MHB Vector Analysis: $\varphi$, $\mathbf{v}$, and Their Cross Products

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The discussion focuses on the vector analysis of the scalar field $\varphi$ and the vector field $\mathbf{v}$. It examines why the expression $\varphi\nabla\times\mathbf{v} = 0$ holds true, while $\nabla\varphi\times\mathbf{v}$ does not. Participants clarify the roles of the symbols $F_{1}$, $h_{1}$, and $\hat{\mathbf{u}}_{1}$, identifying which are position-dependent variables and which are constants. The distinction between these variables is crucial for understanding the behavior of the vector fields involved. Overall, the analysis emphasizes the importance of the mathematical relationships in vector calculus.
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$\newcommand{\unit}[1]{\hat{\mathbf{#1}}}$
Let $\varphi = F_1h_1$ and $\mathbf{v} = \frac{\unit{u}_1}{h_1}$.
Why is $\varphi\nabla\times\mathbf{v} = 0$ but $\nabla\varphi\times\mathbf{v}$ not?
 
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Of the symbols $F_{1}$, $h_{1}$, and $\hat{\mathbf{u}}_{1}$, which are variable, depending on position, and which are constant?
 

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