Understanding Curl: Is |∇ × 𝐟| = ∇ × 𝐟 · 𝐧 a Valid Expression?

[Jhenrique]

Is correct to affirm that:
$$| \nabla \times \vec{f} | = \nabla \times \vec{f} \cdot \hat{n}$$?


I asked thinking in this definition:

http://en.wikipedia.org/wiki/Curl_(mathematics)


Ie, if the affirmation above is correct, so, is correct to express the definition aboce as:
$$\lim_{A \rightarrow 0} \frac{1}{|A|}\oint _{C} \vec{F}\cdot d\vec{r} = |\nabla \times \vec{f} |$$?

 

[jvicens]

I would say that the affirmation is not entirely correct. While it is true that the magnitude of the curl of a vector field, represented by |∇×𝐟|, is equal to the dot product of the curl and the unit normal vector, represented by ∇×𝐟⋅̂𝑛, this does not mean that the two expressions are equal.

The definition of the curl of a vector field, as stated in the Wikipedia link, involves taking the limit of a surface integral as the surface area approaches zero. This is a mathematical concept and cannot be expressed as an equality between two different expressions.

Furthermore, the second expression in the forum post, \lim_{A \rightarrow 0} \frac{1}{|A|}\oint _{C} \vec{F}\cdot d\vec{r} = |\nabla \times \vec{f} |, is a more general form of the definition of the curl, where the surface integral is taken over any closed curve C. This expression is not equivalent to the first expression, which only applies to the specific case of a unit normal vector.

In conclusion, while the affirmation in the forum post may have some basis in the definition of the curl, it is not entirely correct and cannot be expressed as an equality between two different expressions. it is important to carefully understand and communicate mathematical concepts to avoid confusion and misunderstandings.