Transcript
Vector Calculus
COORDINATE SYSTEMS
• RECTANGULAR or Cartesian
• CYLINDRICAL• SPHERICAL
Choice is based on symmetry of problem
Examples:
Sheets - RECTANGULAR
Wires/Cables - CYLINDRICAL
Spheres - SPHERICAL
Vector Calculus
Cylindrical Symmetry Spherical Symmetry
Cartesian Coordinates
P (x, y, z)
x
y
z
P(x,y,z)
Rectangular CoordinatesOr
z
y
x
A vector A in Cartesian coordinates can be written as
),,( zyx AAA or zzyyxx aAaAaA
where ax,ay and az are unit vectors along x, y and z-directions.
Cylindrical Coordinates
P (ρ, Φ, z)
x= ρ cos Φ, y=ρ sin Φ, z=z
z
Φ
z
ρx
y
P(ρ, Φ, z)
z
20
0
A vector A in Cylindrical coordinates can be written as
),,( zAAA orzzaAaAaA
where aρ,aΦ and az are unit vectors along ρ, Φ and z-directions.
zzx
yyx ,tan, 122
The relationships between (ax,ay, az) and (aρ,aΦ, az)are
zz
y
x
aa
aaa
aaa
cossin
sincos
zz
yx
yx
aa
aaa
aaa
cossin
sincosor
zzyxyx aAaAAaAAA )cossin()sincos(
Then the relationships between (Ax,Ay, Az) and (Aρ, AΦ, Az)are
zz
yx
yx
AA
AAA
AAA
cossin
sincos
z
y
x
z A
A
A
A
A
A
100
0cossin
0sincos
In matrix form we can write
Spherical Coordinates
P (r, θ, Φ)
x=r sin θ cos Φ, y=r sin θ sin Φ, Z=r cos θ
20
0
0
r
A vector A in Spherical coordinates can be written as
),,( AAAr or aAaAaA rr
where ar, aθ, and aΦ are unit vectors along r, θ, and Φ-directions.
θ
Φ
r
z
yx
P(r, θ, Φ)
x
y
z
yxzyxr 1
221222 tan,tan,
The relationships between (ax,ay, az) and (ar,aθ,aΦ)are
aaa
aaaa
aaaa
rz
ry
rx
sincos
cossincossinsin
sincoscoscossin
yx
zyx
zyxr
aaa
aaaa
aaaa
cossin
sinsincoscoscos
cossinsincossin
or
Then the relationships between (Ax,Ay, Az) and (Ar, Aθ,and AΦ)are
aAA
aAAA
aAAAA
yx
zyx
rzyx
)cossin(
)sinsincoscoscos(
)cossinsincossin(
z
y
xr
A
A
A
A
A
A
0cossin
sinsincoscoscos
cossinsincossin
In matrix form we can write
cossin
sinsincoscoscos
cossinsincossin
yx
zyx
zyxr
AAA
AAAA
AAAA
Cartesian CoordinatesP(x, y, z)
Spherical CoordinatesP(r, θ, Φ)
Cylindrical CoordinatesP(ρ, Φ, z)
x
y
zP(x,y,z)
Φ
z
rx y
z
P(ρ, Φ, z)
θ
Φ
r
z
yx
P(r, θ, Φ)
Differential Length, Area and Volume
Differential displacement
zyx dzadyadxadl
Differential area
zyx dxdyadxdzadydzadS
Differential VolumedxdydzdV
Cartesian Coordinates
Cylindrical Coordinates
ρρ
ρ
ρ
ρρ
ρ
ρ
ρρ
ρ
Differential Length, Area and Volume
Differential displacement
zdzaadaddl
Differential area
zadddzaddzaddS
Differential Volume
dzdddV
Cylindrical Coordinates
Spherical Coordinates
Differential Length, Area and Volume
Differential displacement
adrarddradl r sin
Differential area
ardrdadrdraddrdS r sinsin2
Differential Volume
ddrdrdV sin2
Spherical Coordinates
Line, Surface and Volume Integrals
Line Integral
L
dlA.
Surface Integral
Volume Integral
S
dSA.
dvpV
v
Gradient, Divergence and Curl
• Gradient of a scalar function is a vector quantity.
• Divergence of a vector is a scalar quantity.
• Curl of a vector is a vector quantity.
• The Laplacian of a scalar A
f Vector
A.
The Del Operator
A
A2
Del Operator
Cartesian Coordinates
zyx az
ay
ax
Cylindrical Coordinates
Spherical Coordinates
zazaa
1
a
ra
rar r
sin
11
The gradient of a scalar field V is a vector
that represents both the magnitude and
the direction of the maximum space rate
of increase of V.
Gradient of a Scalar
zyx az
Va
y
Va
x
VV
zaz
Va
Va
VV
1
a
V
ra
V
ra
r
VV r
sin
11
The divergence of A at a given point P is
the outward flux per unit volume as the
volume shrinks about P.
Divergence of a Vector
v
dSA
AdivA S
v
.
lim.0
z
A
y
A
x
AA
.
z
AAAA z
1)(
1.
The curl of A is an axial vector whose magnitude is the maximum circulation of A per unit area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the circulation maximum.
Curl of a Vector
nL
sa
S
dlA
AcurlA
max
0
.
lim
Where ΔS is the area bounded by the curve L and an is the unit vector normal to the surface ΔS
zyx
zyx
AAAzyx
aaa
A
z
z
AAAz
aaa
A
1
ArrAA
r
arraa
rA
r
r
sin
sin
sin
12
Cartesian Coordinates Cylindrical Coordinates
Spherical Coordinates
The divergence theorem states that the
total outward flux of a vector field A
through the closed surface S is the same
as the volume integral of the divergence
of A.
Divergence or Gauss’ Theorem
V
AdvdSA ..
L S
dSAdlA ).(.
Stokes’ TheoremStokes’s theorem states that the circulation of a
vector field A around a closed path L is equal to
the surface integral of the curl of A over the open
surface S bounded by L, provided A and
are continuous on S
A
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