# Scalar field and vector field

Let $U$ be a region of $\mathbb{R}^3$, then a scalar field $f$ is a function $$\begin{array}{ccc} f:U \subseteq \mathbb{R} ^3 & \longrightarrow & \mathbb{R} \\ (x,y,z) & \longrightarrow & f(x,y,z)\end{array}$$ in such way that it assigns to every point $(x, y, z)$ of the region $U$ only one real value $f(x, y, z)$.

On the other hand, let $V$ be a region of $\mathbb{R}^3$, then a vector field $F$ is a function $$\begin{array}{ccc} F:V \subseteq \mathbb{R} ^3 & \longrightarrow & \mathbb{R}^3 \\ (x,y,z) & \longrightarrow &(F_{1}(x,y,z),F_{2}(x,y,z),F_{3}(x,y,z))\end{array}$$ in such a way that it assigns to every point $(x, y, z)$ of the region $U$ of the space another point of the space.

The following examples are scalar fields $$f(x,y,z)=x^{y}+3\cdot z$$ $$f(x,y,z)=4 \cdot x-\frac{y}{\sqrt{z^2}}+3$$

The following examples are vector fields: $$F(x,y,z)=(3\cdot x \cdot z, x-y, z-y)$$ $$F(x,y,z)=(4 \cdot \sin (x^2 \cdot y), \sqrt z, y \cdot x-z)$$