Answer
$${\mathbf{F}(x, y)=2 x \mathbf{i}-\frac{1}{2} y \mathbf{j}}$$
Work Step by Step
Given
$$f(x, y)=x^{2}-\frac{1}{4} y^{2}$$
Since \begin{align}\mathbf{F}(x,y)&=M \mathbf{i}+N\mathbf{j}\\
&=f_{x}(x, y)\mathbf{i}+f_{y}(x, y)\mathbf{j}
\end{align}
As, we have
\begin{array}{l} {f_{x}(x, y)=\frac{\partial f(x,y)}{\partial x}=2 x} \\ {f_{y}(x, y)=\frac{\partial f(x,y)}{\partial y}=-\frac{1}{2} y} \\ \end{array}
So, we get
$${\mathbf{F}(x, y)=2 x \mathbf{i}-\frac{1}{2} y \mathbf{j}}$$