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