Lecture notes were made while watching the recorded lectures assigned to watch. These notes include theory (theorems) and worked out examples from the lecturer.
These specific notes cover Line integrals.
3. 2. Line integrals
◦
integrate over a curve in #% =
line integrals
1123
I. Stds if f- real-valued
¢ function defined on
the curve @
p vector field
2. J F. di if É is a vector field
on the curve c
c
'
have to first
' describe
To determine a line integral we
the curve that is find a vector valued function
-
#
Parametrize
if can 't parametrize
FCH E- C- [q ,b ] you a
, curve =
can't do line integral
C :
Fct ) cxct), > 1- C- [ 9lb ] AY
yctl 2- It )
=
y☒•Ñ•
, ,
Ifaf initial point of curve
=
•
F (b) =
terminal (end) point of curve Feb)
r
•
>n
study guide
~
Standard Parametrization s
from A Cal / 92,93)
① Line
segment to B Cbi bi bs
, , ) in 1122 or /Rˢ
r (f) =
< di
, di d} > + t < bi -
91 , bz -92 bs -93 > t c- [ 91 ]
,
,
=
[ die tlbi -91 ) ] it [ 92 Case 1- Cbs -93)]K
1- ( bi 92 )
)j TE cost ]
'
- -
+
◦ B
7 7
◦
A Direction of parametrization
is from A to B
, ② circle with Center ( a / b) and radius C
F) rlt) Cd b) CC cost Sint > 1- C- [0,21T ]
=
→
/ ,
a b
Direction of parametrization
is from la, f- (d) → ( b f- (b) )
,
Reversal of parametrization
Tfg C has orientation
curve d
given ,
then -
C is the
curve with the same
points as C but with opposite
orientation
[ → ( t) ⇐
< 11-1 ,
yltl / b)
r ✗ 2- ( t ) > 1- C- [ a
,
C r.lt ) rlzt )
:
- →
=
< ✗
1--1-7 , yczt) zczt) > ,
1- C- C- b , -
9)
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