I was studying about conservative forces from a physics book (NCERT, a standard Indian textbook) and came to a para which is as follows:
A force is conservative if it can be derived from a scalar quantity $V(x)$ by the relation given by $dV(x)=-F(x)dx$. The three dimensional generalisation requires the use of vector derivative, which is outside the scope of this book.
What I didn't understand is what does it mean by 3D generalisation?
It means that your field exists in 3D space (ie it has an x, a y, and a z component). To get the force, you need to know the derivative of a field that points along x, the derivative that points along y, and the derivative that points along z. The generalization of that concept is that it is the 3D vector force is conservativ if it can be written as a gradient of some scalar field V(x,y,z).
The $3D$ generalization is nothing but $$F=-\nabla V$$One another necessary condition is $$\nabla×\vec{F}=0$$
