Basics
What does Stokes’ Theorem relate: a line integral around a closed curve to what kind of integral over a surface?
The surface integral of the curl of a vector field over the surface bounded by the curve.
What is the curl of F = ⟨y, −x, 0⟩
∇ × F = ⟨0, 0, −2⟩
Let
\vec{F} = \langle y, 0, 0 \rangle
and let S be the upper half of the unit disk
x^2 + y^2 \leq 1
in the plane z = 0 , oriented upward. Use Stokes’ Theorem to compute .
0 nabla x F = 0 so the integral is zero
Give a real-world example of a situation modeled by Stokes’ Theorem.
Amperes Law
State the mathematical form of Stokes’ Theorem.
∮_C F ⋅ dr = ∬_S (∇ × F) ⋅ dS, where C is the boundary of surface S.
Compute the unit normal vector to the plane z = 1 − x − y.
The gradient is ⟨1, 1, 1⟩ (upward), so the unit normal is (1/√3)⟨1, 1, 1⟩.
Let S and
\vec{F} = \langle -y, x, 0 \rangle
be the part of the plane z=0 inside the circle
x^2 + y^2 = 4
, oriented upward. Use Stokes’ Theorem to compute
\intint_S (\nabla \times \vec{F}) \cdot d\vec{S}
8pi
If the surface is a flat disk, how does Stokes’ Theorem simplify?
It reduces to Green’s Theorem in the plane, relating the line integral to a 2D curl (∂N/∂x − ∂M/∂y).
What condition must the vector field F satisfy for Stokes’ Theorem to apply?
F must be continuously differentiable (i.e., F ∈ C¹) in a region containing the surface S and its boundar
Given F = ⟨yz, xz, xy⟩ and a surface S: part of the paraboloid z = 4 − x² − y², oriented upward, what is ∇×F.
∇ × F = ⟨x − z, y − z, 0⟩
Let S be the portion of the plane
z = x + 2y
that lies above the disk
x^2 + y^2 \leq 1
, oriented upward. Let
\vec{F} = \langle y, -x, z \rangle
Use Stokes’ Theorem to compute .
\int\int_S (\nabla \times \vec{F}) \cdot d\vec{S}
-2pi
A circular wire loop of radius 1 lies flat in the xy-plane, centered at the origin. The vector field
\vec{F}(x, y, z) = \langle -y, x, 0 \rangle
represents the velocity field of a fluid.
Use Stokes’ Theorem to find the total circulation of the fluid around the wire loop.
2pi
This is the special case of Stokes on xy plane -> Greens Theorem
Explain the orientation convention used in Stokes’ Theorem.
The orientation of the boundary curve C must follow the right-hand rule relative to the surface’s normal vector.
Parametrize the hemisphere x² + y² + z² = 1, z ≥ 0, and find the differential surface vector dS.
Use spherical coordinates: r(θ, φ) = ⟨sinφcosθ, sinφsinθ, cosφ⟩, φ ∈ [0, π/2], θ ∈ [0, 2π]; dS = r_θ × r_φ dθ dφ points outward (upward).
Let
\vec{F} = \langle yz, xz, xy \rangle
and S be the part of the plane x+y+z=1 in the first octant, bounded by the coordinate planes. Orient S upward. Use Stokes’ Theorem to compute
\intint_S (\nabla \times \vec{F}) \cdot d\vec{S}
.
0
Prove Stokes’ Theorem for a planar surface in ℝ³ using Green’s Theorem.
Convert ∮C F ⋅ dr = ∬_R (∂N/∂x − ∂M/∂y) dA into ∬_S (∇ × F) ⋅ n dS by embedding in ℝ³ with upward normal vector.
How does Stokes’ Theorem generalize Green’s Theorem in the plane?
Green’s Theorem is a special case of Stokes’ Theorem when the surface lies entirely in the xy-plane and the curl reduces to a scalar.
A surface is non-orientable (e.g., Möbius strip). Can Stokes’ Theorem be applied? Justify.
No, because Stokes’ Theorem requires an orientable surface to define a consistent normal vector field for dS.
Let
\vec{F} = \langle e^y, xz, y^2 \rangle
and S be the surface of the paraboloid
z = 4 - x^2 - y^2
above the xy-plane, oriented upward. Use Stokes’ Theorem to compute
\intint_S (\nabla \times \vec{F}) \cdot d\vec{S}
-19.9885
A flat, triangular metal plate is placed in the first octant, lying in the plane x+y+z=1, with its edges along the coordinate axes. The plate is in a magnetic field described by
\vec{B}(x, y, z) = \langle y, z, x \rangle
(in teslas).
Suppose a wire is bent to follow the boundary of the plate (the triangle in the plane), and a current of 2 amperes flows counterclockwise around the wire as viewed from above.
(a) Use Stokes’ Theorem to find the total electromotive force (emf) induced around the wire by the magnetic field. The magnetic field is increasing at a rate of 3 teslas per second everywhere
emf= -\frac{d}{dt} \intint_S \vec{B} \cdot d\vec{S}
.5 Volts