Question about the curl of a vector field.

Question

When calculating the curl of a vector field $\vec{F}=M(x,y)\vec{i}+N(x,y)\vec{j}$ why does difference of the partial derivatives $\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}$ tell me if the vector field is rotating or not? I don't understand the meaning behind this difference.

Answer 1

I was struggling with this too. There is a good strategy to tackle this out:

Let $\vec{F}:\mathbb{R}^2\rightarrow \mathbb{R}^3$ be a vector field, of the form (this is the standard form of bidimensional vector fields in physics): $$\vec{F}(x,y)=(a(x,y),b(x,y),c(x,y))$$ Where $\{a,b,c\}:\mathbb{R}^2\rightarrow \mathbb{R}$ are functions of two variables to the real numbers. This only means that for every point in the cartesian plane $(x,y)\in \mathbb{R}^2$, there exists a vector $\vec{v}$ in the vector space $V(\mathbb{R}^3)$ such that: $$\vec{v}=(a,b,c)$$ Where $a,b,c$ are real numbers given by the functions ($c$ is typically zero). $\nabla$ is a dummy vector of $V(\mathbb{R}^3)$ such that $$\nabla=(\frac{\partial}{\partial x},\frac{\partial}{\partial y},0)$$ ($\nabla$ would have also a partial derivative with respect to $z$ if the original vector field were $\vec{F}:\mathbb{R}^3\rightarrow \mathbb{R}^3$, instead of $\vec{F}:\mathbb{R}^2\rightarrow \mathbb{R}^3$).

$\nabla$ is not a real vector, but the linear transformations associated with it function the same way a real vector would: $$\nabla\cdot\vec{v}=\frac{\partial}{\partial x}a+\frac{\partial}{\partial y}b+0$$ And $\nabla\cdot\vec{v}\in\mathbb{R}$. Also $$\vec{w}=\nabla\times\vec{v}=(\frac{\partial}{\partial y}c,-\frac{\partial}{\partial x}c,\frac{\partial}{\partial x}b-\frac{\partial}{\partial y}a)$$ Where $\vec{w}\in V(\mathbb{R}^3)$

Note that if $\vec{F}$ has no $c$ component, then $\vec{w}=(0,0,\frac{\partial}{\partial x}b-\frac{\partial}{\partial y}a)$ has no $\hat{i}$ or $\hat{j}$ components. This is expected, as the cross product is supposed to be orthogonal to both multiplied vectors. It is probable that you do not understand some of these things. If you don't, you will need more abstract linear algebra to get this post.