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Transcript of Dr. Hugh Blanton ENTC 3331 Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 2 Fields and...
Dr. Hugh Blanton
ENTC 3331
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 2
Fields and Waves
VECTORS and VECTOR CALCULUS
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 3
VECTORS
Today’s Class will focus on:
• vectors - description in 3 coordinate systems
• vector operations - DOT & CROSS PRODUCT
• vector calculus - AREA and VOLUME INTEGRALS
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 4
VECTOR REPRESENTATION
3 PRIMARY COORDINATE SYSTEMS:
• RECTANGULAR
• CYLINDRICAL
• SPHERICAL
Choice is based on symmetry of problem
Examples:
Sheets - RECTANGULAR
Wires/Cables - CYLINDRICAL
Spheres - SPHERICAL
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 5
Orthogonal Coordinate Systems: (coordinates mutually perpendicular)
Spherical Coordinates
Cylindrical Coordinates
Cartesian CoordinatesP (x,y,z)
P (r, Θ, Φ)
P (r, Θ, z)
x
y
zP(x,y,z)
θ
z
rx y
z
P(r, θ, z)
θ
Φ
r
z
yx
P(r, θ, Φ)
Page 108
Rectangular Coordinates
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 6
-Parabolic Cylindrical Coordinates (u,v,z)-Paraboloidal Coordinates (u, v, Φ)-Elliptic Cylindrical Coordinates (u, v, z)-Prolate Spheroidal Coordinates (ξ, η, φ)-Oblate Spheroidal Coordinates (ξ, η, φ)-Bipolar Coordinates (u,v,z)-Toroidal Coordinates (u, v, Φ)-Conical Coordinates (λ, μ, ν)-Confocal Ellipsoidal Coordinate (λ, μ, ν)-Confocal Paraboloidal Coordinate (λ, μ, ν)
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 7
Parabolic Cylindrical CoordinatesParabolic Cylindrical Coordinates
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 8
Paraboloidal CoordinatesParaboloidal Coordinates
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 9
Elliptic Cylindrical CoordinatesElliptic Cylindrical Coordinates
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 10
Prolate Spheroidal CoordinatesProlate Spheroidal Coordinates
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 11
Oblate Spheroidal CoordinatesOblate Spheroidal Coordinates
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 12
Bipolar CoordinatesBipolar Coordinates
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 13
Toroidal CoordinatesToroidal Coordinates
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 14
Conical CoordinatesConical Coordinates
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 15
Confocal Ellipsoidal CoordinateConfocal Ellipsoidal Coordinate
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 16
Confocal Paraboloidal CoordinateConfocal Paraboloidal Coordinate
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 17
Cartesian CoordinatesP(x,y,z)
Spherical CoordinatesP(r, θ, Φ)
Cylindrical CoordinatesP(r, θ, z)
x
y
zP(x,y,z)
θ
z
rx y
z
P(r, θ, z)
θ
Φ
r
z
yx
P(r, θ, Φ)
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 18
VECTOR NOTATION
VECTOR NOTATION:
zzyyxx aAaAaAA ˆˆˆ Rectangular or
Cartesian Coordinate
System
x
z
y
zzyyxx BABABABA
Dot Product
zyx
zyx
zyx
BBB
AAA
aaa
BA
ˆˆˆ
Cross Product
2
1222zyx AAAA
Magnitude of vector
(SCALAR)
(VECTOR)
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 19
Cartesian Coordinates
zyx AzAyAxA ˆˆˆ
Page 109
x
y
z
Z plane
y planex plane
222zyx AAAAAA
xyz
x1
y1
z1
Ax
Ay
Az
( x, y, z)Vector representation
Magnitude of A
Position vector A
),,( 111 zyxA
111 ˆˆˆ zzyyxx
Base vector properties
0ˆˆˆˆˆˆ
1ˆˆˆˆˆˆ
xzzyyx
zzyyxx
yxz
xzy
zyx
ˆˆˆ
ˆˆˆ
ˆˆˆ
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 20
x
y
z
Ax
Ay
AzA
B
Dot product:
zzyyxx BABABABA
Cross product:
zyx
zyx
BBB
AAA
zyx
BA
ˆˆˆ
Back
Cartesian Coordinates
Page 108
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 21
VECTOR REPRESENTATION: CYLINDRICAL COORDINATES
Cylindrical representation uses: r ,, z
zzrr aAaAaAA ˆˆˆ
zzrr BABABABA
UNIT VECTORS:
zr aaa ˆˆˆ
Dot Product(SCALAR)
r
z
P
x
z
y
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 22
VECTOR REPRESENTATION: SPHERICAL COORDINATES
r
P
x
z
y
Spherical representation uses: r ,, UNIT VECTORS:
aaar ˆˆˆ
aAaAaAA rr ˆˆˆ
BABABABA rr
Dot Product(SCALAR)
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 23
x
z
y
VECTOR REPRESENTATION: UNIT VECTORS
yaxa
za Unit Vector Representation for Rectangular
Coordinate System
xaThe Unit Vectors imply :
ya
za
Points in the direction of increasing x
Points in the direction of increasing y
Points in the direction of increasing z
Rectangular Coordinate System
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 24
r
z
P
x
z
y
VECTOR REPRESENTATION: UNIT VECTORS
Cylindrical Coordinate System
za
a
ra
The Unit Vectors imply :
za
Points in the direction of increasing r
Points in the direction of increasing
Points in the direction of increasing z
ra
a
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 25
VECTOR REPRESENTATION: UNIT VECTORS
Spherical Coordinate System
r
P
x
z
y
a
a
ra
The Unit Vectors imply :
Points in the direction of increasing r
Points in the direction of increasing
Points in the direction of increasing
ra
aa
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 26
zr aaa ˆˆˆ aaar ˆˆˆ zyx aaa ˆˆˆ
RECTANGULAR Coordinate Systems
CYLINDRICAL Coordinate Systems
SPHERICAL Coordinate Systems
NOTE THE ORDER!
