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$F(x,y,z)$ is a vector field in space and $f(x,y,z)$ is a scalar field in space.

  1. $\nabla \times (\nabla(\nabla \cdot F))$

  2. $\nabla \times (\nabla \cdot (\nabla f))$

  3. $ \nabla (\nabla \cdot (\nabla \times F))$

  4. $\nabla(\nabla \times (\nabla \cdot F))$

  5. $\nabla \cdot (\nabla \times (\nabla f))$

  6. $\nabla \cdot (\nabla (\nabla \times f))$

I'm trying to study for a multivariable final and I am having trouble understanding when and why these expressions become undefined.

Hosein Rahnama
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Neil Philip
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    For each operator, you should write down what it takes as input, and what it outputs. For example, div takes in a vector field and outputs a scalar. – angryavian Dec 15 '15 at 05:27
  • I applied that strategy, but I'm just having trouble understanding when the expressions become undefined. Like what about the 2nd, 4th, and 6th expressions make these expression undefined? – Neil Philip Dec 15 '15 at 05:32

1 Answers1

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For example, in expression 2:

  • $f$ is a scalar field
  • $\nabla f$ is its gradient: it is a vector field
  • $\nabla \cdot \nabla f$ is the divergence of this vector field, it is a scalar field
  • $\nabla \times (\nabla \cdot \nabla f)$ is the curl of this scalar field: this is undefined as the curl operator takes a vector field as input.
Kuifje
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  • Ahh, so that's it! So you can take the gradient of a scalar field but you cannot take the curl of it? – Neil Philip Dec 15 '15 at 16:54
  • Indeed. Carefully check the definitions of gradient, curl, divergence. It should help you manipulate them, and understand their geometrical/physical meaning. – Kuifje Dec 15 '15 at 17:07