
Vector Representation
A
a
A
A ˆ
r
r
=
Dengan :
A
r adalah besar vektor A atau
panjang vektor A (magnitude vektor A)
A
â adalah unit vektor A atau vektor
satuan searah A
Vektor satuan atau unit vektor menyatakan
arah vektor, besarnya satu.
A
A
aA r
r
=
ˆ
1
ˆ =
A
a

13. Vector Algebra
Vector Algebra
C
B
A
r
r
r
=
+
If we know the rules of vector operations,
we can analyze, manipulate, and simplify
vector operations!
Vector Addition
1. Vector additionis commutative-> A+B = B+A
2. Vector additionis associative-> (X+Y) + Z = X + (Y+Z)

14. Vector Algebra
Vector Algebra
Head-to-tail Vektor Composition
e
d
c
b
a
f
r
r
r
r
r
r
+
+
+
+
=
a
r
b
r
c
r
f
r
d
r e
r
e
r
a
r
b
r
c
r
d
r

15. Vector Algebra
Vector Algebra
Vector Substraction
the negative of a vector is a vector with equal magnitude but opposite
direction.
A-B = B-A ?

16. Vector Algebra
Vector Algebra
Scalar-Vectormultiplication
the product of a scalar and a vector—is a vector!
B
a
B
a
C
B
a ˆ
r
r
r
=
=
1. scalar-vector multiplication is distributive:
a B+ b B = (a+b) B
a B+ a C= a (B+C)
2. scalar-vector multiplication is Commutative:
aB= B a
3. Multiplication of a vector by a negative scalar:
-a B = a(-B)
4. Division of a vector by a scalar is the same as
multiplyingthe vector by the inverse of the scalar
B
a
a
B r
r
÷
ø
ö
ç
è
æ
=
1

17. Vector Algebra
Vector Algebra
Dot Product / PerkalianSkalar
The dot product of two vectors is defined as:
IMPORTANTNOTE: The dot product is an
operationinvolvingtwo vectors, but the result is
a scalar
A.B.C = ???
1. the dot product is commutative
A
B
B
A
r
r
r
r
·
=
·
2. The dot product is distributive with addition
( ) C
A
B
A
C
B
A
r
r
r
r
r
r
r
·
+
·
=
+
·
3.
2
0
cos A
A
A
A
A =
°
=
·
r
r
4.
0
=
· B
A
r
r
5.
B
A
B
A
B
A
r
r
r
r
r
r
=
°
=
· 0
cos
6.
B
A
B
A
B
A
r
r
r
r
r
r
-
=
°
=
· 180
cos

18. Vector Algebra
Vector Algebra
CrossProduct
the product ofa scalar and a vector—is a vector!
IMPORTANTNOTE: The cross product is an
operationinvolvingtwo vectors, and the result is
alsoa vector
1.
B
A
a
B
A
a
B
A n
n
r
r
r
r
r
r
ˆ
90
sin
ˆ =
°
=
´
2.
0
180
sin
ˆ
0
sin
ˆ =
°
=
°
=
´ B
A
a
B
A
a
B
A n
n
r
r
r
r
r
r
3. The cross product is not commutative
A
B
B
A
r
r
r
r
´
¹
´
4. The cross product is also not associative
( ) ( )
C
B
A
C
B
A
r
r
r
r
r
r
´
´
¹
´
´
5. the cross product is distributive
( ) ( ) ( )
C
A
B
A
C
B
A
r
r
r
r
r
r
r
´
+
´
=
+
´
Bagaimana
menentukanarah
vector A x B ???

