Chapter 7 :
Integrals Over Paths and surfaces
. 1
7 :
The path Integral
Definition :
the integral of f(x , Y , z) along the path :
I :
(a , b)
ftds : f(x(t) , y(t) ,t))llc'Hilldt
: fltl'l
ex :
T :
I =
10 in] ,
[ (cos(t) sin(t) +)
=
, ,
'(t) =
(-sin(t) cost) 1) , ,
115(till :
in (t) + cos2 (t) + 1 =
f(x y , z) ,
=
X2 + y2 + z
2
f([(t))
"
=
cos(t) + Sin (t) +t =
1+ +
fixiY ,ids
: (Redt :
R
+ Flin +
Ou :
IRIt
7 2
.
:
Line Integrals
↳F
.
j =
(F(TC) - '(t) at
7 [(t) :
(sin(t) cos(t) ,+ ) ,
F (x ,Y , 7) :
(X , 4 , z)
- (t) =
/COS(t) ,
-
Sin() 1) ,
F([(t)) :
(sin(t) cos(t) +) , ,
F([(t) [ ·
(t) =
sin(t)cos(t) -
cos(t) Sin(t) + + =
+
(tdt :
ex :
<(f) (los 10) =
,
sin (8) 0) , ,
0: 7
/ Sin (z)dx +
cos(z)dy -
(xy)" dz
# = -3Cs" (8) sin(t) +dX =
-3 Cos 10) Sin (8) do
If
3sin(t)cos()e dy 3sin(Ecos(E
My
:
=
=
Ie dz =
7/2
( 3(5(8)sin (A) do + 3sin(t)cP(t)dE-cos(Elsin(E) do
-
U
-Costsin()do /-cos()U .-fud---s =
U =
sin(t) -
>
du =
cos(0) &f
Change of Parametrization :
orientation preserving
/F as :F
-
-
:
orientation-reversing :
/F d5 -1. F as
.
=
ex : F(X , Y , z) =
(Yz , Xz , XY)
[(t)
=
=
(t + , , +3) >
-
7 5 , 107
[ It) =
(1 , It , 3t2)
, F([(t)) =
(+
5
,
+" +3) ,
/F d5 /" ·
=
-
j
(t)(1) + (t")(2+) ( + 3)(3 +2)dt +
/ 6tdt :
to =
984 37 ,
Line Integrals of GradientFields :
F =
Lof10X ofidy , ,
of 1dz) =
Of
as
(5t f(((b)) fle(a)
· =
-
EX : [ (t) =
(tY/ ,
sin (th(2) 0) , ,
+ECO 17 ,
Of .
d5 =
yax + Xdy + 0dz =
(4 , X 0) (dX dy , dz)
,
.
,
: f(x , y , z)
=
XY
( yax +
xay f([(1)) f(i(0))
= -
=
f(/ ,
1
,
07 -
+10 ,
0 , 0)
t
0 0
+ 1
- =
=
.
.
7 3:
.
Parametrized surfaces
Definition :
S :
image of D
S =
P(D)
& (u v) ,
:
(x (U ,
v) , y(4 , v) , z(4 , V)
>
- P :
A(X-Xo) + B(Y-Yo) + C(E 20) 0
-
:
Parametric eg . of p :
Plu v)
,
=
au + V j
+
Tangent Vectors to parametrized surfaces :
map :
[(t) $(uo =
,
+)
↑v Vl
=: O (o Volit &YU VoJt , ,
map : [ (t) =
It , vol
Fu Vl
OD OX(o Voit &Co
Voit
= :
, ,
Ou
Regular Surfaces :
FuxY + 0
Tangent Plane to a parametrized surface :
u xFv =
n
(X-Xo Y -Yo , ,
z
-
z0)
· =
g
(X-Xo , y -Yo ,
z -
z0) (4 ,, 12 M3) ·
,
:
0
Integrals Over Paths and surfaces
. 1
7 :
The path Integral
Definition :
the integral of f(x , Y , z) along the path :
I :
(a , b)
ftds : f(x(t) , y(t) ,t))llc'Hilldt
: fltl'l
ex :
T :
I =
10 in] ,
[ (cos(t) sin(t) +)
=
, ,
'(t) =
(-sin(t) cost) 1) , ,
115(till :
in (t) + cos2 (t) + 1 =
f(x y , z) ,
=
X2 + y2 + z
2
f([(t))
"
=
cos(t) + Sin (t) +t =
1+ +
fixiY ,ids
: (Redt :
R
+ Flin +
Ou :
IRIt
7 2
.
:
Line Integrals
↳F
.
j =
(F(TC) - '(t) at
7 [(t) :
(sin(t) cos(t) ,+ ) ,
F (x ,Y , 7) :
(X , 4 , z)
- (t) =
/COS(t) ,
-
Sin() 1) ,
F([(t)) :
(sin(t) cos(t) +) , ,
F([(t) [ ·
(t) =
sin(t)cos(t) -
cos(t) Sin(t) + + =
+
(tdt :
ex :
<(f) (los 10) =
,
sin (8) 0) , ,
0: 7
/ Sin (z)dx +
cos(z)dy -
(xy)" dz
# = -3Cs" (8) sin(t) +dX =
-3 Cos 10) Sin (8) do
If
3sin(t)cos()e dy 3sin(Ecos(E
My
:
=
=
Ie dz =
7/2
( 3(5(8)sin (A) do + 3sin(t)cP(t)dE-cos(Elsin(E) do
-
U
-Costsin()do /-cos()U .-fud---s =
U =
sin(t) -
>
du =
cos(0) &f
Change of Parametrization :
orientation preserving
/F as :F
-
-
:
orientation-reversing :
/F d5 -1. F as
.
=
ex : F(X , Y , z) =
(Yz , Xz , XY)
[(t)
=
=
(t + , , +3) >
-
7 5 , 107
[ It) =
(1 , It , 3t2)
, F([(t)) =
(+
5
,
+" +3) ,
/F d5 /" ·
=
-
j
(t)(1) + (t")(2+) ( + 3)(3 +2)dt +
/ 6tdt :
to =
984 37 ,
Line Integrals of GradientFields :
F =
Lof10X ofidy , ,
of 1dz) =
Of
as
(5t f(((b)) fle(a)
· =
-
EX : [ (t) =
(tY/ ,
sin (th(2) 0) , ,
+ECO 17 ,
Of .
d5 =
yax + Xdy + 0dz =
(4 , X 0) (dX dy , dz)
,
.
,
: f(x , y , z)
=
XY
( yax +
xay f([(1)) f(i(0))
= -
=
f(/ ,
1
,
07 -
+10 ,
0 , 0)
t
0 0
+ 1
- =
=
.
.
7 3:
.
Parametrized surfaces
Definition :
S :
image of D
S =
P(D)
& (u v) ,
:
(x (U ,
v) , y(4 , v) , z(4 , V)
>
- P :
A(X-Xo) + B(Y-Yo) + C(E 20) 0
-
:
Parametric eg . of p :
Plu v)
,
=
au + V j
+
Tangent Vectors to parametrized surfaces :
map :
[(t) $(uo =
,
+)
↑v Vl
=: O (o Volit &YU VoJt , ,
map : [ (t) =
It , vol
Fu Vl
OD OX(o Voit &Co
Voit
= :
, ,
Ou
Regular Surfaces :
FuxY + 0
Tangent Plane to a parametrized surface :
u xFv =
n
(X-Xo Y -Yo , ,
z
-
z0)
· =
g
(X-Xo , y -Yo ,
z -
z0) (4 ,, 12 M3) ·
,
:
0
