The Central Idea
Let a vector field be
\[ \mathbf{F}(x,y) = \langle P(x,y),Q(x,y)\rangle. \]A curve
\[ \mathbf{r}(t)=\langle x(t),y(t)\rangle \]follows the field when its velocity equals the field vector at its current location:
\[ \boxed{ \mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t)) }. \]The curve therefore moves in the direction prescribed by the field at every instant.
From Arrows to Paths
A vector-field diagram shows many arrows at many points.
A flow line answers a different question:
\[ \boxed{\text{What path would follow those local directions?}} \]Instead of reading one arrow at a time, we connect the directional information into a continuous curve.
Velocity along a Flow Line
If
\[ \mathbf{r}(t)=\langle x(t),y(t)\rangle, \]then its velocity is
\[ \mathbf{r}'(t) = \langle x'(t),y'(t)\rangle. \]The flow-line condition is
\[ \langle x'(t),y'(t)\rangle = \langle P(x(t),y(t)),Q(x(t),y(t))\rangle. \]Therefore,
\[ \boxed{ x'(t)=P(x(t),y(t)), \qquad y'(t)=Q(x(t),y(t)) }. \]A System of Differential Equations
Flow lines are therefore solutions of a system of differential equations.
For the field
\[ \mathbf{F}(x,y)=\langle x,y\rangle, \]the flow equations are
\[ x'=x, \qquad y'=y. \]Solving these equations determines the paths generated by the field.
A Simple Constant Field
Consider
\[ \mathbf{F}(x,y)=\langle 2,1\rangle. \]The flow equations are
\[ x'=2, \qquad y'=1. \]Integrating gives
\[ x(t)=2t+C_1, \] \[ y(t)=t+C_2. \]The trajectories are straight lines.
Direction of the Constant-Field Trajectories
Eliminating \(t\) from
\[ x=2t+C_1, \qquad y=t+C_2 \]gives lines whose direction vector is
\[ \langle 2,1\rangle. \]Every trajectory is therefore parallel to the field vectors.
This is exactly what we should expect from a constant vector field.
Initial Conditions Select One Trajectory
A vector field usually has an entire family of flow lines.
An initial condition such as
\[ \mathbf{r}(0)=\langle x_0,y_0\rangle \]selects the trajectory passing through the chosen starting point.
The initial position therefore determines which member of the family we follow.
Example with an Initial Condition
For
\[ \mathbf{F}(x,y)=\langle 2,1\rangle, \]suppose
\[ \mathbf{r}(0)=\langle 3,-2\rangle. \]Then
\[ x(0)=3, \qquad y(0)=-2. \]Therefore,
\[ \boxed{ \mathbf{r}(t) = \langle 3+2t,-2+t\rangle }. \]Flow Lines and Trajectories
The terms flow line and trajectory are closely related but emphasize different ideas.
A flow line emphasizes the geometric curve that follows the vector field.
A trajectory emphasizes the time-dependent motion of an object along such a curve.
In an autonomous velocity field, the same parameterized solution can often be interpreted both ways.
Integral Curves
A curve satisfying
\[ \mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t)) \]is also called an integral curve of the vector field.
The name reflects the fact that the curve is obtained by solving, or integrating, differential equations determined by the field.
Streamlines
In fluid mechanics, a curve tangent to the instantaneous velocity field is commonly called a streamline.
At a fixed instant, a streamline shows the direction the fluid velocity is pointing at each point along the curve.
For steady flows, streamlines and particle trajectories align particularly naturally.
Steady Fields
A vector field is steady when it does not depend explicitly on time:
\[ \mathbf{F}(x,y). \]The field pattern remains fixed while a particle moves through it.
The particle changes position, but the vector assigned to any fixed location does not change with time.
Time-Dependent Fields
An unsteady or time-dependent field has the form
\[ \mathbf{F}(x,y,t). \]A trajectory must then satisfy
\[ \boxed{ \mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t),t) }. \]The motion depends both on where the object is and on when it is there.
Autonomous Systems
A system such as
\[ x'=P(x,y), \qquad y'=Q(x,y) \]is called autonomous because the right-hand side has no explicit time variable.
The velocity is determined entirely by the current position.