19. Vector Algebra
Vector Algebra
TripleProduct
The triple product is simply a combination of
the dot and cross products.
IMPORTANTNOTE: The triple product
results in a scalar value.
( )
C
B
A
C
B
A
r
r
r
r
r
r
´
·
=
´
·
C
B
A
r
r
r
´
·
The Cyclic Property
A
C
B
B
A
C
C
B
A
r
r
r
r
r
r
r
r
r
´
·
=
´
·
=
´
·
The cyclical rule means that the
triple product is invariant to
Shifts in the order of the vectors

20. Vector Algebra
Vector Algebra
IMPORTANT NOTE: If the
expression initially results
in a vector (or scalar), then
after each manipulation, the
result must also be a vector
(or scalar)
Let’s test your vector algebraic skills!
Can you evaluate the following expressions,
and determine whether the result is a
scalar (S), a vector(V), or neither (N) ??

21. Find the baby
Intermezo
Intermezo

22. Coordinate Systems
Coordinate Systems
• A set of 3 scalar values that define position and a set of 3 unit
vectors that define direction form a Coordinate system
• The 3 scalar values used to define position are called
coordinates
• All coordinates are defined with respect to an arbitrary point
called the origin.
• The 3 unit vectors used to define direction are called base
vectors
Coordinates system
Cylindrical Coordinate
Cartesian Coordinate Spherical Coordinate

23. Coordinate Systems
Coordinate Systems
• Kita dapat menentukan posisi (koordinat) titik
P dengan 3 scalar values yaitu x, y, dan z
• coordinate values in the Cartesian system
effectively represent the distance from a
plane intersecting the origin
Cartesian Coordinates system
y
z
x
z
x
y
)
,
,
( z
y
x
P
origin
x
y
z
1
=
x
1
-
=
y
1
=
z
Q: But how do we
specify direction of
vector in 3-D space??
• to specify the direction of a vector
quantity is by using a well defined
orthornormal set of vectors known as
base vectors.
• Orthonormal :
a. Each vector is a unit vector:
a. Each vector is mutually orthogonal:
• Additionally,a set of base vectors
must be arranged such that:
1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ =
·
=
·
=
· z
z
y
y
x
x a
a
a
a
a
a
z
y
x a
a
a ˆ
,
ˆ
,
ˆ
0
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ =
·
=
·
=
· z
x
z
y
y
x a
a
a
a
a
a
y
x
z
x
z
y
z
y
x a
a
a
a
a
a
a
a
a ˆ
ˆ
ˆ
;
ˆ
ˆ
ˆ
;
ˆ
ˆ
ˆ =
´
=
´
=
´

24. Coordinate Systems
Coordinate Systems
• Kita dapat menentukan posisi (koordinat) titik
P pada koordinat tabung dengan 3 scalar
values yaitu
• Keterangan :
1)The value ρ indicates the distance of the point
from the z-axis (0 ρ < )
2)The value φ indicates the rotation angle around
the z-axis (0 φ < 2π)
3)The value z indicates the distance of the point
from the x-y (z = 0) plane (-8 < z < 8)
Cylindrical Coordinates system
)
,
,
( z
P j
r
origin
y
z
x j
r z
z
,
,j
r
r
â
j
â
z
â • to specify the direction of a vector quantity is by using a
base vectors
• Not like Cartesian base vectors, the cylindrical base
vectors are dependent on position ( )
• set of base vectors must be arranged such that:
z
a
a
a ˆ
ˆ
ˆ j
r
j
r a
a ˆ
ˆ
z
z
z a
a
a
a
a
a
a
a
a ˆ
ˆ
ˆ
;
ˆ
ˆ
ˆ
;
ˆ
ˆ
ˆ =
´
=
´
=
´ j
r
j
r
r
j

