After read this stretch https://en.wikipedia.org/wiki/Closed_and_exact_forms#Vector_field_analogies, my doubts increased exponentially...(adsbygoogle = window.adsbygoogle || []).push({});

1. A scalar field correspond always to a 0-form?

1.1. The laplacian of 0-form is a 2-form?

1.2. But the laplacian of sclar field is another scalar field.

1.3. If yes, we have a scalar field 2-form, thus scalar field can be k-form.

2. Exist (-1)-form?

3. Exist 4-form?

4. How know if a vector field (and a scalar field too) is a 0-form, 1-form, 2-form, 3-form or k-form?

5. If the gradient of a 0-form is zero, so, this 0-form is closed?

5.1. Which the name given for a scalar field that have gradient zero?

5.3. Which are your the properties?

6. A 3-form can be exact?

7. A 3-form can be closed?

8. Why the laplacian wasn't mentioned in the stretch? The laplacian can be used in the theory of closed and exact forms?

8.1. If the laplacian of a vector field is zero, thevector fieldis called harmonic; if the laplacian of a scalar field is zero, so thescalar fieldis called harmonic, correct?

9. d²=0 but ∇²≠0

10. If a vector field F can be expressed as F=∇²G, so, what is G?

11. If a scalar field f can be expressed as f=∇²g, so, what is g?

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# Differential forms and differential operators

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