Vector Fields

Definition Let $D$ be a set in $\mathbb{R}^2$. A vector field on $D$ is a function $\mathbf{F}$ that assigns to each point $(x, y)$ in $D$ a vector in $\mathbb{R}^2$,

A vector field on a set $E$ in $\mathbb{R}^3$ assigns to each $(x, y, z)$ in $E$ a vector $\langle P, Q, R \rangle$ in $\mathbb{R}^3$. The scalar functions $P$, $Q$ and $R$ are the component functions. The field is continuous, or $C^1$, on an open set when all its component functions are continuous, or have continuous first partial derivatives, there. To picture $\mathbf{F}$, draw the arrow $\mathbf{F}(x, y)$ with its tail at the point $(x, y)$.

Definition A steady fluid flow in a region $E$ is described by its velocity field $\mathbf{v}$, where $\mathbf{v}(x, y, z)$ is the velocity of the fluid particle passing through $(x, y, z)$, the same at every moment. A particle carried by the flow moves along a path $\mathbf{r}(t)$ with

for every $t$ in an interval. Its speed at time $t$ is $\lvert \mathbf{v}(\mathbf{r}(t)) \rvert$.

Definition A force field assigns to each point the force that a given test object would feel if placed there. The standard example is the gravitational field of a mass $M$ at the origin acting on a mass $m$. With $\mathbf{x} = \langle x, y, z \rangle$ and $k = GMm > 0$, where $G$ is the gravitational constant,

The vector points toward the origin and has magnitude $\frac{k}{\lvert \mathbf{x} \rvert^2}$, so it obeys an inverse square law. The electric force between two charges has the same form, with a constant that may have either sign.

Definition If $f$ is a scalar function with first partial derivatives on an open set $D$, its gradient field is

and similarly $\nabla f = \langle f_x, f_y \rangle$ in the plane. At a point where $\nabla f$ is nonzero and $f$ is differentiable, the gradient points in the direction in which $f$ increases fastest and is perpendicular to the velocity of every differentiable curve that stays in the level set of $f$ and passes through that point. When $f$ is $C^1$ that level set is, near the point, a smooth curve or surface.

Definition A vector field $\mathbf{F}$ on an open set $D$ is conservative, or a gradient field, if there is a differentiable function $f$ on $D$ with $\nabla f = \mathbf{F}$ at every point of $D$. Such an $f$ is called a potential function for $\mathbf{F}$. If $D$ is also connected, any two potentials differ by a constant. Physicists often use the potential energy $V = -f$, so that $\mathbf{F} = -\nabla V$. Not every field has a potential. If $\mathbf{F} = \langle P, Q \rangle$ is $C^1$ on $D$ and has a potential there, then the potential has continuous second partials and Clairaut’s theorem forces $P_y = Q_x$ on $D$.

Definition A field line, also called a flow line or integral curve, of a continuous vector field $\mathbf{F}$ on an open set $D$ is a differentiable curve $\mathbf{r}(t)$ in $D$ with

for every $t$ in an interval. In the plane this is the system $x' = P(x, y)$, $y' = Q(x, y)$. If $\mathbf{F}$ is $C^1$ on $D$, the existence and uniqueness theorem for differential equations gives, for each point $p$ of $D$, exactly one maximal field line with $\mathbf{r}(0) = p$. So two field lines that share a point lie along the same curve, and different field lines never cross. At a point where $\mathbf{F}$ is zero the field line is constant.

Definition The domain of a vector field given by a formula is the set of points where every component is defined, unless a smaller domain is stated. A singularity is a point, or a set of points such as a line, where the field is not defined but which is surrounded by points of the domain, typically because the magnitude grows without bound as you approach it. Statements about potentials and field lines always refer to a specific domain, and the answer can change when the domain changes.