25. Coordinate Systems
Coordinate Systems
• Kita dapat menentukan posisi (koordinat) titik
P pada koordinat tabung dengan 3 scalar
values yaitu
• Keterangan :
1)The value r (0 = r <8) expresses the distanceof
the point from the origin.
2)Angle θ (0 θ π) represents the angle formed
with the z-axis
3)Angle φ (0 φ< 2π) represents the rotation angle
around the z-axis
Spherical Coordinates system
)
,
,
( j
q
r
P
origin
j
q ,
,
r
r
â
j
â
q
â
• to specify the direction of a vector quantity is by
using a base vectors
• the Spherical base vectors are dependent on position
( )
• set of base vectors must be arranged such that:
q
j a
a
ar
ˆ
ˆ
ˆ
y
z
x j
q
r
q
j a
a
ar
ˆ
ˆ
ˆ
j
q
q
j
j
q a
a
a
a
a
a
a
a
a r
r
r
ˆ
ˆ
ˆ
;
ˆ
ˆ
ˆ
;
ˆ
ˆ
ˆ =
´
=
´
=
´

26. Coordinate Systems
Coordinate Systems
1. Posisi/coordinate suatu titik pada sistem koordinat
Titik A berposisi pada x=2, y=3, dan
z=4 atau A(2,3,4) pada koordinat
kertesian
Titik B berposisi pada ρ =2, φ =90o,
dan z=4 atau A(2,3,4) pada koordinat
tabung
Y
Z
2
4
3
A
X
Y
Z
φ=90o
Z=4
ρ =2
B
X
Coba gambar koordinat
titik C pada koordinat bola
dimana r=4, φ =90o, θ=45o

27. Coordinate Systems
Coordinate Systems
2. Position Vector pada sistem koordinat
a vector beginning at the origin and extending outward to a point—is a
very important and fundamental directed distance known as the position
vector r Using the Cartesian coordinate system, the position
vector can be explicitly written as:
Contoh :
Sehingga dengan mudah kita bisa langsung menetukan
koordinat titik yang dituju. X=1, y=2, z=-3
z
y
x a
a
a
r ˆ
3
ˆ
2
ˆ -
+
=
The magnitude of r NEVER, EVER express the position vector in either
of Cylindrical base vector or Spherical base vector
Vektor satuan searah vektor r
2
2
2
ˆ
ˆ
ˆ
ˆ
z
y
x
a
z
a
y
a
x
r
r
a
z
y
x
r
+
+
+
+
=
= r
r

28. Coordinate Systems
Coordinate Systems
Latihan Soal
Titik A(3,4,5) terletak dalam koordinat Cartesius. Tentukan :
a) Gambar vektor posisi A
b) Tulis vektor posisi A
c) Hitung besar/magnitude vektor A
d) Tentukan vektor satuan searah A
Jarak antar 2 titik (koordinat)
q Position Vector berguna saat kita ingin menentukan jarak antar 2 titik dalam suatu sistem
koordinat
q Misalkan dua buah titik P dan Q
q Jarak antara dua titik P dan Q adalah magnitudo dari
perbedaan vektor P dan Q
q Jarak titik P dan Q

29. Coordinate Systems
Coordinate Systems
3. Scalar Field pada sistem koordinat
q a scalar field is a scalar quantity that is a function of position and/or
time
q Scalar field can be written as :
)
,
,
(
)
,
,
( z
y
x
f
z
y
x
A =
Contoh
yz
y
x
z
y
x
g )
(
)
,
,
( 2
2
+
= Coba Anda Gambarkan
scalar field disamping
pada koordinat kartesian
)
,
,
(
)
,
,
( z
f
z
A j
r
j
r =
)
,
,
(
)
,
,
( j
q
j
q r
f
r
A =

