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1.3 Integral Calculus. 1.3.1 Line, Surface, Volume Integrals. a) line integral:. Example 1.6. For a given boundary line there many different surfaces, on which the surface integral depends. It is independent only if. If the surface is closed:. b) surface integral:. 2. 2. 2. - PowerPoint PPT Presentation
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1.3 Integral Calculus
1.3.1 Line, Surface, Volume Integrals
a) line integral:
Vvexcept path, theon Depends
ncirculatio if
,
P
P
lvba
lvb
a
d
d
b
a
lF dW
workl Mechanica:Example
Example 1.6
. to from pathes two thealongˆ)1(2ˆ of integral line theCalculate 2
bayxv yxy
patch. thislar toperpendicu is
surface, theofpatch malinfinitesian is
,flux
aa
av
d
d
d S
b) surface integral:
If the surface is closed: S
av d
For a given boundary line there manydifferent surfaces, on which the surface integral depends. It is independent only if
Avv 0
Example 1.7
)(
2 )ˆ)3(ˆ)2(ˆ2(viexclude
dzyxxz azyx
22
2
volume integral:
V
),,( dxdydzddzyxT
dvdvdvd zyx zyxv ˆˆˆ
Example 1.8
prism
dxyz 2
1.3.3 Fundamental Theorem for Gradients
0 if ),()( PP
lbaablb
a
dTTTdT
The line integral does not depend on the path P.
)()(W
workl Mechanica:Example
abllFb
a
b
a
VVdVd
Example 1.9
b
a
ldxy )( 2along I-II and III
1.3.4 Fundamental Theorem for Divergences
(also Gauss’s or Green’s theorem)
V S
avv dd)(
The surface S encloses the volume V.
dx
dy
dz
Example 1.10
Check the divergence theorem for
zyxv ˆ)2(ˆ)2(ˆ 22 xyzxyy
1.3.5 Fundamental Theorem for Curls
(also Stokes’ theorem)
The path P is the boundary of the surface S.The integral does not depend on S.
S P
lvav dd)(
0)( av d
dz
dy
You must do it in a consistent way!
Example 1.11
zyv ˆ)4(ˆ)32( 22 yzyxz
Check Stokes’ Theorem for