07 Vector Fields

Vector Fields and Their Representations

A vector field assigns a vector to every point in a region. In two dimensions, it has the form

In three dimensions, it has the form

The component functions determine the direction and magnitude of the vector at each point. A diagram represents the field with arrows: arrow direction shows the field direction, and arrow length represents its magnitude.

Important applications include:

• velocity fields, which describe fluid motion;

• force fields, which describe forces such as gravity or electric force;

• heat-flow fields, which describe the direction and rate of heat transfer.

A vector field can be evaluated along a parametrized curve $\mathbf{r}(t)$ by substituting the curve into the field, producing $\mathbf{F}(\mathbf{r}(t))$.

Line Integrals and Work

A line integral measures the accumulated tangential effect of a vector field along a curve. For a curve $C$ parametrized by $\mathbf{r}(t)$, with $a\le t\le b$,

The dot product selects the component of the field tangent to the direction of motion. If $\mathbf{F}$ is a force field, this integral represents work:

The orientation of the curve matters. Reversing the direction of traversal changes the sign of the line integral.

Takeaway: A line integral combines the field with the direction and geometry of a path, rather than measuring the field at only one point.

Conservative Vector Fields

A conservative vector field is a field that can be written as the gradient of a scalar function:

The scalar function $f$ is a potential function. In two dimensions, if $\mathbf{F}(x,y)=\langle P(x,y),Q(x,y)\rangle$, then conservativeness means that

In three dimensions, if $\mathbf{F}=\langle P,Q,R\rangle$, then

The gradient points in the direction of greatest increase of $f$, and its magnitude gives the rate of increase in that direction.

For a conservative field, the Fundamental Theorem for Line Integrals gives

where $A$ and $B$ are the initial and terminal points. Thus, the value depends only on the endpoints. This is path independence. For a closed curve, $A=B$, so

Takeaway: Recognizing a conservative field can replace a difficult path computation with evaluation of a potential function at two endpoints.

Testing Conservativeness

For a two-dimensional field $\mathbf{F}=\langle P,Q\rangle$ with continuous first partial derivatives, compare the cross-partials:

On a simply connected domain, this condition is sufficient for the field to be conservative. A simply connected domain has no holes that allow paths to wind around a missing point or region.

For a three-dimensional field $\mathbf{F}=\langle P,Q,R\rangle$, the corresponding conditions are

These conditions are equivalent to

The domain must be checked before applying this test. A field may have zero curl away from a missing point or line and still fail to be conservative because the domain is not simply connected.

To find a potential function in two dimensions, integrate one component and then determine the remaining single-variable function. For example, let

Here, $P_y=2x$ and $Q_x=2x$, and the domain is all of $\mathbb{R}^2$, so the field is conservative. Integrating $f_x=P$ with respect to $x$ gives

Differentiating with respect to $y$ and matching $f_y=Q$ gives $g'(y)=4y$, so one potential function is

For a path from $A=(0,1)$ to $B=(2,3)$,

Takeaway: Use the cross-partial or curl test together with the domain, then construct a potential function when the field is conservative.

Potential Functions and Energy

Level curves and level surfaces of a potential function are sets on which $f$ is constant. If $\mathbf{v}$ is tangent to a level set, then

Therefore, the gradient is perpendicular to the level set. This geometric fact explains why the field $\mathbf{F}=\nabla f$ points across potential contours rather than along them.

In physical models, potential energy is often denoted by $U$, with force defined by

The negative sign means that the force points toward decreasing potential energy. If a particle moves from $A$ to $B$, the work is

This is consistent with the endpoint-only nature of conservative work.

Velocity Fields and Streamlines

A time-dependent velocity field $\mathbf{v}(x,y,z,t)$ gives the instantaneous velocity of the fluid particle located at $(x,y,z)$ at time $t$. Its magnitude $\lvert\mathbf{v}\rvert$ is the fluid speed, and its direction is the direction of motion.

A streamline is a curve whose tangent direction is parallel to the velocity field. If $\mathbf{r}(s)$ parametrizes a streamline, then

For a two-dimensional velocity field $\mathbf{v}=\langle P,Q\rangle$, streamlines can be found from

Streamlines show the instantaneous pattern of the flow. If the field changes with time, they do not necessarily trace the trajectory of one particular fluid particle.

Takeaway: Velocity fields describe local motion, while streamlines organize that local information into visible flow patterns.

Divergence and Curl

Divergence measures local expansion or compression. For a three-dimensional field $\mathbf{F}=\langle P,Q,R\rangle$,

For a velocity field:

• $\nabla\cdot\mathbf{v}>0$ indicates local expansion or source-like behavior;

• $\nabla\cdot\mathbf{v}<0$ indicates local compression or sink-like behavior;

• $\nabla\cdot\mathbf{v}=0$ indicates locally source-free flow.

An incompressible flow satisfies

This means that the flow has no local net creation or removal of volume.

Curl measures local rotation. In three dimensions,

For the two-dimensional rotational field

its scalar curl is

so the field has nonzero local rotation and circulates counterclockwise around the origin.

By contrast, the radial field

has scalar curl $Q_x-P_y=0$ and points outward. It is conservative on $\mathbb{R}^2$ because

Takeaway: Divergence describes outward or inward flow, whereas curl describes local rotational tendency.