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Scalars , Vectors
By : A. Hesami
AdvisorDr. B. Dabir
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Physical Quantities
Physical Quantities can be categorized as : Scalar Quantities ( time, volume, pressure , )
Vector Quantities ( velocity, force, momentum ,)
Tensor Quantities ( stress, )
=
zzzyzx
yzyyyx
xzxyxx
FTime = 20 Sec
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Scalar Quantities
Scalar Quantity determined only by Magnitude An example is Temperature.
r nc p es o sca ars ro uct Commutative
Associative
Distributive
rssr =
)()( srqsrq =
)()()()( rsqspsrqps ++=++
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Vector Quantities
A vector has both magnitude and direction
An example is position vector
Vectors Summation
Resultant vector Not the sum of the magnitudes
Vectors add head-to-tail
R = A + B
A
B
R
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Vector Quantities
Properties of Vector summation Commutative )()( vwwv
rrrr+=+
rrrrrr Associative
Negative of a Vector is the Same in Magnitude
but in Opposite Direction
uwvuwv ++=++
vr
vr
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Vector Quantities
Vector Multiplication
Scalar rsr
Dot
Cross
wvrr
wvrr
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Vector Quantities
Multiplication of Scalar in a Vector Vector Magnitude Multiplied by Scalar , No Changing in
Direction
Properties of Product
Commutative
Associative
Distributive
v v
svvsrr
=
vrsvsrrr
)()( =
vsvrvqvsrqrrrr
++=++ )(
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Vector Quantities
Dot Productvr
rr
cosr
( )vrrvrv rr
rrcos=
The Result of Dot Product is a SCALAR
Properties of Dot Product
Commutative Not Associative
Distributive
)()( wvuwvurrrrrr
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Vector Quantities
Cross Product
vwwv
wv
( ) vwvw nwvwvrrr
)sin( =
The Result of Cross Product is a Vector
Properties of Cross Product
Not Commutative
Not Associative
Distributive
vwwvrrrr
wvuwvurrrrrr
][][
][][][ wvwuwvurrrrrrr
+=+
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Vector Quantities
Algebraic Representation of Vectorx
y
z
ij
kxr
r r
zyxx ++=
Unit Vectorsparallel toCoordinateAxis Direction
Scalar Quantities
,, zyxx=
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Vector Quantities
Definition : kronecker delta
1ij = +
i j=i f
Definition : permutation symbol
ij
11
0
ijk
ijk
ijk
= +=
=
ifif
i
123, 231, 312
321,132, 213ijkijk
=
=
Any two index be alike
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Vector Quantities
Properties of kronecker delta andpermutation symbol
ijk=
3 3
1 1
2ijk hjk ilt j k
= =
= 3
1
ijk mnk im jn in jm
k
=
=
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Vector Quantities
Suppose , and being the unit vectoralong x , y and z axis
3
= =r r r rr
321
1i =
1 1 2 2 3 3
1 2 2 3 3 1
1 1 2 2 3 3
1 2 3
2 1 3
( . ) ( . ) ( . ) 1
( . ) ( . ) ( . ) 0
[ ] [ ] [ ] 0
[ ] ;
[ ] ;
= = =
= = =
= = =
=
=
3 1 2
1 3 2
[ ]
[ ]
=
=
ijji =
=
=
3
1
][k
kijkji
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Vector Quantities
Vector Summation and Subtraction
( )i i i i i i ii i i
v w v w v w = = r r rr r
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Vector Quantities
BBy R = Rxi + Ryj
Vector Summation
Ax Bx
Ay
Rx= Ax + Bx
Ry
=Ay
A
B
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Vector Quantities
Algebraic representation of vector multiplications
Multiplication by scalars
{ }i i i ii i
sv s v sv
= =
r rr
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Vector Quantities
Dot Product
( . ) . ( . )i i j j i j i jv w v w v w
= = r r r rr r
ij i j i i
i j i
v w v w= =
332211
332211
wwww
vvvv
++=
++=
r
r
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Vector Quantities
Cross Product
r rr rj j k k
j kv w v v=
[ ]j k j k ijk i j kj k i j k
v w v w = = r r r
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Tensor Quantities
In 3D Domain a tensor is determined by 9component
=
zzzyzx
yzyyyx
xzxyxx
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x
y
zTensor Quantities
1F 2F
Each of the six faces has a direction.For example, this faceand this face
are normal to the y direction
A force acting on any face can act in the x, y and z directions.
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x
y
z
yy yz
yx
Tensor Quantities
The face is in the direction y.
The force per unit face area acting in the x direction on that face is thestress yx (first face, second stress).
The forces per unit face area acting in the y and z directions on thatface are the stresses yy and yz.
