tuesday sept 21st: vector calculus derivatives of a scalar field: gradient, directional derivative,...
TRANSCRIPT
Tuesday Sept 21st: Vector Calculus
•Derivatives of a scalar field: gradient, directional derivative, Laplacian
•Derivatives of a vector field: divergence, curl
21 ˆ( )tu u u u p gk u
0u
A homogeneous fluid on a rotating sphere
Why we need it
21 ˆ( )tu u u u p gk u
0u
A homogeneous fluid on a rotating sphere
Why we need it
vectordivergence
derivative
crossproduct
gradient unit vector
Laplacian
Scalar fieldsno directionality, e.g. temperature, oxygen content
( , )
or
, ,
( , )
x y z
x
t
t
i
j
kx
ˆˆ ˆx x y zi j k
Cartesiancoordinates
e.g. surface pressure = P(longitude, latitude)
Differentiating a scalar: directional derivative, gradient
,
where { , , }
and
( )
( , , , ) or ( , )
, ,
·
x y z t x
x y z
d dxdt t dt
t
t x y zt x y z
t xt
x x y z
x x t
gradient
directionalderivative
Pressure gradient
Gradient is a VECTOR
Pressure gradient force
Pressure gradient force1F p
Examples
2
2
2
( )
gradient:
( ) 2
( )
( )
time derivative:
( , ) sin( ); 0
( , ) sin( );
( , ) sin( ); 2
( , , ) sin( ); 2 , 0
( , , ) sin( );
y z
x x
y
y
x x y z
x xe
x e
x t x ct x
x t x ct x ct
x t x ct x ct
x y t e x ct x ct y
x y t e x ct x
2 , ct y x
,
( , , , ) or
,
·
( , )
x y z
d 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”,
ˆˆ ˆ
SO: , , , ,
OR: i j kx y z
x y z x y z
The Laplacian
2 2 222 2 2
x y z
Laplacian is a SCALAR
2,3 dimensional PDEs
22
2 2 22
2
2 2 2
2 2
t tx
c ct x t
Diffusion eq’n
Wave eq’n
x
y
z
( , , )u u x y z
Vector fieldse.g. velocity, acceleration, gradient
( ) { ( ), ( ), ( )} , ,
( , ) { ( , ), ( , ), ( , )} , ,
or ,
du du dv dwu x u x v x w xdx dx dx dx
u u v wu x t u x t v x t w x tx 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 wux y z x y z
Divergence is a SCALAR.
Curlˆˆ ˆ
ˆˆ ˆ( ) ( ) ( )
i j kw v w u v uu i j k
x y z y z x z x yu v w
Which way does the curl vector point?
Example: river flow
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
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
2
1
2
1
2
sin
sin0
( ) 0
( ) 0
: 0
: 2 0
t yy
t yy
yy
y
u g Ku
gu u a
K
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
ua
h
u yu a y h y h u
h h
Example: river flow
y h
y h
2
0 21 yu uh
ˆˆ ˆ( ) ( ) ( )ˆ =
zy x x x y
y
u i w v j w u k v u
ku
curl
The Laplacian
2 2 222 2 2
· , , , ,x y z x y x x y z
Horizontal divergence· 0
: 0
u v wux y z
w u vBUTz x y
Modeling rain( 2 )
( 2 )
1. Set 0, compute convergence.
2. Set , compute divergence.
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 z z H
3. Compute for all .x yu v z