Transcript Section 17.3 - Gordon State College

Section 17.3
The Fundamental Theorem for
Line Integrals
THE FUNDAMENTAL THEOREM
FOR LINE INTEGRALS
Theorem: Let C be a smooth curve given by the
vector function r(t), a ≤ t ≤ b. Let f be a
differentiable function of two or three variables
whose gradient vector f is continuous on C.
Then

C
f  dr  f r(b)   f r(a) 
This says we can evaluate the line integral of a
conservative vector field simply by knowing the
value of f at the endpoints of C.
EXAMPLE
Evaluate
 F  dr , where C is any smooth
C
curve from (−1, 4) to (1, 2) and
F(x, y) = 2xyi + (x2 − y)j.
INDEPENDENCE OF PATH
If F is a continuous vector field with domain D,
we say that the line integral ∫C F · dr is
independent of path if

C1
F  dr   F  dr
C2
for any two paths C1 and C2 connecting the initial
point A and the terminal point B.
NOTE: A path is a piecewise-smooth curve
between two points.
CLOSED CURVE
A curve is closed if its terminal point coincides
with its initial point; that is, r(a) = r(b).
A THEOREM
∫C F · dr is independent of path in D if and only if
∫C F · dr = 0 for every closed path C in D.
OPEN SETS; CONNECTED SETS
• A set D is open if for every point P in D there
is a disk with center P that lies entirely in D.
(So, D does not contain any of its boundary
points).
• A set D is connected if any points in D can be
joined by a path that lies entirely in D.
INDEPENDENCE OF PATH
THEOREM
Theorem: Suppose that F is a vector field that
is continuous on an open connected set D. Then
the line integral  F  dr is independent of path if
C
and only if F is a conservative vector field on D;
that is, there exists a function f such that F  f .
A THEOREM
If F(x, y) = P(x, y)i + Q(x, y)j is a conservative
vector field, where P and Q have continuous
partial derivates on a domain D, then throughout
D we have
P Q

y x
SIMPLE CURVES;
SIMPLY-CONNECTED REGIONS
• A simple curve is a curve that does not
intersect itself anywhere between its endpoints.
• A simply-connected region in the plane is a
connected region D such that every simple
closed curve in D encloses only points that are
in D. Intuitively speaking, a simply-connected
regions contains no holes and cannot consist of
more than one piece.
A THEOREM
Let F = P i + Q j be a vector field on an open
simply-connected region D. Suppose that P and
Q have continuous first-order derivatives and
P Q

y x
Then F is conservative.
throughout D
FINDING THE POTENTIAL FUNCTION
OF A CONSERVATIVE VECTOR FIELD
Let F = Pi + Qj be a conservative vector field. Then P = fx and Q = fy
where f is the potential function. To find f:
1. Find f by integrating P(x, y) with respect to x, while holding y
constant. We can write
f ( x, y )   P( x, y ) dx  g ( y )
(1)
where the arbitrary function g(y) is the “constant” of integration.
2. Differentiate (1) with respect to y and set equal to Q(x, y). This yields,
after solving for g′(y),
g ( y )  Q( x, y ) 

P( x, y ) dx.

y
( 2)
3. Integrate (2) with respect to y and substitute the result into (1). The
result is the potential function f (x, y, z)
EXAMPLES
1. Show the vector field
y2
F( x, y ) 
i  2 y arctanx j
2
1 x
is conservative, and find the potential function f.
2. Calculate the following line integral, where C
the path given by r(t) = t2i + (t + 1)j + (2t − 1)k,
0 ≤ t ≤ 1.
C (2 xz  y ) dx  2 xy dy  ( x  3z ) dz
2
2
2
WORK REVISITED
Let F be a continuous force field that moves an
object along the path C given by r(t), a ≤ t ≤ b,
where r(a) = A is the initial point and r(b) = B is
the terminal point of C. Then the force F(r(t)) at
a point on C is related to the acceleration
a(t) = r″(t) by the equation
F(r(t)) = m r″(t)
WORK (CONTINUED)
The work done by the force on the object can be
simplified to
W   F  dr
C
b
  mr(t )  r(t ) dt
a
 m | v (b) |  m | v (a ) |
1
2
2
1
2
where v = r′ is velocity.
2
KINETIC ENERGY
The quantity 12 m | v(t ) |2 is called the kinetic
energy of the object. Thus, work can be
written as
W = K(B) − K(A)
which says the work done by the force field
along C is equal to the change in the kinetic
energy at the endpoints of C.
POTENTIAL ENERGY
Suppose F is a conservative force field, that is,
we can write F  f . In physics, the potential
energy of an object at point (x, y, z) is defined as
P (x, y, z) = − f (x, y, z), so we have F   P .
Work can be written as
W   F  dr    P  dr
C
C
 Pr(b)   Pr (a) 
 P( A)  P( B)
LAW OF CONSERVATION OF
ENERGY
By setting the two expressions for work W equal,
we find that
P(A) + K(A) = P(B) + K(B)
which says that if an object moves from one point
A to another point B under the influence of a
conservative force field, then the sum of its
potential energy and kinetic energy remains
constant. This is called the Law of Conservation
of Energy and it is the reason the vector field is
called conservative.