30. • Menentukan scalar
component :
• Sehingga :
• Each of these three vectors point in one of the three
orthogonal directions
• The magnitude of each of these three vectors are
determined by the scalar values Ax, Ay, and Az that
called the scalar components of vector A.
• The vectors are called the vector
components of A.
Coordinate Systems
Coordinate Systems
4. Vector pada sistem koordinat
any vector can be written as a sum of three vectors!
z
z
y
y
x
x a
A
a
A
a
A ˆ
,
ˆ
,
ˆ
z
z a
A
a
A
a
A
A ˆ
ˆ
ˆ +
+
= j
j
r
r
r
q
q
j
j a
A
a
A
a
A
A r
r
ˆ
ˆ
ˆ +
+
=
r
• Menentukan Magnitude Vector:
• Menentukan vektor satuan searah vektor A
2
2
2
2
2
2
2
2
2
j
q
j
r
A
A
A
A
A
A
A
A
A
A
r
z
z
y
x
+
+
=
+
+
=
+
+
=
r
2
2
2
ˆ
ˆ
ˆ
ˆ
z
y
x
z
z
y
y
x
x
A
A
A
A
a
A
a
A
a
A
A
A
a
+
+
+
+
=
= r
r
z
z
y
y
x
x a
A
a
A
a
A
A ˆ
ˆ
ˆ +
+
=
r
( )
x
x
z
z
x
y
y
x
x
x
x
z
z
y
y
x
x
x
A
a
a
A
a
a
A
a
a
A
a
a
A
a
A
a
A
a
A
=
·
+
·
+
·
=
·
+
+
=
·
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
r
x
x a
A
A ˆ
·
=
r
z
z
y
y
x
x
a
A
A
a
A
A
a
A
A
ˆ
ˆ
ˆ
·
=
·
=
·
=
r
r
r
z
z
y
y
x
x a
a
A
a
a
A
a
a
A
A ˆ
)
ˆ
(
ˆ
)
ˆ
(
ˆ
)
ˆ
( ·
+
·
+
·
=
r
r
r
r

31. Coordinate Systems
Coordinate Systems
Contoh
0
ˆ
5
,
2
ˆ
0
ˆ
=
·
=
=
·
=
=
·
=
q
q
j
j
a
A
A
a
A
A
a
A
A r
r
r
r
r
j
a
A ˆ
5
,
2
=
r
0
ˆ
5
,
1
ˆ
2
ˆ
=
·
=
=
·
=
=
·
=
z
z
y
y
x
x
a
A
A
a
A
A
a
A
A
r
r
r
5
,
2
5
,
2
5
,
2
25
,
6
5
,
1
2
2
2
2
2
2
2
2
2
2
=
=
+
+
=
=
=
+
=
+
+
=
j
q A
A
A
A
A
A
A
r
z
y
x
r
j
j
a
a
A
A
a
a
a
a
a
A
A
a
A
y
x
y
x
A
ˆ
5
,
2
ˆ
5
,
2
ˆ
ˆ
6
,
0
ˆ
8
,
0
5
,
2
ˆ
5
,
1
ˆ
2
ˆ
=
=
=
+
=
+
=
=
r
r
r
r
y
x
z
z
y
y
x
x
a
a
a
A
a
A
a
A
A
ˆ
5
,
1
ˆ
2
ˆ
ˆ
ˆ
+
=
+
+
=
r
q
â
j
â
A
r A
r
j
â
5
,
2
A
r

32. Coordinate Systems
Coordinate Systems
Latihan soal
1. Gambarkan vektor berikut dalam sistem koordinat silinder
a) A = 3ar + 2af + az berpangkal di M(2,0,0)
b) B = 3ar + 2af + az berpangkal di N(2,p/2,0)
2. Gambarkan vektor berikut pada sistem koordinat bola
a) C= 3ar + aq + 2af berpangkal di M(2, p/2, 0)
b) D= 3ar + aq + 2af berpangkal di N(2, p/2, p/2)

