Derivatives of scalar fields and vector fields

A scalar field is a function that assigns a number to every position in space. Temperature, pressure, density, molecular concentration, electrostatic potential, and charge density are scalar fields. A vector field is a function that assigns a vector to every position in space. Electric fields and magnetic fields are vector fields. The gradient of a scalar field $\phi$ is a vector field. Minus the gradient of the pressure points in the direction that the wind blows (high pressure to low pressure). Minus the gradient of temperature points in the direction that the heat flows (high temperature to low temperature). Minus the gradient of the electrostatic potential points in the direction of the electric field (from high potential to low potential).

The divergence of a vector field $\vec{A}$ is a scalar field. The divergence tells us if the vectors in the vector field are moving apart or moving together. Imagine a small sphere about some point in space. If more vectors on the surface of the sphere are pointing outwards, the divergence is positive. If more vectors on the surface of the sphere are pointing inwards, the divergence is negative.

The curl of a vector field describes how the vectors are rotating about a certain point.

The gradient of a scalar field is a vector field,

\begin{equation} \nabla \phi = \frac{\partial \phi }{\partial x}\hat{x}+\frac{\partial \phi }{\partial y}\hat{y}+\frac{\partial \phi }{\partial z}\hat{z}. \end{equation}

$\phi(x,y,z)=$  
$\nabla \phi = $ () $\hat{x}$ + () $\hat{y}$ + () $\hat{z}$

Generally, the electrostatic potential depends on $x$, $y$, and $z$. To take the partial derivative $\frac{\partial \phi }{\partial x}$, differentiate with respect to $x$ while treating $y$ and $z$ as constants.

The divergence of a vector field is a scalar field,

\begin{equation} \nabla \cdot \vec{A} = \left( \frac{\partial }{\partial x}\hat{x} +\frac{\partial }{\partial y}\hat{y} +\frac{\partial }{\partial z}\hat{z}\right)\cdot \vec{A}=\frac{\partial A_x}{\partial x} +\frac{\partial A_y}{\partial y} +\frac{\partial A_z}{\partial z}. \end{equation}

$\vec{A}(x,y,z)=$  $\hat{x}$ +  $\hat{y}$ +  $\hat{z}$  
$\nabla\cdot \vec{A} = $ () + () + ()

The curl of a vector field is a vector field,

\begin{equation} \nabla\times\vec{A}=\left(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}\right)\hat{x}+ \left(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}\right)\hat{y}+ \left(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}\right)\hat{z}. \end{equation}

$\vec{A}(x,y,z)=$  $\hat{x}$ +  $\hat{y}$ +  $\hat{z}$  
$\nabla\times \vec{A} = $ ()$\hat{x}$ + ()$\hat{y}$ + ()$\hat{z}$