Saturday Sept 19th: Vector Calculus

•Vector derivatives of a scalar field: gradient, directional derivative, Laplacian•Vector derivatives of a vector field: divergence, curl

i

j

kx

Scalar fields

ˆˆ ˆx x y zi j k

unit vectors

( , )

or

, ,

( , )

x y z

x

t

t

e.g. temperature oxygen content

e.g. surface pressure = P(longitude, latitude)

Differentiating a scalar: directional derivative, gradient

· ,

where { , , }

and

( , , , ) or

( )

( ,

,

·

)

,x y x

d dxdt t d

x y

t x y zt x y z

z t

t xt

x x y

tt

z

x

x t

x

gradient

directionalderivative

Pressure gradient

Pressure gradient force

Pressure gradient force1F p

Examples

2 2

2

2

2

( )

( )

gradient:

( ) 2

( )

( )

( )

( ) sin( )

time derivative:

( , ) sin( ); 0

( , ) sin( );

( , ) sin( ); 2

( , , ) sin( );

y z

x x

x y

y

x x y z

x x yz y

x xe

x e

x e x z

x t x ct x

x t x ct x ct

x t x ct x ct

x y t e x ct x

2

2 , 0

( , , ) sin( ); 2 , y

ct y

x y t e x ct x ct y x

,

( , , , ) or

,

·

( , )

x y xd dx

f x y z

d

t

t dt

t

t

x

A vector differential operator

ˆˆ ˆ, , i j kx y z x y z

“Del”, or “Nabla”,

The Laplacian

2 2 222 2 2 · , , , ,

x y z x y x x y z

2,3 dimensional PDEs

2 222 20; e.g. tent: 0x y

2

2

2 22

t

ct

Laplace’s eq’n

Diffusion eq’n

Wave eq’n

x

y

z

( , , )u u x y z

Vector fields

( ) { ( ), ( ), ( )} , ,

( , ) { ( , ), ( , ), ( , )} , ,

or ,

du du dv dwu x u x v x w x dx dx dx dx

u u v wu x t u x t v x t w x t x x x x

u u vt t

,

( , , , ) ( , ) , ,

or , ,

wt t

u u v wu x y z t u x ty y y y

u u v wz z z z

Differentiating a vector fieldexamples

Divergence of a vector field

, ,· ,, u v w u v wu x y z x y z

Modeling rain

( 2 )

( 2 )

1. Set 0, compute .

2. Set , compute .

x y

x y

u y H z x

v x H z y

z u v

z H u v

4. 0 ; 0 at 0.

Solve for ( , , ) in 0 .

zx yu v w w z

w x y z z H

3. Compute for all .x yu v z

Curl

ˆˆ ˆˆˆ ˆ( ) ( ) ( )z zy x x yx y z

i j ku i w v j w u k v u

u v w

Example: river flow

y h

y h

Example: river flowsinyytu Ku g

Diffusion(friction)

Concentration ofvelocity diffuses away

Example: river flowsinyytu Ku g

gsing

gravity

Example: river flowsin yytu g Ku

2

2

2

2

1

2

1

21

2

1

2

sin

sin0

( ) 0

( ) 0

: 0

: 2 0

t yy

t yy

yy

y

u g Ku

gu u aK

u a

u ay b

u ay by c

u h ah bh c

u h ah bh c

ADD ah c

SUBT bh

2

2 2 2 2

20

02

22 2 2 20

02 2

1

2

1 1 1

2 2 2

1

2

1

2

1

2

0,

0 ( )

At 0,

( ) ( ) 1

b c ah

u ay ah a y h

y u u ah

uah

u yu a y h y h uh h

Example: river flow

y h

y h

20 21 yu u

h

ˆˆ ˆ( ) ( ) ( )ˆ =

zy x x x y

y

u i w v j w u k v u

ku

curl

Identities of vector calculus

2

2 2 2

·( ) 0

( ) 0

( ) ( · )

( )

·( ) · ·

( )

·( ) ·( ) ·( )

( ) 2 ·

u

u u u

u u u

u u u

u v v u u v