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