Vector Lecture

7/31/2019 Vector Lecture

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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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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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Z is scalar Function of x , y

XY

Z


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0>

x

z

0z

Gradient Tell what is the Path of greatest Growth

XY

Z


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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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Physical Interpretation of Divergence

U xdxduU + )(

AUVx =&

))(( xdx

du

UAV xx+=

+

&

))(())((dx

dudv

dx

duxAVV xxx ==+

&&


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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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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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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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Cartesian and Cylindrical Coordinate Systemrelate as follow

2 2

siny r

z z

r x y

z z

=

=

= +

=


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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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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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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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2r

rrrrr

===

dd

d

rrrr 120lim)(

1r


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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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Integration

dddrrdV =

V dVzrf ),,(


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Spherical Coordinate

Spherical Coordinate Representationz

Pr

x

y),,(


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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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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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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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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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Del Operator in Spherical Coordinate

1 1

sinr

r r r

= + +

r r rr


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Integration))()()()((sin drdrdrdV =

))()()((sin2 drddr =

vdVrf ),,(


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