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The scalar field $ \ \nabla \cdot \mathbf{F} \ $ is known as the divergence of the vector field $\mathbf{F}$.

Why can we use the del operator $ \nabla $ as a vector here? What do the components $ \partial / \partial x \ $, $ \partial / \partial y \ $ and $ \ \partial / \partial z \ $ of $ \nabla $ mean in this context?

They are clearly not variables or constants, so I'm not sure what they represent. They seem to be placeholders, but of what?

I'm okay with the usage of $ \nabla $ as "an operator", but I'm not seeing what it means to use it as a vector.

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Let $f:\mathbb{R}^n\to\mathbb{R}$ be a differentiable function and let $\gamma:(-\varepsilon,\varepsilon)\to\mathbb{R}^n$ be a smooth curve in $\mathbb{R}^n$ such that $\gamma(0)=p$ and $\gamma'(0)=v$ where $\gamma(t):=(x_1(t),...,x_n(t))$ then the derivative of $f$ at $p$ in the direction of $v$ can be computed as follows $$\frac{d(f\circ\gamma)(t)}{\,dt}\Big|_{t=0}=\frac{df(x_1(t),...,x_n(t))}{\,dt}\Big|_{t=0}=\sum_{k=1}^n\frac{\partial f}{\partial x_k}\frac{dx_k}{dt}\Big|_{t=0}=\sum_{k=1}^n\frac{\partial f}{\partial x_k}x_k'(0)$$ this can be rewritten as $$\sum_{k=1}^n\frac{\partial f}{\partial x_k}x_k'(0)=\Big(\sum_{k=1}^nx_k'(0)\frac{\partial }{\partial x_k}\Big)f$$ So you can think of {$\partial/\partial x_k$} as a basis for the linear operator $$A(t):=\sum_{k=1}^nx_k'(t)\frac{\partial }{\partial x_k}$$ Note $A(0)=v$. In this respect you can regard $\nabla:=(\partial/\partial x_1,...,\partial/\partial x_n)$ as a vector whose entries are the basis element $\{\partial/\partial x_k\}$ and this basis acts as an operator on a differentiable function by what is known as differentiation.

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Saying that $\nabla$ is the "vector" $\left < \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right >$ is simply notational shorthand. It's so that we can write out the equations for divergence and curl as $\nabla \cdot \textbf{F}$ and $\nabla \times \textbf{F}$, without having to write out the actual definition every time.

Specifically, the curl is a somewhat complicated formula, namely: if $$\textbf{F}(x,y,z) = \langle u(x,y,z), v(x,y,z), w(x,y,z) \rangle, $$ then

$$\begin{align*} \nabla \times \textbf{F} & = \left\langle w_y - v_z, \; u_z - w_x, \; v_z - u_y\right\rangle \\ & = \left\langle \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}, \; \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}, \; \frac{\partial v}{\partial z} - \frac{\partial u}{\partial y} \right\rangle, \end{align*}$$ which is the definition of the curl of $\textbf{F}$, and which looks a lot like a cross product.

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