Why is CurlF a vector field

Question

I am having a really hard time understanding the concept of CurlF. Could someone help explain what it is and how we know that it's a vector field?

Answer 1

For 3 dimensional space the definition is $$ \text{curl}\overrightarrow{F}=\overrightarrow{\nabla}\times\overrightarrow{F}=\left(\frac{\partial F_{z}}{\partial y}-\frac{\partial F_{y}}{\partial z}\right)\hat{x}+\left(\frac{\partial F_{x}}{\partial z}-\frac{\partial F_{z}}{\partial x}\right)\hat{y}+\left(\frac{\partial F_{y}}{\partial x}-\frac{\partial F_{x}}{\partial y}\right)\hat{z} $$ And by definition it is a vector-field

Answer 2

Some treatments of this subject distinguish between polar vectors, which are conventional vectors defining a quantity and a direction, and axial vectors (sometimes called pseudovectors) which describe a rotational quantity.

An axial vector is like this.

Imagine a wheel spinning round an axle. This can be described by an axial vector. How fast the wheel is spinning is identified by the magnitude of the vector. The direction of the axle is then identified with the direction of the vector. The direction is defined by the right hand rule: if you imagine your hand curled around the axle with your thumb sticking out, and your fingers pointing in the direction the wheel is spinning, the positive direction is the direction in which your thumb is pointing.

Now, you can loosely describe the curl of a field as an axial vector defining the spinning motion of a small element of the field. If this small element is turning, then there is an axial vector defining that turn. And the curl of a vector field is the amount of turning at each point.

The above is possibly inaccurate in order to give you a general feel for what is going on.