Here yy is a normal stress (acts normal, or perpendicular to the face)
and yx and yz are shear stresses (act parallel to the face)
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Tensor Quantities
Algebraic representation of Tensors Definition : Dyad (Pairs of Unit Vectors)
jirr
1 1 11 1 2 12 1 3 13 = + +r r r r r rr
2 1 2 1 2 2 2 2 2 3 2 3
3 1 3 1 3 2 3 2 3 3 3 3
+ + +
+ + +
r r r r r r
r r r r r r
3 3
1 1i j ij
i j
= =
=
r r
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Tensor Quantities
Dyad Mathematical Operations Definition
( ) ( ) ( ): . .i j k l j k i l j k i l = =r r r r r r r r
r r r r r r r
( )
{ } ( )
{ }
{ }
3
1
. .
. .
. .
i j k i j k i j k
i j k i j k i j k
i j k l i j k l j k i l
i j k i j k j k l i l
l
i j k i j
=
= =
= =
= =
= =
=
r r r r r r r
r r r r r r r r r r
r r r r r r r r
r r r r r r 3
1
k i j k l k
l
=
= r r
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Tensor Quantities
Summation
( )i j ij i j ij i j ij iji j i j i j
+ = + = + r r r r r rr r
Multiplication by a Scalar
{ }i j ij i j iji j i j
s s s = = r r r rr
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Tensor Quantities
Scalar Product ( Double Dot )
( ): : :i i k l kl i k l i kl
= = r r r r r r r rr r
The Result is Scalar
i j k l i j k l
il jk ij kl ij ij
i j k l i j
= =
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Tensor Quantities
Dot Product
{ } {. . .i j ij k l kl i j k l ij kl
= = r r r r r r r rr r
The Result is Tensor
i j k l i j k l
jk i j ij kl i l ij jl
i j k l i l j
= =
r r r r
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Tensor Quantities
Dot product of Tensor and Vectors
[ ]. . .i j ij k k i j k ij k v v v = =
r r r r r rr r
The Result is Vector
i jk ij k i ij j
i j k i j
v v
= =
r r
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Tensor Quantities
Cross product of two tensors
r r r r r rr r
The result is tensor
i j ij k k i j k ij k
i j k i j k
jkl i l ij k i l jkl ij k
i j k l i l j k
v v
= =
r r r r
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Tensor Quantities
Tensor + Tensor Tensor Scalar * Tensor Tensor
Tensor . Tensor Tensor
Vector . Tensor Vector
Tensor x Tensor Tensor
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Tensor , Vector , Scalar
Order of Multiplication Scalars can be assumed as
zero order Tensors
Vectors can be assumed as
Product Type Result order
No sign
Cross -1
first order Tensors
Dot -2Double Dot -4
= Sum of orders
Scalarwv =+ 02)11(rr
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Differential Operations
Del Operator
r
1 2 3 i
= + + =r r r rr
= Unit Vectors = Cartesian Axis
Del Operator can applied on Scalars , Vectors or
Tensors quantities
1 2 3i
i
x x x x
i
r
321,, xxx
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Differential Operations
Del Operation on 3D Scalar Filed ( Gradient )
1 2 3
1 2 3
i
i i
s s s ss
x x x x
= + + =
r r r rr
Properties of Gradient
( ) ( )
( )
s s
r s rs
r s r s
+ = +
r r
r r
r r r
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Differential Operations
Z is scalar Function of x , y
XY
Z
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Differential Operations
0>
x
z
0z
Gradient Tell what is the Path of greatest Growth
XY
Z
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Differential Operations
Dot Product of Del and vector Field (divergence)
( ) . ( . )i j j i j ji j i ji i
v v vx x
= =
r r r rr r
Properties of Divergence
iij j
i j ii i
vvx x
= =
r
( . ) ( . )
( . ) ( . )
( .{ }) ( . ) ( . )
v v
sv s v
v w v w
+ = +
r rr r
r rr r
r r rr r r r
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Differential Operations
Physical Interpretation of Divergence
U xdxduU + )(
AUVx =&
))(( xdx
du
UAV xx+=
+
&
))(())((dx
dudv
dx
duxAVV xxx ==+
&&
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Differential Operations
Summing the rate of Volume change
))((z
u
y
u
x
udvChangeVolofRate z
yx
+
+
=
))(( udv =
nContractioFluiduif
ExpansionFluiduif
0
0
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Differential Operations Cross Product of Del and Vector ( Curl )
[ ] j k kj kj
v vx
=
r rr r
r r r
j k k ijk i k
j k i j kj j
v v
x x
= =
1 2 3
1 2 3
1 2 3
x x x
v v v
=
r r r
3 32 1 2 1
1 2 3
2 3 3 1 1 2
v vv v v v
x x x x x x
= + +
r r r
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Differential Operations
Curl is a Vector Operator that shows a vectorfield rate of rotation
Curl Can be described as Circulation Density
A vector field which has zero curl everywhere is
called irrotational
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Differential Operations
Del Operation on Vector Field
= =
r r r rr r
The Result is Tensor
i j j i j j
i j i ji ix x
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Differential Operation
( . ) ( . ) ( . )
( . [ ] ) ( . [ ] ) ( . [ ] )
[ ] [ ] [ ]
r s r s s r
s v s v s v
v w w v v w
s v s v s v
= +
= +
=
= +
r r r
r r rr r r
r r rr r r r r r
r r rr r r
r r r r r rr r r. .
