Today s lecture - University of Saskatchewanphysics.usask.ca/~alex/ep356/EP356session2c.pdf · 3....
Transcript of Today s lecture - University of Saskatchewanphysics.usask.ca/~alex/ep356/EP356session2c.pdf · 3....
III. Integral calculus: • Line integrals • Surface integrals • Volume integrals • Fundamental theorem of calculus • Fundamental theorem for gradients • Fundamental theorem for divergences • Fundamental theorem for curls • Integration by parts
Today’s lecture
Line integral of vector function from point a to point b along path P: If the path forms a closed loop (a=b) one writes:
Integral calculus
∫ ⋅b
Paldf
∫ ⋅P
ldf
circulation of f around P
Example: (Griffith 1.28)
Calculate integral of
From the origin to the point (1,1,1) by :
a. route
b. straight line
( ) ( ) ( ) ( )1,1,10,1,10,0,10,0,0 →→→
( )22 ,2, yyzxf =
Surface integral of vector function over surface S: is (in each point) the normal vector and therefore perpendicular to the surface S. If the surface is closed one writes:
Integral calculus
∫∫∫ ⋅=⋅SS
adfadf
∫∫∫ ⋅=⋅SS
adfadf
ad
flux of f through S
dzdydxd ⋅⋅=τ
∫∫∫∫ ⋅=⋅VV
dfdf ττ
Volume integral of scalar field (or vector field)
over the volume element dτ is
The volume element dτ is
(in Cartesian coordinates):
Example: (Griffith 1.30)
Calculate the volume integral of the function f=z2 over the tetrahedron with corners at (0,0,0), (1,0,0), (0,1,0), and (0,0,1).
1. The fundamental theorem of calculus:
Integral over derivative equal to value of function f(x) (at boundaries) or Instead of adding the infinitesimal changes df of function f between two points A and B, one can subtract the values of f at the two points.
Fundamental theorems 1
( ) ( ) )(: afbfdxxFdfdxdxdf b
a
b
a
b
a−=⋅==⋅ ∫∫∫
( ) )(afbfdxdxdfb
a−=⋅∫
3. The divergence theorem: (also Gauss’s or Green’s theorem)
Volume integral of derivative equal to • value of function at boundaries (closed surface) or • total flux of vector field through closed surface S. • It is not necessary to know the function at all points, It is sufficient to know the function on a (closed) surface.
Fundamental theorems 2
( ) ∫∫∫∫∫ ⋅=⋅⋅∇SV
adfdf τ
2. The gradient theorem:
• Special property of the gradient: Line integrals of gradients do not depend on path of integration P!
• Note that (since a=b)
)()( afbfldfb
Pa
−=⋅∇∫
0=⋅∇∫ ldf
4. The curl theorem: (also Stokes’ theorem) (open) surface integral of derivative equal to • value of function at boundaries (closed path). • It is sufficient to know the field along the boundary line enclosing the (open) surface. • Important: All theorems are valid only for Problems with continuity – especially on surfaces - will be discussed later (boundary problems).
Fundamental theorems 3
( ) ∫∫∫ ⋅=⋅×∇PS
ldfadf
Soncontinuousff
Voncontinuousff
×∇
⋅∇
,
,
( ) 0=⋅×∇∫∫ adf
Integration of the product rule for derivatives Example:
Integration by parts
( )
( )
∫∫
∫ ∫∫
+=
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛==
+=
b
a
b
a
ba
b
a
b
a
b
a
ba
dxgfdxgffg
dxdxdfgdx
dxdgffgdxfg
dxd
dxdfg
dxdgffg
dxd
''
∫2
0
2 )cos(
π
dxxx
I. Vector calculus • How to transform vectors: the transformation matrix Coordinate systems: • Spherical and cylindrical coordinate systems • Gradient, divergence and curl in different coordinate
