1. Fundamental Theorem of Line Integrals
The line integral concept is derived from the physical concept of work done by
R
Rb
a force field, F(x, y). The notation: is computed as C F · dr = a F(r(t)) · r0 (t) dt,
where the curve C is parametrized by r(t), a ≤ t ≤ b. If the vector field F is a
∂f
gradient vector field, Of = ∂f
∂x i + ∂y j, then the following fundamental theorem
holds and the line integral depends only on the value of f at the endpoints of the
curve:
Z
Of · dr = f (r(b)) − f (r(a))
C
This is true for the gradient vector field of the function f by the chain rule of
the gradient along a path: ∇(φ(c(t))) = (∇(φ))(c(t))) · C 0 (t) and the fundamental
theorem of calculus. Note the subtle placement of parentheses. You should compare
this with the fundamental theorem of first calculus relating a definite integral to an
anti-derivative evaluated at the limits.
2. Conservative Vector Fields
A vector field is conservative if it is path independent on it domain, D, that
is, if C1 (t) and C2 (t) are paths with the same endpoints ( C1 (a) = C2 (a) and
C1 (b) = C2 (b)) then
Z
Z
F · dr =
F · dr
C1
C2
Equivalently, a vector field Ris conservation if the line integral along all closed
paths (C(a) = C(b)) is zero: C F · dr = 0. In the case that the vector field is
a gradient vector field, F = ∇φ, φ is called a potential function for F . By the
fundamental result above, line integrals of a gradient vector field depends only on
the value of the potential function at the endpoints of the curve, i.e. the vector
field is independent of path and hence conservative.
A converse is true provided any two paths with the same endpoints can be
smoothly deformed to each other, i.e. the domain of definition simply connected.
A necessary condition for a vector field F = P i + Qj to be a gradient vector field is
∂Q
that ∂P
∂y = ∂x . If this condition is satisfied continuously and the region is simply
connected, then a potential function can often be constructed by anti-differentiation
(by Theorem 3, p. 966, Chapter 16).
1
2
3. Fundamental Theorems of Integration in Vector Calculus
The fundamental theorem of the calculus of functions of one variable expresses
the relationship between the integral of a derivative over an interval and the (signed)
values of the function at the endpoints.
b
Z
f 0 [x] dx = f [x]|ba = f [b] − f [a]
a
There are two crucial concepts hidden in this relationship that must be elucidated
and generalized to higher dimensional situations: boundary and orientation.
The (point-set topological) boundary of a set S (see Chapter 14: p. 852), denoted
∂S, consists of those points for which every open ball centered at the given points
intersects both S and its complement, i.e. contains points in S and points not in S.
There is a second, more subtle notion of boundary for curves and surfaces in space is
associated with the parametrization (covered in Advanced Calculus: Math 4035).
We will only speak informally about orientation, which formalizes the notion of
“right handedness.” In the one dimensional case above, the boundary of the closed
interval, [a, b] consists of the endpoints {a, b} and the orientation assigns a negative
sign to the lower limit, a. The fundamental theorems of vector calculus in Chapter
17: Green’s, Gauss’s and Stoke’s all follow the pattern that integration over the
boundary of a set S is related to the integral over S of another object related to the
boundary integrand by some differentiation procedure.
4. Integration over Curves, Surfaces and Regions
Dimension
Integral
Application
Zero
f [x]|ba
Function Evaluation
One
R
Two
RR
Three
C
F[C[t]] • C0 [t] dt =
R
R
C
F · ds
Work by F along C
F • n dA
Flux thru R
F dV
Divergence
RRR
S