Math 21a
The fundamental theorem for line integrals
Fall 2016
1 Let f (x, y) = ex + xy and F~ = ∇f be its gradient vector field on R2 .
2
2
(a) Let C be the curve
R is R parametrized by ~r(t) = ht, t i, 0 ≤ t ≤ 1. Compute directly
the line integral C F~ · d~r.
(b) What is f (~r(t))? Did you use this anywhere when you computed the line integral in
(a)? Can you explain why this happened?
2+
(c) Now let C be the curveR given by the parametrization ~r(t) = h(sin t)ecos t
with 0 ≤ t ≤ π. Find C F~ · d~r.
√
t
, sin t + cos ti
Fundamental theorem of line integrals
Let C be a smooth curve given by the parametrization ~r(t), a ≤ t ≤ b. Let f be a
smooth function and F~ = ∇f its gradient vector field. Then
Z
F~ · d~r = f (~r(b)) − f (~r(a)).
C
2 Use the above theorem to compute the line integral
R
C
F~ · d~r, where
(a) F~ (x, y) = hln y + 2xy 3 , 3x2 y 2 + xy i and C is the parabola ~r(t) = ht, t2 + 1i, 0 ≤ t ≤ 2.
(b) F~ (x, y, z) = hyz, xz, xy + 2zi and C is the line segment from (−1, 0, 2) to (4, 6, 3).
Summary
Consider the following four properties for a smooth vector field F~ :
(A) F~ = ∇f is a gradient vector field with
R potential function f .
(B) F~ is conservative: the line integral C F~ · d~r depends only on the endpoints of C.
(C) F~ has the closed loop property: the line integral of F~ along a closed loop is zero.
(D) F~ is irrotational: its curl ∇ × F~ is zero (Clairaut’s test).
• In 2D, if F~ = hP, Qi, this means Qx − Py = 0.
• In 3D, if F~ = hP, Q, Ri, this means hRy − Qz , Pz − Rx , Qx − Py i = ~0.
We have the following implications:
1. (A) ⇒ (B): follows from the fundamental theorem of line integrals.
2. (B) ⇒ (A): follows by direct construction of the potential function f .
3. (B) ⇔ (C): follows by splicing or concatenating and reversing paths.
4. (A) ⇒ (D): follows from Clairaut’s theorem.
5. (D) ⇒ (C): if the domain of F~ is open and simply connected, then this follows from
Green’s theorem (covered next week). Otherwise, we can find counterexamples (problem
5).
In conclusion, properties (A), (B), and (C) are equivalent and they imply (D). If the domain
of F~ is open and simply connected, then all four properties are equivalent.
3 Let F~ be a vector field on R2 . In each part, what can you conclude from the given
information about F~ ? Is F~ definitely conservative, definitely not conservative, or is there
not enough information to tell?
R
(a) C F~ · d~r = 1, where C is the unit circle, traversed once counterclockwise.
(b)
R
C
F~ · d~r = 0, where C is the unit circle, traversed once counterclockwise.
4 Decide whether you can conclude that F~ is conservative or not from Clairaut’s test.
(a) F~ (x, y) = hx2 y, xy 2 i
(b) F~ (x, y, z) = h1 + 2xy + zexz , x2 + 3y 2 , xexz i
y
x
(c) F~ (x, y) = h− x2 +y
2 , x2 +y 2 i
5 Let F~ (x, y) = h x2−y
, x i.
+y 2 x2 +y 2
(a) You should have seen in the previous exercise that F~ is irrotational. It turnsRout that
F~ is not conservative, however. Prove this by showing that the line integral C F~ · d~r,
where C is the unit circle oriented counterclockwise, is nonzero.
(b) Show that if f (x, y) = arctan( xy ), then ∇f = F~ . Why does your result in (a) not
contradict the fundamental theorem for line integrals?