systems
Today’s lecture
How can a vector be represented in different coordinate systems? Key point: Projection of onto unit vector is Another way of expressing this in a more elegant way by using matrixes:
Transformation Matrix
The Transformation matrix
''''' kAjAiAA
kAjAiAA
zyx
zyx
⋅+⋅+⋅=
⋅+⋅+⋅=
'iA⋅A
'i
'''''
'''''
'''''
zzyx
yzyx
xzyx
AkkAkjAkiAkA
AjkAjjAjiAjA
AikAijAiiAiA
=⋅⋅+⋅⋅+⋅⋅=⋅
=⋅⋅+⋅⋅+⋅⋅=⋅
=⋅⋅+⋅⋅+⋅⋅=⋅
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⋅⋅⋅
⋅⋅⋅
⋅⋅⋅
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
z
y
x
z
y
x
AAA
kkkjki
jkjjji
ikijii
AAA
'''
'''
'''
'
''
1. Cartesian coordinates x, y, z
( )∞<<∞−
=++=
zyxwithzyxuzuyuxr zyx
,,
,,
( )( )
dzdydxddydxdzdxdzdyudydxudzdxudzdyad
dzdydxudzudyudxld
zyx
zyx
=
=++=
=++=
τ
,,
,,
xuzu
yu
x
y
z
dz
dx
dy
r
z
xy
2. Cylindrical coordinates s, φ, z
( ) zzss uAuAuAzsr ++= φφφ ,,
dzddssduddssudzdsudzdsad
udzudsudsld
zs
zs
φτ
φφ
φ
φ
φ
=
++=
++=
zzsysx
=
=
=
φ
φ
sincos
zz
sy
sx
zz
yx
yxs
uuuuuuuu
uuuuuuuu
=
+=
−=
=
+−=
+=
φ
φ
φ
φφ
φφ
φφ
φφ
cossinsincos
cossinsincos
s
z
z
y
xsu
φu
zuφsd
dz
ds
s φ
z
With 0 ≤ s < ∞ 0 ≤ φ ≤ 2π - ∞ < z < ∞
3. Spherical coordinates r, ϑ, φ
φφϑϑ uAuAuAr rr
++=
( )
φϑϑτ
ϑφϑφϑϑ
φϑϑφϑϑ
φϑ
φϑ
dddrrd
uddrruddrruddrad
drdrdrudrudrudrld
r
r
sin
sinsin
sin,,sin
2
2
=
++=
=++=
ϑ
φϑ
φϑ
φ
ϑ
ϑϑ
φφϑφϑ
φφϑφϑ
φφ
ϑφϑφϑ
ϑφϑφϑ
uuuuuuuuuuu
uuuuuuuuuuu
rz
ry
rx
yx
zyx
zyxr
sincoscossincossinsinsincoscoscossin
cossinsinsincoscoscoscossinsincossin
−=
++=
−+=
+−=
−+=
++=
ϑφϑφϑ cossinsincossin rzryrx ===
dr
φϑ drsin
z
y
xφ
ϑ r
With 0 ≤ r < ∞ 0 ≤ ϑ ≤ π 0 ≤ φ ≤ 2π
φu
ϑuφd
ru
The transformation matrixes
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
−
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
−=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
φ
ϑ
φ
ϑ
ϑϑ
φφϑφϑ
φφϑφϑ
φφ
ϑφϑφϑ
ϑφϑφϑ
AAA
AAA
AAA
AAA
r
z
y
x
z
y
xr
0sincoscossincossinsinsincoscoscossin
0cossinsinsincoscoscoscossinsincossin
For cylindrical coordinates:
For spherical coordinates:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛ −
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−=⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
z
s
z
y
x
z
y
x
z
s
AAA
AAA
AAA
A
AA
φφφ
φφ
φφ
φφ
φ
1000cossin0sincos
1000cossin0sincos
The infinitesimal surface elements
For cylindrical coordinates:
For spherical coordinates:
ds
Gradient, divergence, curl, Laplacian in cylindrical coordinates s, φ, z
zs uzfuf
susff
δδ
δφδ
δδ
φ ++=∇1
( )zff
ssf
ssf z
s δδ
δφ
δ
δδ φ ++=∇
11
( ) zszs
sz ufsf
ssu
sf
zfu
zff
sf
⎥⎦
⎤⎢⎣
⎡−+⎟
⎠⎞
⎜⎝⎛ −+⎟⎟
⎠
⎞⎜⎜⎝
⎛−=×∇
δφδ
δδ
δδ
δδ
δ
δ
δφδ
φφφ 11
2
2
2
2
22 11
zff
ssfs
ssff
δδ
δφδ
δδ
δδ
++⎟⎠
⎞⎜⎝
⎛=Δ=∇
Curl
Gradient
Divergence
Laplacian
Gradient, divergence, curl, Laplacian in spherical coordinates r, ϑ, φ
φϑ δφδ
ϑδϑδ
δδ uf
ruf
rurff r
sin11
++=∇
( ) ( )δφ
δ
ϑϑ
δϑδ
ϑδδ φ
ϑ
fr
fr
frrr
f r sin1sin
sin11 2
2 ++=∇
( )
( ) ( ) φϑϑφ
ϑφ
δϑδ
δδ
δδ
δφδ
ϑ
δφδ
ϑδϑδ
ϑ
uffrrr
ufrr
fr
uffr
f
rr
r
⎥⎦
⎤⎢⎣
⎡−+⎥
⎦
⎤⎢⎣
⎡−+
⎥⎦
⎤⎢⎣
⎡−=×∇
1sin11
sinsin1
2
2
2222
22
sin1sin
sin11
δφδ
ϑδϑδ
ϑδϑδ
ϑδδ
δδ f
rf
rrfr
rrf +⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=∇
Curl
Gradient
Divergence
Laplacian