Visualizing a Vector field and its curl

Let $$ \mathbf{F}(x, y, z) = \begin{pmatrix} P(x, y, z) \\ Q(x, y, z) \\ R(x, y, z) \end{pmatrix} = P(x, y, z) \,\mathbf{i} + Q(x, y, z) \, \mathbf{j} + R(x, y, z) \, \mathbf{k} $$ be a vector field. The curl of the vector field $$ curl \, \mathbf{F} = \nabla \times \mathbf{F} = \det \begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial_x & \partial_y & \partial_z \\ P & Q & R \end{pmatrix}. $$ For example
$$ \mathbf{F} = y \,\mathbf{i} - x \,\mathbf{j} + z \,\mathbf{k}, \quad curl \, \mathbf{F} = 2 \,\mathbf{k}. $$ To visualize, we think of $curl \, \mathbf{F}$ as pointing towards the axis of rotation, when rotations exists in $\mathbf{F}$. In fluid mechanics, when $\mathbf{F}$ denote the velocity field of a fluid, it's curl $curl \, \mathbf{F}$ is called the vorticity field.

See the following picture.

Another good way to visualize vector field is to look at flow lines. These are curves that are everywhere tangent to the vector field. To be more precise, these are the solution to the ordinary differential equation $$ \dot{\mathbf{r}}(t) = \mathbf{F}(\mathbf{r}(t)) . $$