Nonautonomous Systems
A system such as
\[ x'=P(x,y,t), \qquad y'=Q(x,y,t) \]is nonautonomous.
Two particles passing through the same point at different times may receive different velocity vectors.
A Radial Outward Field
Consider
\[ \mathbf{F}(x,y)=\langle x,y\rangle. \]The flow equations are
\[ x'=x, \qquad y'=y. \]Their solutions are
\[ x(t)=C_1e^t, \qquad y(t)=C_2e^t. \]Geometry of the Radial Flow
If \(C_1\neq0\), then
\[ \frac{y(t)}{x(t)} = \frac{C_2}{C_1}, \]which is constant.
Therefore, nonzero trajectories lie on straight rays through the origin.
As \(t\) increases, their distance from the origin grows exponentially.
An Inward Radial Field
Consider
\[ \mathbf{F}(x,y)=\langle -x,-y\rangle. \]The equations are
\[ x'=-x, \qquad y'=-y. \]Their solutions are
\[ x(t)=C_1e^{-t}, \qquad y(t)=C_2e^{-t}. \]As \(t\) increases, trajectories approach the origin.
Equilibrium Points
A point \((x_0,y_0)\) where
\[ \mathbf{F}(x_0,y_0)=\mathbf{0} \]is an equilibrium point for the autonomous flow.
If a particle begins exactly there, then
\[ \mathbf{r}'(t)=\mathbf{0}, \]so the constant path
\[ \mathbf{r}(t)=\langle x_0,y_0\rangle \]is a trajectory.
The Origin as an Equilibrium
For both
\[ \mathbf{F}(x,y)=\langle x,y\rangle \]and
\[ \mathbf{F}(x,y)=\langle -x,-y\rangle, \]we have
\[ \mathbf{F}(0,0)=\mathbf{0}. \]Therefore, the origin is an equilibrium point in both fields.
The surrounding behavior, however, is very different.
Stable-Looking and Unstable-Looking Behavior
In the inward field
\[ \langle -x,-y\rangle, \]nearby trajectories move toward the origin.
In the outward field
\[ \langle x,y\rangle, \]nearby trajectories move away from the origin.
This previews the later study of stability in dynamical systems.
A Rotational Field
Consider
\[ \mathbf{F}(x,y)=\langle -y,x\rangle. \]The flow equations are
\[ x'=-y, \qquad y'=x. \]The field is tangent to circles centered at the origin.
Why the Rotational Trajectories Stay on Circles
Along a trajectory,
\[ \frac{d}{dt}(x^2+y^2) = 2xx'+2yy'. \]Substitute
\[ x'=-y, \qquad y'=x: \] \[ \frac{d}{dt}(x^2+y^2) = 2x(-y)+2y(x)=0. \]Therefore,
\[ x^2+y^2 \]remains constant along each trajectory.
Circular Trajectories
Because
\[ x^2+y^2=C, \]every nonzero trajectory of
\[ \mathbf{F}(x,y)=\langle -y,x\rangle \]lies on a circle centered at the origin.
The direction of the field determines the direction of travel around those circles.
A Parametric Circular Solution
One solution is
\[ x(t)=R\cos t, \qquad y(t)=R\sin t. \]Differentiate:
\[ x'(t)=-R\sin t=-y(t), \] \[ y'(t)=R\cos t=x(t). \]Therefore, the path satisfies the rotational flow equations.
Speed along a Trajectory
If a trajectory follows the vector field exactly, then
\[ \mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t)). \]Therefore, its speed is
\[ \boxed{ \|\mathbf{r}'(t)\| = \|\mathbf{F}(\mathbf{r}(t))\| }. \]The vector-field magnitude gives the instantaneous speed when the field is interpreted as velocity.
Direction and Speed Play Different Roles
The direction of the field determines the tangent direction of a trajectory.
The magnitude determines how quickly the parameterized motion proceeds through that direction when the field represents velocity.
Two fields can have the same flow-line shapes but different speeds along those curves.
Same Paths, Different Speeds
Compare
\[ \mathbf{F}(x,y)=\langle -y,x\rangle \]with
\[ \mathbf{G}(x,y)=2\langle -y,x\rangle. \]Both fields point tangent to the same family of circles.
