The Uniqueness Theorem for Vector Fields

The Uniqueness Theorem for Vector Fields is a cornerstone of physics and engineering, proving that a vector field is uniquely defined by its "fingerprint"—a combination of its divergence (sources), curl (circulations), and normal component on the boundary. This is intuitively demonstrated by showing that any two fields with matching fingerprints must be identical, as an animation can smoothly transform one into the other, visually confirming the theorem's validity.

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✍️Mathematical Proof

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The Power of the Laplacian

The proof hinges on the property that if a vector field has both zero divergence and zero curl, it can be expressed as the gradient of a scalar potential ( $u=\nabla \phi$ ), which is a harmonic function ( $\nabla^2 \phi=0$ ). This connection between a divergence- and curl-free field and the Laplacian operator is a cornerstone of potential theory.

The Role of the Boundary Condition

The given boundary condition, $n \cdot v=n \cdot w$, simplifies to $n \cdot u=0$. When combined with the Divergence Theorem, this condition forces the integral of $|\nabla \phi|^2$ over the volume to be zero. Because a squared magnitude is nonnegative, the only way for its integral to be zero is for the term itself to be zero everywhere.

A "Fingerprint" for Vector Fields

The theorem implies that the divergence, curl, and normal boundary component act like a unique "fingerprint" for a vector field. Knowing these three properties is enough to fully identify the field everywhere within the volume. This is highly significant in physics and engineering, particularly in electromagnetism and fluid dynamics, where fields are often defined by their sources (divergence), circulations (curl), and boundary conditions.

✍️Mathematical Proof

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🧄Proof and Derivation-2

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