Let \(\mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\) be a vector field, where \(P\), \(Q\), and \(R\) are functions of the coordinates \(x\), \(y\), and \(z\), and \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) are the unit vectors in the \(x\), \(y\),
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