Imagine the air around you. At every single point in the room, the air has a specific velocity—a direction it is moving and a speed. This is a vector field. Unlike a scalar field, which might just tell you the temperature at each point, a vector field "fills" space with arrows that describe dynamic physical phenomena like wind, ocean currents, or the invisible pull of gravity.
Formal Definitions
To analyze these fields mathematically, we use the following foundational definitions:
Definition 1 (2D Vector Field): Let $D$ be a set in $\mathbb{R}^2$. A vector field on $\mathbb{R}^2$ is a function $\mathbf{F}$ that assigns to each point $(x, y)$ in $D$ a two-dimensional vector:
$$\mathbf{F}(x, y) = P(x, y)\mathbf{i} + Q(x, y)\mathbf{j} = \langle P(x, y), Q(x, y) \rangle$$
where $P$ and $Q$ are scalar fields (functions of two variables).
Definition 2 (3D Vector Field): For a subset $E$ of $\mathbb{R}^3$, the field is defined as: $$\mathbf{F}(x, y, z) = P(x, y, z)\mathbf{i} + Q(x, y, z)\mathbf{j} + R(x, y, z)\mathbf{k}$$
Definition 2 (3D Vector Field): For a subset $E$ of $\mathbb{R}^3$, the field is defined as: $$\mathbf{F}(x, y, z) = P(x, y, z)\mathbf{i} + Q(x, y, z)\mathbf{j} + R(x, y, z)\mathbf{k}$$
Physical Interpretations
- Velocity Fields: Represent fluid flow or wind patterns. For example, Figure 1 shows San Francisco Bay wind patterns, while Figure 13 models fluid through a converging pipe.
- Force Fields: Newton’s Law of Gravitation defines a field where magnitude $|\mathbf{F}| = \frac{mMG}{r^2}$. In vector form: $\mathbf{F}(\mathbf{x}) = -\frac{mMG}{|\mathbf{x}|^3}\mathbf{x}$. Note: Physicists often use $\mathbf{r}$ instead of $\mathbf{x}$.
- Electric Fields: Defined as $\mathbf{E}(\mathbf{x}) = \frac{\varepsilon Q}{|\mathbf{x}|^3}\mathbf{x}$, representing the force per unit charge.
The Geometry of Gradient Fields
If $f$ is a scalar function, its gradient $\nabla f$ creates a special kind of vector field. In 3D, this is expressed as:
$$\nabla f(x, y, z) = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}$$☸ Geometrical Insight
As illustrated in Figure 15, gradient vectors are always perpendicular to the level curves (or level surfaces) of the original function $f$ and point in the direction of the greatest rate of increase.
Example 1: The Rotating Field
Consider $\mathbf{F}(x, y) = -y\mathbf{i} + x\mathbf{j}$. At $(1, 0)$, we have $\langle 0, 1 \rangle$. At $(0, 1)$, we have $\langle -1, 0 \rangle$. Plotting these reveals a circular flow around the origin—the mathematical backbone for modeling vortices and mechanical rotation.