33. Coordinate Systems
Coordinate Systems
5. Vector Fields pada sistem koordinat
q a vector field is a vector quantity that is a function of position and/or time
q When we express a vector field using orthonormal base vectors, the scalar
component of each direction is a scalar field—a scalar function of position!
q Often, vector fields must be written as variables of three spatial coordinates, as
well as a time variable t and known as a dynamic vector field.
Contoh
z
z a
A
a
A
a
A
A ˆ
ˆ
ˆ +
+
= j
j
r
r
r
q
q
j
j a
A
a
A
a
A
A r
r
ˆ
ˆ
ˆ +
+
=
r z
z a
z
A
a
z
A
a
z
A
z
A ˆ
)
,
,
(
ˆ
)
,
,
(
ˆ
)
,
,
(
)
,
,
( j
r
j
r
j
r
j
r j
j
r
r +
+
=
r
q
q
j
j q
j
q
j
q
j
q
j a
r
A
a
r
A
a
r
A
r
A r
r
ˆ
)
,
,
(
ˆ
)
,
,
(
ˆ
)
,
,
(
)
,
,
( +
+
=
r
Contoh
z
z
y
y
x
x a
A
a
A
a
A
A ˆ
ˆ
ˆ +
+
=
r
z
z
y
y
x
x a
z
y
x
A
a
z
y
x
A
a
z
y
x
A
z
y
x
A ˆ
)
,
,
(
ˆ
)
,
,
(
ˆ
)
,
,
(
)
,
,
( +
+
=
r
z
y
x a
y
a
y
xz
a
y
x
z
y
x
A ˆ
)
3
(
ˆ
ˆ
)
(
)
,
,
( 2
2
-
+
+
+
=
r
z
y
x a
t
y
a
t
y
xz
a
t
y
x
t
z
y
x
A ˆ
)
4
3
(
ˆ
ˆ
)
(
)
,
,
,
( 2
2
2
+
-
+
+
+
=
r

34. Coordinate Systems
Coordinate Systems
Gallery of Vector Scalar and Vector Field
x
a
A ˆ
=
r
x
a
x
A ˆ
=
r
y
x a
y
a
x
A ˆ
ˆ +
=
r
Coba gambarkan vector
field beikut :
r
a
A ˆ
=
r

35. Coordinate Systems
Coordinate Systems
Point Location/Coordinate and scalar field Transformation
Transformation of expressing point location/Coordinate and scalar field in terms
of cartesian coordinate to cylindrical or spherical coordinates or verse versa
Cartesian ke Cylindrical
)
,
,
(
)
,
,
( z
z
y
x j
r
Þ
z
z
x
y
y
x
=
÷
ø
ö
ç
è
æ
=
+
=
-1
2
2
tan
j
r
Cylindrical ke Cartesian
)
,
,
(
)
,
,
( z
y
x
z Þ
j
r
z
z
y
x
=
×
=
×
=
j
r
j
r
sin
cos
Cartesian ke Spherical
)
,
,
(
)
,
,
( j
q
r
z
y
x Þ
÷
÷
ø
ö
ç
ç
è
æ +
=
÷
ø
ö
ç
è
æ
=
+
+
=
-
-
z
y
x
x
y
z
y
x
r
2
2
1
1
2
2
2
tan
tan
q
j
j
z
q
r
x
y
r
r
q
j
q
j
q
cos
sin
sin
cos
sin
×
=
×
×
=
×
×
=
r
z
r
y
r
x
Spherical ke Cartesian
)
,
,
(
)
,
,
( z
y
x
r Þ
j
q
Cylindrical ke Spherical
)
,
,
(
)
,
,
( j
q
j
r r
z Þ
Spherical ke Cylindrical
)
,
,
(
)
,
,
( z
r j
r
j
q Þ
j
j
r
q
r
=
÷
ø
ö
ç
è
æ
=
+
=
-
z
z
r
1
2
2
tan
q
j
j
q
r
cos
sin
r
z
r
=
=
=

36. Coordinate Systems
Coordinate Systems
Latihan Soal
1. Titik A(-3,-3,2) terletak dalam koordinat Cartesius. Tentukan :
a. Tentukan koordinat titik tersebut pada koordinat tabung
b. Tentukan koordinat titik tersebut pada koordinat bola
2. Suatu scalar field
tulis fungsi g di atas dengan expresi koordinat kartesian
j
r
j
r sin
)
,
,
( 3
z
z
g =

