SME
Calculus III (Multivariable) · Problem type
Stokes’ theorem round a circle
Use this for “Use Stokes’ theorem to find the circulation of F = ⟨−2y, 2x, 0⟩ round the circle x² + y² = 4, z = 0.”
Diagram
The circle of radius 3 at z = 0 runs counterclockwise seen from above.
curl F = ⟨0, 0, 2⟩, so any cap with this edge has flux 2 × π × 3² = 56.55 = ∮ F·dr.
This is an example. Type your own values to start a new problem.
F
=
k
⟨−y,
x,
0⟩
x²
+
y²
=
R
2
∮
F
·
dr
=
∬
curl
F
·
n
dS
=
(curl F)₃
×
A
=
∬ curl F · n dS
∮ F · dr
Circulation round the circle
worked out (not typed)
Formulas
(curl
F)₃
=
k
−
(−
k
)
2 = 1 − (−1)
A
=
π
×
R
2
9π
=
π
×
3
2
∬
curl
F
·
n
dS
=
((curl
F)₃)
×
A
18π = 2 × 9π
∮
F
·
dr
=
2π
∫
0
(
k
×
R
2
) dt
18π
=
2π
∫
0
(1 ×
3
2
) dt
Assumptions
•
F = k⟨−y, x, 0⟩, and the circle of radius R runs counterclockwise seen from above, so n points up.
•
Stokes’ theorem: ∮ F · dr = ∬ curl F · n dS over any surface with the circle as its edge, a flat disk or a dome.
•
r(t) = ⟨R cos t, R sin t, 0⟩ for t from 0 to 2π runs once round the circle.
Step-by-step
WE KNOW
k = 1, R = 3
FIND
curl F, z part ((curl F)₃), area of the disk (A), flux of the curl (∬ curl F · n dS), circulation round the circle (∮ F · dr)
Step 1 · Find curl F, z part ((curl F)₃)
Use (curl F)₃ = k − (−k)
F
=
⟨−ky,
kx,
0⟩.
The
x
and
y
parts
of
curl
F
are
0;
the
z
part
is
∂(kx)/∂x
−
∂(−ky)/∂y
=
k
−
(−
k
)
=
2
k
.
(curl F)₃ = 1 − (−1)
(curl F)₃ = 2
Step 2 · Find area of the disk (A)
Use A = π × R²
The flat disk inside the circle is the simplest surface with the circle as its edge.
A
=
π
×
3
2
A
=
9
×
π
A
=
9π
(≈
28.27)
Step 3 · Find flux of the curl (∬ curl F · n dS)
Use ∬ curl F · n dS = ((curl F)₃) × A
On the disk n = ⟨0, 0, 1⟩, so curl F · n is the steady z part, times the area. A dome on the same rim gives the same flux.
∬ curl F · n dS = 2 × 9π
∬ curl F · n dS = 18π (≈ 56.55)
Step 4 · Find circulation round the circle (∮ F · dr)
Use ∮ F · dr = ∫ from 0 to 2π of (k × R²) dt
Check by walking round: F · r′(t) = kR² sin² t + kR² cos² t = kR² at every t. It matches the flux of the curl, as Stokes’ theorem says.
∮
F
·
dr
=
2π
∫
0
(1 ×
3
2
) dt
2π
∫
0
9 dt
=
9
×
2π
∮ F · dr = 18π (≈ 56.55)
Check
Put the numbers back in. Both sides should match.
2 = 1 − (−1) ✓
9π
=
π
×
3
2
✓
18π = 2 × 9π ✓
18π
=
2π
∫
0
(1 ×
3
2
) dt
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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