For the potential function $\varphi$ and points $A, B, C,$ and D on the level curve $\varphi(x, y)=0,$ complete the following steps.

a. Find the gradient field $\mathbf{F}=\nabla \varphi.$

b. Evaluate $\mathbf{F}$ at the points $A, B, C,$ and $D.$

c. Plot the level curve $\varphi(x, y)=0$ and the vectors $\mathbf{F}$ at the points $A$ $B, C,$ and $D.$

$$\begin{aligned}

&\varphi(x, y)=\frac{1}{2} x^{2}-y ; A(-2,2), B(-1,1 / 2), C(1,1 / 2), \text { and } D(2,2)

\end{aligned}$$