A vector field is uniquely determined within a volume if its divergence (sources and sinks) and curl (rotational flow) are specified throughout that volume, provided that the normal component of the field is fixed on the boundary. By examining the difference between two such fields, we find that the difference must be both solenoidal and irrotational, which allows it to be represented as the gradient of a scalar potential satisfying Laplace's equation. Given that the normal derivative of this potential vanishes at the boundary, Green's First Identity forces the gradient-and thus the difference between the two original fields-to be zero. This result is a specific application of the Helmholtz Decomposition Theorem, confirming that the internal structure and boundary flux together leave no room for variation in the field's configuration.
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%% Proof and Derivation
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