1[ . ] ( . ) [ [ ] ]
2
[ . ] [ . ] ( . )
(
v v v
v v v v v v
v w v w w v
=
=
= +
r r rr r r r r r
r r rr r r r r r
: ) ( . )[ . ]
[ . ] [ . ] [ . ]
( . ) [ ( ) . ] [ ( ) . ]
s v s vs s
s s s
v w v w w v
= =
= +
= +
r r rr r
rr r
r r rr r r
r r rr r r r r r
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The Gauss DivergenceThe Gauss DivergenceTheoremTheorem If V represent a Volume , enclosed by
Surface S then
n represent the normal unit vector on S
( . ) ( . )V S
v dV n v dS =rr r
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The Gauss Divergence TheoremThe Gauss Divergence Theorem
There are two similar theorem for scalarsand tensors
r
[ . ] [ . ]
V S
V S
s ns
dV n dS
=
=
r rr r
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The Stockes Curl TheoremThe Stockes Curl Theorem
If S represent a Surface enclosed by curve C then
dcvtdsvn = ).()).((rrrr n
r
n represent the normal unit vector
t represent the tangent unit vector on C in integration direction
s c
c
Vr
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The Stockes Curl TheoremThe Stockes Curl Theorem
Stockes theorem Tell Us that :all line integral in z plane must vanish dlv
r
for the field
Because
jxiyv +=r
0=
v
r
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Leibniz Formula
If S represent the scalar filed of position andtime then
( . )sV V S
d ssdV dV s v n dS
dt t
= +
rr
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Cylindrical Coordinate
Cylindrical Coordinate RepresentationzP
x
y
z
r
),,( zr
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Cylindrical Coordinate
Cartesian and Cylindrical Coordinate Systemrelate as follow
2 2
siny r
z z
r x y
z z
=
=
= +
=
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Cylindrical Coordinate
Using chain rule the derivatives can transferredfrom Cartesian to Cylindrical coordinate system
( )
(cos ) (0)
cos(sin ) (0)
(0) 0 (1)
x r r z
y r r z
z r z
= + +
= + +
= + +
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Cylindrical Coordinate
Unit Vectors inCylindrical Coordinate
(cos ) (sin ) (0)
( sin ) (cos ) (0)
(0) (0) (1)
r x y z
x y z
z x y z
= + +
= + +
= + +
r r r r
r r r r
r r r r
y r
x
( , , ) ( , , )P x y z or P r z
y
r
xr
r
r
(cos ) ( sin ) (0)
(sin ) (cos ) (0)
(0) (0) (1)
x r z
y r z
z r z
= + +
= + +
= + +
r r r r
r r r r
r r r r
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Cylindrical Coordinate
Differential Operations
r r r
0
0 0 0
r z
r r z
r z
r r r
z z z
= = =
= = =
r r r r r
r r r
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Cylindrical Coordinate
2r
rrrrr
===
dd
d
rrrr 120lim)(
1r
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Cylindrical Coordinate
Del Operator in Cylindrical Coordinate
x y zx y z
= + +
r r rr
( )
( )
sin
cos sin cos
cossin cos sin
r
r z
r r
r r z
=
+ + + +
r r
r r r
1r z
r r z
= + +
r r rr
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Cylindrical Coordinate
Integration
dddrrdV =
V dVzrf ),,(
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Spherical Coordinate
Spherical Coordinate Representationz
Pr
x
y),,(
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Spherical Coordinate
Cartesian and Spherical Coordinate Systemrelate as follow
sin cosx r =
( )( )
2 2 2
2 2
sin sincos
arctan
arctan
y rz r
r x y z
x y z
y x
=
=
= + +
= +
=
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Spherical Coordinate
Using chain rule the derivatives can transferredfrom Cartesian to Spherical coordinate system
( )
cos cos s n
(sin cos ) sin
cos sin cos(sin sin )
sin
sincos (0)
x r r r
y r r r
z r r
= + +
= + +
= + +
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Spherical Coordinate
Differential Operation
= = =
r r r
0
sin cos sin cos
r
r r
r r
r r r
= = =
= = =
r r r r r
r r r r r r r
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Spherical Coordinate
Unit Vector in Spherical Coordinate
(sin cos ) (sin sin ) (cos )r x y z = + +r r r r
r r r r
( ) ( )sin cos (0)
x y z
x y z
= + +r r r r
( )
( )
( ) ( )
(sin cos ) (cos cos ) sin
(sin sin ) (cos sin ) cos
cos sin (0)
x r
y r
z r
= + +
= + +
= + +
r r r r
r r r r
r r r r
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Spherical Coordinate
Del Operator in Spherical Coordinate
1 1
sinr
r r r
= + +
r r rr
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Spherical Coordinate
Integration))()()()((sin drdrdrdV =
))()()((sin2 drddr =
vdVrf ),,(
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End