Understanding Gradient and Curl: Equations and Directions

[Jhenrique]

If the direction of the gradient of f in a point P is the direction of most/minor gradient, so a direction of the curl of f in a point P is the direction of most/minor curl too, correct?

Also, if the gradient of f in the direction t is given by equation: ∇f·t, so the curl of f in the direction n is given by equation: ∇×f·n, correct?

 

[chogg]

I don't understand what you mean by "minor".

The direction of the gradient is the direction where $f$ changes the fastest.

I believe the direction of the curl is the axis about which a sphere would spin, if it were fixed in place and torqued by $\vec{f}$.

Of course, the "vector" "curl" only works in 3D; it's really just a disguised bivector, which works in any 2+-dimensional space. So in general, the plane of the (bivector) curl would be the plane of rotation for that fixed sphere.

Interestingly, in 4+ dimensions, there could be multiple such planes simultaneously!

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As to your second question, I believe you're correct: $(\nabla \times \vec{f}) \cdot \vec{n}$ gives the amount of rotation for a fixed axis (fixed along $\vec{n}$).