37. Coordinate Systems
Coordinate Systems
Vector Transformation
Transformation of expressing Vector field in terms of cartesian coordinate to
cylindrical or spherical coordinates or verse versa
( ) ( ) ( ) z
z
z
z
a
a
z
y
x
A
a
a
z
y
x
A
a
a
z
y
x
A
a
z
A
a
z
A
a
z
A
z
A
ˆ
ˆ
)
,
,
(
ˆ
ˆ
)
,
,
(
ˆ
ˆ
)
,
,
(
ˆ
)
,
,
(
ˆ
)
,
,
(
ˆ
)
,
,
(
)
,
,
(
·
+
·
+
·
=
+
+
=
r
r
r
r
j
j
r
r
j
j
r
r j
r
j
r
j
r
j
r
Vektor Field di Koordinat Cartesian
Vektor Field di Koordinat Silinder
Ingat!!!
( ) ( ) ( ) j
j
q
q
j
j
q
q j
q
j
q
j
q
j
q
a
a
z
y
x
A
a
a
z
y
x
A
a
a
z
y
x
A
a
r
A
a
r
A
a
r
A
r
A
r
r
r
r
ˆ
ˆ
)
,
,
(
ˆ
ˆ
)
,
,
(
ˆ
ˆ
)
,
,
(
ˆ
)
,
,
(
ˆ
)
,
,
(
ˆ
)
,
,
(
)
,
,
(
·
+
·
+
·
=
+
+
=
r
r
r
r
Vektor Field di Koordinat Bola

38. Coordinate Systems
Coordinate Systems
Vector Transformation
r
â
j
â
q
â
r
â
j
â
z
â
Cartesian ke Cylindrical
)
,
,
(
)
,
,
( z
z
y
x j
r
Þ
Cylindrical ke Cartesian
)
,
,
(
)
,
,
( z
y
x
z Þ
j
r
Cartesian ke Spherical )
,
,
(
)
,
,
( j
q
r
z
y
x Þ
Spherical ke Cartesian )
,
,
(
)
,
,
( z
y
x
r Þ
j
q
z
z
y
x
y
x
A
A
A
A
A
A
A
A
=
+
-
=
+
=
j
j
j
j
j
r
cos
sin
sin
cos
z
z
y
x
A
A
A
A
A
A
A
A
=
+
=
-
=
j
j
j
j
j
r
j
r
cos
sin
sin
cos
j
j
q
q
j
q
j
q
q
j
q
j
j
q
cos
sin
sin
cos
sin
cos
cos
cos
sin
sin
sin
cos
y
x
z
y
x
z
y
x
r
A
A
A
A
A
A
A
A
A
A
A
+
-
=
-
+
=
+
+
=
q
q
j
j
q
j
q
j
j
q
j
q
q
j
q
j
q
sin
cos
cos
sin
cos
sin
sin
sin
cos
cos
cos
sin
A
A
A
A
A
A
A
A
A
A
A
r
z
r
y
r
x
-
=
+
+
=
-
+
=
z
r
r
z
z
z
z
y
y
x
x
a
A
a
A
a
A
r
A
a
A
a
A
a
A
z
A
a
A
a
A
a
A
z
y
x
A
j
j
q
q
j
j
r
r
j
q
j
r
ˆ
ˆ
ˆ
)
,
,
(
ˆ
ˆ
ˆ
)
,
,
(
ˆ
ˆ
ˆ
)
,
,
(
+
+
=
+
+
=
+
+
=
r
r
r
Cylindrical ke Spherical )
,
,
(
)
,
,
( j
q
j
r r
z Þ
j
j
r
q
r
q
q
q
q
A
A
A
A
A
A
A
A
z
z
r
=
-
=
+
=
sin
cos
cos
sin
Spherical ke Cylindrical )
,
,
(
)
,
,
( z
r j
r
j
q Þ
q
q
q
q
q
j
j
q
r
sin
cos
cos
sin
A
A
A
A
A
A
A
A
r
z
r
-
=
=
+
=

