Transcript 1.3 Integral Calculus 1.3.1 Line, Surface, Volume Integrals

1.3 Integral Calculus
1.3.1 Line, Surface, Volume Integrals
a) line integral:
b
 v  dl ,
aP
if a  b
 v  dl  circulatio n
P
Depends on the path,
except v  V
Example : Mechanica l work
b
W    F  dl
a
Example 1.6
Calculate the line integral of v  y 2xˆ  2 x( y  1) yˆ
along the two pathes from a to b.
b) surface integral:
 v  da  flux
,
S
da is an infinitesi mal patch of the surface,
da is perpendicu lar to this patch.
For a given boundary line there many
different surfaces, on which the surface
integral depends. It is independent only if
v  0
v  A
If the surface is closed:
 v  da
S
Example 1.7
2
ˆ
ˆ
(
2
xz
x

(
x

2
)
y

y
(
z
 3)zˆ )  da

exclude ( vi )
2
2
2
volume integral:
 T ( x, y, z )d
d  dxdydz
V
 vd  xˆ  v d  yˆ  v d zˆ  v d
x
y
z
Example 1.8
2
xyz
 d
prism
1.3.3 Fundamental Theorem for
Gradients
b
 T  dl  T (b)  T (a),
if a  b
aP
 T  dl  0
P
The line integral does not depend on the path P.
Example : Mechanica l work
b
b
a
a
W    F  dl   V  dl  V (b)  V (a)
Example 1.9
b
2

(
xy
)  dl

a
along I-II and III
1.3.4 Fundamental Theorem for
Divergences
(also Gauss’s or Green’s theorem)
 (   v ) d   v  da
V
S
The surface S encloses the volume V.
dz
dy
dx
Example 1.10
Check the divergence theorem for
2
ˆ
v  y x  (2 xy  z ) yˆ  (2 xy)zˆ
2
1.3.5 Fundamental Theorem for
Curls
(also Stokes’ theorem)
 (  v)  da   v  dl
S
P
The path P is the boundary of the surface S.
The integral does not depend on S.
 (  v)  da  0
dz
dy
You must do it in a consistent way!
Example 1.11
Check Stokes’ Theorem for
2
ˆ
v  (2 xz  3 y ) y  (4 yz )zˆ
2