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.
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✍️Mathematical Proof
$\complement\cdots$Counselor
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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.
✍️Mathematical Proof
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- 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
- Dot Cross and Triple Products
- 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
- The Vanishing Curl Integral
- 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
🧄Proof and Derivation-2
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