Let's break down this vector calculus equation step by step:

Goal:

To verify or interpret the expression:

$\nabla \times (\psi \hat{\mathbf{k}}) = (\operatorname{grad} \psi) \times \hat{\mathbf{k}}$

where:

• $\nabla \times$ is the curl operator,
• $\psi$ is a scalar field,
• $\hat{\mathbf{k}}$ is the unit vector in the $z$-direction,
• $\operatorname{grad} \psi = \nabla \psi$ is the gradient of $\psi$.

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The Left-Hand Side (LHS): $\nabla \times (\psi \hat{\mathbf{k}})$

Using the vector calculus identity: $\nabla \times (f \mathbf{A}) = (\nabla f) \times \mathbf{A} + f (\nabla \times \mathbf{A}),$ where $f$ is a scalar field ($\psi$) and $\mathbf{A}$ is a vector ($\hat{\mathbf{k}}$), we calculate:

1. First term: $(\nabla \psi) \times \hat{\mathbf{k}}$.

   • Here, $\nabla \psi$ is the gradient of the scalar field $\psi$, which gives a vector.

2. Second term: $\psi (\nabla \times \hat{\mathbf{k}})$.

   • The curl of $\hat{\mathbf{k}}$ is zero because $\hat{\mathbf{k}}$ is a constant unit vector (its components do not depend on position).

Thus: $\nabla \times (\psi \hat{\mathbf{k}}) = (\nabla \psi) \times \hat{\mathbf{k}}.$

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The Right-Hand Side (RHS): $(\operatorname{grad} \psi) \times \hat{\mathbf{k}}$

This is simply a rephrasing of the term $(\nabla \psi) \times \hat{\mathbf{k}}$, as $\operatorname{grad} \psi = \nabla \psi$ by definition.

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Conclusion:

The equation is correct: $\nabla \times (\psi \hat{\mathbf{k}}) = (\operatorname{grad} \psi) \times \hat{\mathbf{k}}.$

This is consistent with the vector calculus identity used earlier.

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Let me know if you'd like further clarifications or a worked example!

Related Questions:

1. What is the physical interpretation of the curl operator in vector calculus?
2. How is the curl of a vector field related to fluid dynamics?
3. Can this result be extended to any arbitrary unit vector (not just $\hat{\mathbf{k}}$)?
4. What is the divergence of the vector field $\psi \hat{\mathbf{k}}$?
5. How do you calculate the gradient, divergence, and curl in cylindrical coordinates?

Tip:

Always check if vector components (like $\hat{\mathbf{k}}$) are constant or spatially varying; this simplifies the application of vector calculus identities.