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Saturday Sept 19th: Vector Calculus. Vector derivatives of a scalar field : gradient, directional derivative, Laplacian Vector derivatives of a vector field : divergence, curl. Scalar fields. unit vectors. e.g. temperature oxygen content. - PowerPoint PPT Presentation
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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