39. Coordinate Systems
Coordinate Systems
Latihan soal
1. Vektor A = 3âr+4âj+5âz berada pada sistem koordinat silinder dengan
titik pangkal di (10,p/2,0). Tentukan penulisan vektor ini pada sistem
koordinat kartesian!
2. Suatu vektor tentukan:
a. Expresi vektor tersebut pada koordinat tabung!
b. Expresi vektor tersebut pada koordinat bola!
z
y
x a
z
a
x
a
y
z
y
x
B ˆ
ˆ
ˆ
)
,
,
( +
-
=
r

40. Coordinate Systems
Coordinate Systems
Vektor Algebra pada coordinate systems
Misalnya kita memiliki dua buah
vektor pada koordinat kartesian :
Addition and Subtraction
Vektor-scalar multiplication
Dot Product
Cross Product
ú
ú
ú
û
ù
ê
ê
ê
ë
é
=
´
z
y
x
z
y
x
z
y
x
B
B
B
A
A
A
a
a
a
B
A
ˆ
ˆ
ˆ
r
r x
â y
â
y
A
x
A
x
B y
B
Å Å Å
- - -

41. Coordinate Systems
Coordinate Systems
Latihan soal
1. Diberikan tiga vektor pada sistem koordinat Kartesian dibawah ini :
tentukan hasil dari operasi-operasi vektor dibawah ini :
z
y
z
y
x
y
x
a
a
C
a
a
a
B
a
a
A
ˆ
2
ˆ
ˆ
2
ˆ
2
ˆ
ˆ
ˆ
+
=
-
+
=
+
=
r
r
r
C
B
A
f
B
A
e
B
A
d
C
c
C
B
b
B
A
a
r
r
r
r
r
r
r
r
r
r
r
r
´
·
´
·
-
+
)
)
)
4
)
)
)

42. Coordinate Systems
Coordinate Systems
Elemen Perpindahan, Elemen Luas, dan Elemen Volume
Elemen perpindahan
Elemen luas
Elemen volume
z
y
x a
dz
a
dy
a
dx
L
d ˆ
ˆ
ˆ +
+
=
r
x
x a
dydz
S
d
r
r
=
y
y a
dxdz
S
d
r
r
=
z
z a
dxdy
S
d
r
r
=
dxdydz
dV =
Elemen perpindahan
Elemen luas
Elemen volume
Elemen perpindahan
Elemen luas
Elemen volume
z
a
dz
a
d
a
d
L
d ˆ
ˆ
ˆ +
+
= f
r j
r
r
r
r
r j
r a
dz
d
S
d ˆ
=
r
j
j r a
dz
d
S
d ˆ
=
r
z
z a
d
d
S
d ˆ
j
r
r
=
r
dz
d
d
dV j
r
r
=
j
q j
q
q a
d
r
a
rd
a
dr
L
d r
ˆ
sin
ˆ
ˆ +
+
=
r
r
r a
d
d
r
S
d ˆ
sin j
q
q
=
r q
q j
q a
drd
r
S
d ˆ
sin
=
r
j
j q a
rdrd
S
d ˆ
=
r
j
q
q d
drd
r
dV sin
2
=

43. Calculus of scalar and Vector
Calculus of scalar and Vector
Gradient
q Gradien dari suatu scalar field adalah suatu vektor yang magnitudenya
menunjukkan perubahan maksimum scalar field tersebut dan arahnya
menunjukkan arah dari peningkatan tercepat scalar field tersebut
q Ilustrasi 1
q Ilustrasi 2
Suhu adalah scalar field. Jika terukur suhu pada
suatu titik X dari sebuah sumber lilin, maka gradien
terhadap suhu di X adalah vektor A dan bukan vektor
B
Ketinggian/kontur
permukaan bumi adalah
scalar field h(x,y).
Jika kita melakukan
operasi gradient
terhadap h(x,y), maka
Akan menghasilkan
vector field seperti
gambar disamping