The second field has twice the magnitude at each point, so motion under \(\mathbf{G}\) proceeds twice as fast along corresponding circular paths.
Multiplying by a Positive Scalar Field
If
\[ \mathbf{G}(x,y) = a(x,y)\mathbf{F}(x,y) \]with
\[ a(x,y)>0, \]then \(\mathbf{G}\) points in the same local direction as \(\mathbf{F}\).
Under suitable conditions, the geometric flow lines may be the same even though the rate of travel along them changes.
Negative Scaling Reverses Orientation
If
\[ \mathbf{G}=-\mathbf{F}, \]then every field vector is reversed.
The same geometric curves may be followed in the opposite time direction.
Orientation is therefore part of the trajectory information.
Flow Lines without Solving for Time
In two dimensions, suppose
\[ x'=P(x,y), \qquad y'=Q(x,y). \]Where \(P\neq0\),
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{Q(x,y)}{P(x,y)}. \]Therefore, flow-line shapes can sometimes be found from
\[ \boxed{ \frac{dy}{dx} = \frac{Q}{P} }. \]Example: Eliminate the Time Parameter
For the radial field
\[ \mathbf{F}(x,y)=\langle x,y\rangle, \]we have
\[ \frac{dy}{dx} = \frac{y}{x}. \]Separate variables:
\[ \frac{dy}{y} = \frac{dx}{x}. \]Integrating gives
\[ \ln|y| = \ln|x|+C. \]Therefore,
\[ y=Cx, \]confirming that the flow lines are straight lines through the origin.
Example: Rotational Field without Time
For
\[ \mathbf{F}(x,y)=\langle -y,x\rangle, \]where \(y\neq0\),
\[ \frac{dy}{dx} = \frac{x}{-y} = -\frac{x}{y}. \]Rearranging,
\[ y\,dy=-x\,dx. \]Integrating gives
\[ \frac12y^2 = -\frac12x^2+C, \]or
\[ \boxed{x^2+y^2=C}. \]Flow Lines Are Tangent to the Field
The essential geometric condition is local:
\[ \boxed{ \text{trajectory tangent} \parallel \text{field vector} }. \]When the trajectory is parameterized so that
\[ \mathbf{r}'(t)=\mathbf{F}(\mathbf{r}(t)), \]the tangent vector is not merely parallel to the field; it is exactly equal to it.
A Field Diagram Suggests Flow Geometry
Before solving equations, a vector-field plot can suggest:
- outward motion;
- inward motion;
- rotation;
- shearing;
- parallel transport;
- convergence toward equilibria;
- departure from equilibria.
Exact trajectories require mathematical analysis, but the diagram provides geometric intuition.
Trajectories in Three Dimensions
For a three-dimensional vector field
\[ \mathbf{F}(x,y,z) = \langle P,Q,R\rangle, \]a trajectory satisfies
\[ \boxed{ x'=P(x,y,z), \qquad y'=Q(x,y,z), \qquad z'=R(x,y,z) }. \]The resulting path is a curve through three-dimensional space.
Example: Three-Dimensional Constant Flow
Let
\[ \mathbf{F}(x,y,z)=\langle 1,2,-1\rangle. \]Then
\[ x'=1, \qquad y'=2, \qquad z'=-1. \]Integrating gives
\[ \mathbf{r}(t) = \langle t+C_1,2t+C_2,-t+C_3\rangle. \]Every trajectory is a straight line parallel to \(\langle 1,2,-1\rangle\).
Trajectory versus Path Shape
The geometric path tells us which points are visited.
The parameterization tells us when those points are visited and how fast the motion proceeds.
Two parameterizations may trace the same geometric curve while describing different timing.
Example: Same Circle, Different Parameterization
Compare
\[ \mathbf{r}_1(t) = \langle \cos t,\sin t\rangle \]and
\[ \mathbf{r}_2(t) = \langle \cos 2t,\sin 2t\rangle. \]Both trace the unit circle.
The second completes the circle twice as quickly with respect to the same parameter \(t\).