r,, z r,,
Note: We do not emphasize transformations between coordinate systems
VECTOR REPRESENTATION: UNIT VECTORS
Summary
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 27
METRIC COEFFICIENTS
1. Rectangular Coordinates:
When you move a small amount in x-direction, the distance is dx
In a similar fashion, you generate dy and dz
Unit is in “meters”
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 28
Cartesian Coordinates
Differential quantities:
Length:
Area:
Volume:
dzzdyydxxld ˆˆˆ
dxdyzsd
dxdzysd
dydzxsd
z
y
x
ˆ
ˆ
ˆ
dxdydzdv
Page 109
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 29
METRIC COEFFICIENTS
2. Cylindrical Coordinates:
Distance = r d
x
y
d
r
Differential Distances:
( dr, rd, dz )
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 30
3. Spherical Coordinates:
Distance = r sin d
x
y
d
r sin
Differential Distances:
( dr, rd, r sind )
r
P
x
z
y
METRIC COEFFICIENTS
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 31
Representation of differential length dl in coordinate systems:
zyx adzadyadxld ˆˆˆ
zr adzadradrld ˆˆˆ
adrardadrld r ˆsinˆˆ
rectangular
cylindrical
spherical
METRIC COEFFICIENTS
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 32
AREA INTEGRALS
• integration over 2 “delta” distances
dx
dy
Example:
x
y
2
6
3 7
AREA = 7
3
6
2
dxdy = 16
Note that: z = constant
In this course, area & surface integrals will be on similar types of surfaces e.g. r =constant or = constant or = constant et c….
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 33
Representation of differential surface element:
zadydxsd ˆ
Vector is NORMAL to surface
SURFACE NORMAL
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 34
DIFFERENTIALS FOR INTEGRALS
Example of Line differentials
or or
Example of Surface differentials
zadydxsd ˆradzrdsd ˆ
or
Example of Volume differentials dzdydxdv
xadxld ˆ
radrld ˆ
ardld ˆ
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 35
BaseVectors
A1
r radial distance in x-y plane
Φ azimuth angle measured from the positive x-axis
Z
r0
20
z
Cylindrical Coordinates
ˆˆˆ
,ˆˆˆ
,ˆˆˆ
rz
rz
zr
zr AzAArAaA ˆˆˆˆ
Pages 109-112Back
( r, θ, z)
Vector representation
222zr AAAAAA
Magnitude of A
Position vector A
Base vector properties
11 ˆˆ zzrr
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 36
Dot product:
zzrr BABABABA
Cross product:
zr
zr
BBB
AAA
zr
BA
ˆˆˆ
B A
Back
Cylindrical Coordinates
Pages 109-111
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 37
Cylindrical Coordinates
Differential quantities:
Length:
Area:
Volume:
dzzrddrrld ˆˆˆ
rdrdzsd
drdzsd
dzrdrsd
z
r
ˆ
ˆ
ˆ
dzrdrddv
Pages 109-112
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 38
ˆˆˆ,ˆˆˆ,ˆˆˆ RRR
Spherical Coordinates
Pages 113-115Back
(R, θ, Φ)
AAARA Rˆˆˆ
Vector representation
222 AAAAAA R
Magnitude of A
Position vector A
1ˆRR
Base vector properties
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 39
Dot product:
BABABABA RR
Cross product:
BBB
AAA
R
BA
R
R
ˆˆˆ
Back
B A
Spherical Coordinates
Pages 113-114
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 40
Spherical Coordinates
Differential quantities:
Length:
Area:
Volume:
dRRddRR
dldldlRld R
sinˆˆˆ
ˆˆˆ
RdRddldlsd
dRdRdldlsd
ddRRdldlRsd
R
R
R
ˆˆ
sinˆˆ
sinˆˆ 2
ddRdRdv sin2
dRdl
Rddl
dRdlR
sin
Pages 113-115Back
Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 41
zz
yx
yxr
ˆˆ
cosˆsinˆˆ
sinˆcosˆˆ
zz
yx
yxr
AA
AAA
AAA
cossin
sincos
Back
Cartesian to Cylindrical Transformation
zz
xy
yxr
)/(tan 1
22
Page 115