Gradient field of scalar field functions

Question

The scalar field functions $s$ are defined in space by: $$s(x,y) = x^2y^2 + xy - z + C$$

How do I show that all functions have an identical gradient field and how do I calculate it?

Answer 1

The gradient $\nabla \equiv \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \end{pmatrix}$ contains in its first component the partial derivative with respect to $x$ and in its second component the partial derivative with respect to $y$ (for 2D cartesian coordinates).

For $C \neq C(x, y)$, i.e., $C$ is a constant, the gradient $\nabla s$ is the same for all $C \in \mathbb C$.