Vector Fields

We will now look at a new type of function known as a vector field which maps points $\mathbf{x} \in \mathbb{R}^n$ to $n$-dimensional vectors $\mathbf{F}(\mathbf{x})$.

Definition: Let $D \subseteq \mathbb{R}^2$ and let $E \subseteq \mathbb{R}^3$. A Vector Field on $\mathbb{R}^2$ is a function $\mathbf{F}$ that assigns each element $(x, y) \in D$ to a two-dimensional vector $\mathbf{F}(x, y)$. A Vector Field on $\mathbb{R}^3$ is a function $\mathbf{F}$ that assigns each element $(x, y, z) \in E$ to a three-dimensional vector $\mathbf{F}(x, y, z)$.

If $\mathbf{x} \in D \subseteq \mathbb{R}^2$ then we can define a vector field on $\mathbb{R}^2$ as a function $\mathbf{F}$ that assigns each vector $\mathbf{x} \in D$ to a unique vector $\mathbf{F}(\mathbf{x}) \in \mathbb{R}^2$. Of course this can be extended to real vector fields in higher dimensions.

We can construct a vector field on $\mathbb{R}^2$ by taking some points $(x, y) \in D$ and drawing the corresponding vector $\mathbf{F}(x, y)$ (with respect to its magnitude) with its initial point at $(x, y)$. Now since $\mathbf{F}(x, y)$ is a two-dimensional vector, then we can write $\mathbf{F}(x, y)$ as a linear combination with respect to the standard basis vectors of $\mathbb{R}^2$ which are $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$. For some two variable real-valued functions $P$ and $Q$, we have that:

(1)

\begin{align} \quad \mathbf{F}(x, y) = P(x, y) \vec{i} + Q(x, y) \vec{j} \end{align}

For example, the following is the graph of the vector field $\mathbf{F}(x, y) = -0.1(y - 0.02)^3 \vec{i} + 0.2(x - 0.01)^2 \vec{j}$. The magnitudes of the vectors are not to scale however.

Similarly, we can construct a vector field on $\mathbb{R}^3$ by taking some points $(x, y, z) \in E$ and drawing the corresponding vector $\mathbf{F}(x, y, z)$ with its initial point at $(x, y, z)$. Since $\mathbf{F}(x, y, z)$ is a three-dimensional vector, then we can write $\mathbf{F}(x, y, z)$ as a linear combination with respect to the standard basis vectors of $\mathbb{R}^3$ which are $\vec{i} = (1, 0, 0)$, $\vec{j} = (0, 1, 0)$ and $\vec{k} = (0, 0, 1)$ and for some three variable real-valued functions $P$, $Q$, and $R$ we have that:

(2)

\begin{align} \quad \mathbf{F} (x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{j} \end{align}

For example, the following is the graph of the vector field $\mathbf{F}(x, y, z) = 2xy \vec{i} + yz \vec{j} + 2xz \vec{k}$.

Of course, graphing vector fields on $\mathbb{R}^3$ is impractical, so computer software is useful to compute such fields when necessary.