Trajectories Cannot Usually Cross in an Autonomous Field
If a sufficiently regular autonomous vector field assigns one velocity vector to each point and the corresponding initial-value problem has unique solutions, then two distinct trajectories cannot cross at the same point at the same time-independent state.
A crossing would imply two different future motions starting from the same state, contradicting uniqueness.
Why Uniqueness Matters
The noncrossing rule depends on regularity sufficient to guarantee unique solutions.
In poorly behaved differential equations, uniqueness can fail.
Therefore, the strongest statement is not simply “flow lines never cross,” but rather that unique trajectories of a well-behaved autonomous system do not cross in state space.
Time-Dependent Motion Can Revisit a Location
A trajectory can pass through the same spatial location at different times, especially in periodic motion.
What matters in a time-dependent system is the full state including time.
The same spatial point can carry different field vectors at different instants.
Closed Trajectories
A trajectory is closed if it eventually returns to its starting position and repeats geometrically.
The circular paths generated by
\[ \mathbf{F}(x,y)=\langle -y,x\rangle \]are closed trajectories.
Closed trajectories are associated with periodic motion.
Open Trajectories
Many trajectories do not close.
Straight-line motion in a constant field, outward radial motion, and inward radial motion generally produce open paths.
A flow-line family can therefore contain very different global geometries.
Bounded and Unbounded Trajectories
A trajectory is bounded if it remains inside some finite region.
Circular trajectories are bounded.
Outward exponential trajectories from
\[ \mathbf{F}(x,y)=\langle x,y\rangle \]are unbounded as \(t\to\infty\) unless they begin at the origin.
Trajectories Can Approach Equilibria
For
\[ \mathbf{F}(x,y)=\langle -x,-y\rangle, \]the solution
\[ \mathbf{r}(t) = \langle C_1e^{-t},C_2e^{-t}\rangle \]satisfies
\[ \mathbf{r}(t)\to\langle 0,0\rangle \]as \(t\to\infty\).
The equilibrium acts as an attracting destination for nearby trajectories.
Trajectories Can Move Away from Equilibria
For
\[ \mathbf{F}(x,y)=\langle x,y\rangle, \]nonzero trajectories satisfy
\[ \|\mathbf{r}(t)\| \to\infty \]as \(t\to\infty\).
The origin is an equilibrium, but nearby trajectories move away from it.
Flow through a Scalar Field
A trajectory can move through a scalar field
\[ f(x,y). \]Along the path \(\mathbf{r}(t)\), the experienced scalar value is
\[ f(\mathbf{r}(t)). \]Its rate of change is
\[ \frac{d}{dt}f(\mathbf{r}(t)) = \nabla f(\mathbf{r}(t)) \cdot \mathbf{r}'(t). \]Flow Guided by a Vector Field
If the trajectory follows
\[ \mathbf{r}'(t)=\mathbf{F}(\mathbf{r}(t)), \]then
\[ \boxed{ \frac{d}{dt}f(\mathbf{r}(t)) = \nabla f\cdot\mathbf{F} }. \]The dot product reveals whether the flow moves toward higher, lower, or locally unchanged scalar values.
Flow along a Level Set
If
\[ \nabla f\cdot\mathbf{F}=0 \]along a trajectory, then the scalar field has no first-order change along that motion.
The vector field is tangent to the level sets of \(f\).
The rotational field around circles provides an important example.
Flow Across Level Sets
If
\[ \nabla f\cdot\mathbf{F}>0, \]the trajectory moves locally toward larger values of \(f\).
If
\[ \nabla f\cdot\mathbf{F}<0, \]it moves locally toward smaller values.
Gradient Flow
A particularly important system is
\[ \mathbf{r}'(t)=\nabla f(\mathbf{r}(t)). \]Such trajectories move locally in the direction of greatest increase of \(f\).
The opposite system
\[ \mathbf{r}'(t)=-\nabla f(\mathbf{r}(t)) \]moves locally in the direction of steepest decrease.
Gradient Descent as a Flow Idea
The continuous system
\[ \mathbf{r}'(t)=-\nabla f \]provides a geometric model for descending a scalar landscape.
Discrete optimization algorithms often imitate this idea by taking a sequence of finite steps opposite the gradient.