44. Calculus of scalar and Vector
Calculus of scalar and Vector
Gradient operator pada sistem koordinat
GRADIEN PADA KOORDINAT KARTESIAN
GRADIEN PADA KOORDINAT TABUNG
GRADIEN PADA KOORDINAT BOLA
Contoh

45. Calculus of scalar and Vector
Calculus of scalar and Vector
Divergensi
q Untuk mengestimasi dan meng-kuantisasi vector-vector field, sering dengan cara
mengukur aliran vector field tersebut (atau netto aliran masuk dan keluar)
q Divergensi mengamati unsur volume tertentu yang sangat kecil, mengamati apakah
ada 'sumber' atau tidak di dalam volume tersebut
q Definisi dan simbol: Misalkan vector field yang diamati adalah vektor D
q Hasil operasi divergensi adalah skalar, karena dot product
q Misalkan vector field yang diamati adalah vektor D maka:

46. Calculus of scalar and Vector
Calculus of scalar and Vector
Divergensi pada sistem koordinat
DIVERGENSI PADA KOORDINAT KARTESIAN
DIVERGENSI PADA KOORDINAT TABUNG
DIVERGENSI PADA KOORDINAT BOLA
Contoh
Misalkan suatu vector Field :

47. Calculus of scalar and Vector
Calculus of scalar and Vector
Curl (Pusaran)
q Curl adalah ukuran seberapa besar putaran/pusaran/rotasi dari suatu vector
field di sekitar titik tertentu
q Rotasi/pusaran terjadi jika adanya ketidakseragaman vector field
q Definisi dan simbol : Misalkan vector field yang diamati adalah vektor H
q Curl adalah integral garis yang membatasi luas yang sangat kecil
q Curl digunakan untuk mengetahui vector field menembus permukaan diferensial
yang sangat kecil, yang menyebabkan pusaran medan lain
q Ilustrasi
Rapat arus J yang menembus permukaan dS
menimbulkan suatu pusaran/rotasi medan
magnetik H

48. Calculus of scalar and Vector
Calculus of scalar and Vector
Curl pada sistem koordinat
CURL PADA KOORDINAT KARTESIAN CURL PADA KOORDINAT TABUNG
CURL PADA KOORDINAT BOLA

49. Divergence Theorem/Gauss’s Theorem
Divergence Theorem/Gauss’s Theorem
q Teorema Divergensi
FLUX: adalah netto aliran
yang menembus permukaan
dengan arah normal
terhadap permukaan
Teorema divergensi menyatakan
bahwa jumlah flux dari suatu
vector field yang menembus
suatu permukaan tertutup/close
surface sama dengan jumlah
total / integral volume dari
suatu divergensi vector field
tersebut pada seluruh
area(volume) yang terlingkupi
close surface tersebut
( ) ( )
òòò òò ·
=
·
Ñ
v s
S
d
F
dV
F
r
r
r
Open Surface Close Surface

50. Stoke’s
Stoke’s Theorem
Theorem
q Theorema stoke
Contour/Garis tepi C adalah close
contour yang mengelilingi
permukaan/surface S. arah dari
contour C didefinisikan dari vektor ds
dan aturan putaran tangan kanan. Pada
gambar disamping contour C berputar
berlawanan jarum jam.
Artinya, jika kita ingin
mengintegralkan suatu vector
field terhadap suatu open
surface maka seolah olah kita
ingin menjumlahkan /
mengintegralkan suatu
pusaran di setiap titik pada
suatu permukaan.
Hal tersebut sama saja
dengan secara sederhana
menjumlahkan
/mengintegralkan ‘rotasi’
vector field tersebut pada
sepanjang close contour yang
mengelilingi surface tersebut.
A closed contour adalah
contour/garis tepi yang
dimulai dan diakhiri pada
titik yang sama!

51. ANY QUESTION???