- Agnes Scott College

22 Pages·2012·0.42 MB·English

r
gradient, divergence and curl
Eddie Wilson
[email protected]
Department of Engineering Mathematics
University of Bristol
r
–p.1/11
New questions:
What is r ?
Can it be used in other ways?
Last time : : :
m
R R
Differential calculus of functions f : ! , m = 2; 3
Directional derivatives D f given in terms of vector
n^
gradient rf:
n
D f = rf (cid:1) ^
n^
where
@f=@x
0 1
rf := @f=@y for m = 3:
B C
@f=@z
@ A
r
–p.2/11
Last time : : :
Rm R
Differential calculus of functions f : ! , m = 2; 3
Directional derivatives D f given in terms of vector
n^
gradient rf:
n
D f = rf (cid:1) ^
n^
where
@f=@x
0 1
rf := @f=@y for m = 3:
B C
@f=@z
@ A
New questions:
What is r ?
Can it be used in other ways?
r
–p.2/11
Therefore might say
@=@x
0 1
r := @=@y
B C
@=@z
@ A
Can r be used in other ways?
The vector r (pronounced Del)
Vector gradient
@f=@x @=@x
0 1 0 1
rf := @f=@y = @=@y f
B C B C
@f=@z @=@z
@ A @ A
r
–p.3/11
The vector r (pronounced Del)
Vector gradient
@f=@x @=@x
0 1 0 1
rf := @f=@y = @=@y f
B C B C
@f=@z @=@z
@ A @ A
Therefore might say
@=@x
0 1
r := @=@y
B C
@=@z
@ A
Can r be used in other ways?
r
–p.3/11
u u
Define the divergence r (cid:1) (pronounced div )
@=@x u(x; y; z)
0 1 0 1
u
r (cid:1) := @=@y (cid:1) v(x; y; z)
B C B C
@=@z w(x; y; z)
@ A @ A
@u @v @w
= + + :
@x @y @z
f Rn Rn
NB Similar definitions for : ! for n 6= 3.
=
Divergence r(cid:1)
u R3 R3
Suppose given vector field : ! .
u u x u T x T
= ( ), = (u; v; w) , = (x; y; z) .
r
–p.4/11
f Rn Rn
NB Similar definitions for : ! for n 6= 3.
=
Divergence r(cid:1)
u R3 R3
Suppose given vector field : ! .
u u x u T x T
= ( ), = (u; v; w) , = (x; y; z) .
u u
Define the divergence r (cid:1) (pronounced div )
@=@x u(x; y; z)
0 1 0 1
u
r (cid:1) := @=@y (cid:1) v(x; y; z)
B C B C
@=@z w(x; y; z)
@ A @ A
@u @v @w
= + + :
@x @y @z
r
–p.4/11
=
Divergence r(cid:1)
u R3 R3
Suppose given vector field : ! .
u u x u T x T
= ( ), = (u; v; w) , = (x; y; z) .
u u
Define the divergence r (cid:1) (pronounced div )
@=@x u(x; y; z)
0 1 0 1
u
r (cid:1) := @=@y (cid:1) v(x; y; z)
B C B C
@=@z w(x; y; z)
@ A @ A
@u @v @w
= + + :
@x @y @z
f Rn Rn
NB Similar definitions for : ! for n 6= 3.
r
–p.4/11
u u
Define r (cid:2) (pronounced curl )
@=@x u(x;y;z)
r (cid:2) u (cid:2)
:= @=@y v(x;y;z)
@=@z w(x;y;z)
(cid:0)
@w=@y @v=@z
= @u=@z (cid:0) @w=@x :
(cid:0)
@v=@x @u=@y
f Rn Rn
NB No similar definition for : ! for n > 3.
=
Curl r(cid:2)
u R3 R3
Suppose given vector field : ! .
u u x u T x T
= ( ), = (u; v; w) , = (x; y; z) .
r
–p.5/11
f Rn Rn
NB No similar definition for : ! for n > 3.
=
Curl r(cid:2)
u R3 R3
Suppose given vector field : ! .
u u x u T x T
= ( ), = (u; v; w) , = (x; y; z) .
u u
Define r (cid:2) (pronounced curl )
(cid:0) (cid:5) (cid:0) (cid:5)
@=@x u(x;y;z)
r (cid:2) u := @(cid:1)(cid:2) =@y (cid:6)(cid:2)(cid:2) v(cid:1)(cid:2) (x;y;z) (cid:6)(cid:2)
(cid:1)(cid:4) (cid:6)(cid:4) (cid:1)(cid:4) (cid:6)(cid:4)
(cid:3) (cid:7) (cid:3) (cid:7)
@=@z w(x;y;z)
(cid:0) (cid:0) (cid:5)
@w=@y @v=@z
= @(cid:1)(cid:2) u=@z (cid:0) @w=@x (cid:6)(cid:2):
(cid:1)(cid:4) (cid:6)(cid:4)
(cid:3) (cid:7)
(cid:0)
@v=@x @u=@y
r
–p.5/11