An electric force field is defined by the vector field \[ {\bbox[5px, border:1px solid black, #F9F7ED]{\bbox[#FAF8ED,5pt]{{ \mathbf{F}=\mathbf{F}(x,y,z)=\dfrac{\varepsilon qQ}{x^{2}+y^{2}+z^{2}}\mathbf{u} }}}} \]
where \(\varepsilon \) is a positive constant that depends on the units being used; \(q\) and \(Q\) are the charges of two objects, one located at the origin \((0,0,0)\) and the other at the point \((x,y,z)\); and \(\mathbf{u}\) is the unit vector in the direction from \((0,0,0)\) to \((x,y,z)\). Here, \(\mathbf{F}\) is the electric force exerted by the charge at \((0,0,0)\) on the charge at \((x,y,z)\). For like charges, we have \(qQ>0\) and the force \(\mathbf{F}\) is repulsive. For unlike charges, we have \(qQ<0\) and the force \(\mathbf{F}\) is attractive.
Recall that the gradient \(\boldsymbol\nabla\! f\) of a function \(f=f(x,y) \) in the plane is the vector \[ {\bbox[5px, border:1px solid black, #F9F7ED]{\bbox[#FAF8ED,5pt]{{ \boldsymbol\nabla\! f(x,y)=f_{x}(x,y)\mathbf{i}+f_{y}(x,y)\mathbf{j} }}}} \]
and the gradient \(\boldsymbol\nabla\! f\) of a function \(w=f( x,y,z) \) in space is the vector \[ {\bbox[5px, border:1px solid black, #F9F7ED]{\bbox[#FAF8ED,5pt]{{ \boldsymbol\nabla\! f(x,y,z)=f_{x}(x,y,z)\mathbf{i}+f_{y}(x,y,z) \mathbf{j}+f_{z}(x,y,z)\mathbf{k} }}}} \]
We see that the gradient \(\boldsymbol\nabla\! f\) of a function \(f\) is a vector field, called the gradient vector field.