The closed surface integral $\oint_S x \times d S$ is always zero because the curl of the position vector ( $\nabla \times x$ ) is always zero, a mathematical result that is physically consistent with vectors either being individually zero or cancelling each other out due to a surface's symmetry.
<aside> 🧄
$\gg$Mathematical Structures Underlying Physical Laws
$\complement\cdots$Counselor
</aside>
To calculate the closed surface integral of the cross product of the position vector and the differential surface area vector, a generalized form of the divergence theorem is used. This allows the surface integral to be rewritten as a volume integral.
The generalized divergence theorem for a cross product, $\oint_S F \times d S=-\int_V(\nabla \times F) d V$, provides a method for converting a surface integral into a volume integral.
The curl of the position vector is always the zero vector ( $\nabla \times x=0$ ). This is because the partial derivatives of the independent components of the position vector with respect to each other are all zero.
As a result, the volume integral becomes the integral of the zero vector over the enclosed volume, which always evaluates to zero. This confirms that for any closed surface, the integral $\oint_S x \times d S$ is always zero.
A zero result for a surface integral can be achieved either because all individual vectors are zero (as seen on the sphere), or because non-zero vectors cancel each other out due to symmetry (as seen on the cylinder).
Compare how vectors behave on a sphere and a cylinder
Compare how vectors behave on a sphere and a cylinder
‣
<aside> 🧄
Proving the Cross Product Rules with the Levi-Civita Symbol
Proving the Epsilon-Delta Relation and the Bac-Cab Rule
Simplifying Levi-Civita and Kronecker Delta Identities
Why a Cube's Diagonal Angle Never Changes
How the Cross Product Relates to the Sine of an Angle
Finding the Shortest Distance and Proving Orthogonality for Skew Lines
A Study of Helical Trajectories and Vector Dynamics
The Power of Cross Products: A Visual Guide to Precessing Vectors
Divergence and Curl Analysis of Vector Fields
Unpacking Vector Identities: How to Apply Divergence and Curl Rules
Commutativity and Anti-symmetry in Vector Calculus Identities
Double Curl Identity Proof using the epsilon-delta Relation
The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation
Surface Parametrisation and the Verification of the Gradient-Normal Relationship
Proof and Implications of a Vector Operator Identity
Conditions for a Scalar Field Identity
Solution and Proof for a Vector Identity and Divergence Problem
Kinematics and Vector Calculus of a Rotating Rigid Body
Work Done by a Non-Conservative Force and Conservative Force
The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field
Calculating the Area of a Half-Sphere Using Cylindrical Coordinates
Divergence Theorem Analysis of a Vector Field with Power-Law Components
Total Mass in a Cube vs. a Sphere
Momentum of a Divergence-Free Fluid in a Cubic Domain
Total Mass Flux Through Cylindrical Surfaces
Analysis of Forces and Torques on a Current Loop in a Uniform Magnetic Field
Computing the Integral of a Static Electromagnetic Field
Surface Integral to Volume Integral Conversion Using the Divergence Theorem
Circulation Integral vs. Surface Integral
Using Stokes' Theorem with a Constant Scalar Field
Verification of the Divergence Theorem for a Rotating Fluid Flow
Integral of a Curl-Free Vector Field
Boundary-Driven Cancellation in Vector Field Integrals
Proving the Generalized Curl Theorem
Computing the Magnetic Field and its Curl from a Dipole Vector Potential
Proving Contravariant Vector Components Using the Dual Basis
Verification of Orthogonal Tangent Vector Bases in Cylindrical and Spherical Coordinates
Vector Field Analysis in Cylindrical Coordinates
Vector Field Singularities and Stokes' Theorem
Compute Parabolic coordinates-related properties
Analyze Flux and Laplacian of The Yukawa Potential
Verification of Vector Calculus Identities in Different Coordinate Systems
Analysis of a Divergence-Free Vector Field
The Uniqueness Theorem for Vector Fields
Analysis of Electric Dipole Force Field
</aside>