3. 3 Fundamentals of line int .
.
let c be a smooth curve parametrized
differentiable fn
by Flt ), 1- c- [ dib ] ,f is a
Ofcgradient of f) is continuous oh C
defined on C and
Then final initial
f of -
di -
fifdr =
f- Crib )) -
f- ( Fla) )
The line integral 50 f. di INDEPENDENT of the
is
curve
, only the
c
initial point (Fla ) ) and the endpoint
,
( Fcb) ) are IMPORTANT
Thein between
motsoimpon-antf.is
→
the
{ f- .
di
=
flendptl-fcin.pt#opp-
potential
✓ function
To ask before applying
[Questions ] Fundamental theorem
field is conservative
?
I. HOW do we know a vector
of
How do we know v. f. defined on the curve is actually a
gradient
the fn .
É .
How do potential fn of a conservative v. f.
?
2 .
we determine _
Theorem :
<P, Q > is conservative it and only if Py=Qu
A v. f. f-
=
↓
partial
Note
:
derivatives
for vector fields in 1123 we will discuss theorem w.r.t.ie and y
in Lu 3.5 .
respectively
.
let c be a smooth curve parametrized
differentiable fn
by Flt ), 1- c- [ dib ] ,f is a
Ofcgradient of f) is continuous oh C
defined on C and
Then final initial
f of -
di -
fifdr =
f- Crib )) -
f- ( Fla) )
The line integral 50 f. di INDEPENDENT of the
is
curve
, only the
c
initial point (Fla ) ) and the endpoint
,
( Fcb) ) are IMPORTANT
Thein between
motsoimpon-antf.is
→
the
{ f- .
di
=
flendptl-fcin.pt#opp-
potential
✓ function
To ask before applying
[Questions ] Fundamental theorem
field is conservative
?
I. HOW do we know a vector
of
How do we know v. f. defined on the curve is actually a
gradient
the fn .
É .
How do potential fn of a conservative v. f.
?
2 .
we determine _
Theorem :
<P, Q > is conservative it and only if Py=Qu
A v. f. f-
=
↓
partial
Note
:
derivatives
for vector fields in 1123 we will discuss theorem w.r.t.ie and y
in Lu 3.5 .
respectively