The detailed theory belongs to optimization, but the spatial picture begins with vector-field flow.
Numerical Trajectories
Many flow equations cannot be solved with a simple closed-form formula.
A numerical method can approximate the trajectory by repeatedly using the local field vector to estimate the next position.
One simple idea is
\[ \mathbf{r}_{n+1} \approx \mathbf{r}_n + \Delta t\, \mathbf{F}(\mathbf{r}_n). \]This is the basic structure of an Euler step.
Small Steps Follow the Field More Closely
A numerical method approximates a curved trajectory by many short updates.
Smaller time steps can often follow the changing field direction more closely, although computational cost increases.
More advanced numerical methods can achieve greater accuracy without requiring extremely small steps.
Flow-Line Plots Are Numerical Visualizations
Software often generates streamline or trajectory plots by numerically integrating the field from selected starting points.
The curves shown are therefore usually approximations to mathematical integral curves.
Their quality depends on numerical step size, method, domain, and plotting choices.
Starting Points Affect the Picture
A field contains infinitely many possible starting points.
A visualization displays only a selected set of trajectories.
Sparse starting points may hide important structure, while too many curves may make the diagram unreadable.
A flow-line plot is therefore a representation of the field, not the field itself.
Common Error: Treating Every Vector Field as Velocity
A vector field can represent velocity, but it may also represent force, acceleration, electric field, or another vector quantity.
A physical object follows
\[ \mathbf{r}'=\mathbf{F} \]only when the model says the field itself is the object's velocity.
Common Error: Confusing a Field Vector with a Position Vector
If
\[ \mathbf{F}(2,3)=\langle -1,4\rangle, \]then \((2,3)\) is the location and \(\langle -1,4\rangle\) is the assigned vector.
The field vector is not necessarily the position of the moving object.
Common Error: Drawing a Path That Is Not Tangent to the Field
A legitimate flow line must agree locally with the field direction.
If a proposed curve crosses field arrows at large angles, it is not a flow line of that field.
The tangent condition must hold at every regular point along the path.
Common Error: Ignoring Vector Magnitude
Two fields can point in the same directions but have different magnitudes.
Their geometric flow-line shapes may match while their time-dependent trajectories move at different speeds.
Direction determines shape; magnitude can affect timing.
Common Error: Assuming Every Flow Line Is Closed
Some fields produce circles or other closed curves, but many do not.
Straight lines, spirals, rays, and paths approaching equilibria are all possible.
Closure is a property of a particular flow pattern, not of vector fields in general.
Common Error: Assuming All Trajectories Can Cross
In a well-behaved autonomous system with unique solutions, distinct trajectories cannot cross in state space.
A crossing would assign incompatible motions to the same state.
However, a spatial path may revisit a location at a later time, especially in periodic motion.
Common Error: Confusing Streamlines with Particle Paths in Unsteady Flow
A streamline is tangent to the velocity field at one chosen instant.
A particle trajectory records the actual position of a moving particle over time.
In steady flow these notions align closely. In time-dependent flow they can differ.
A Reliable Procedure for Checking a Proposed Flow Line
- Write the proposed parameterized curve \(\mathbf{r}(t)\).
- Differentiate to find \(\mathbf{r}'(t)\).
- Evaluate the vector field at \(\mathbf{r}(t)\).
- Compare \(\mathbf{r}'(t)\) with \(\mathbf{F}(\mathbf{r}(t))\).
- If they are equal, the parameterized curve follows the field exactly.
- If they are only parallel, the geometric path may match while the parameterization differs.
A Reliable Procedure for Finding Planar Flow-Line Shapes
- Write \(x'=P(x,y)\) and \(y'=Q(x,y)\).
- When valid, form \(\dfrac{dy}{dx}=\dfrac{Q}{P}\).
- Solve the resulting differential equation.
- Interpret the solution family geometrically.
- Apply the initial condition if one is given.
- Check special cases where division by \(P\) may have excluded trajectories.
A Reliable Procedure for a Parameterized Trajectory
- Write the system determined by the vector field.
- Solve the differential equations for the coordinate functions.
- Use initial conditions to determine constants.
- Assemble the coordinate functions into \(\mathbf{r}(t)\).
- Differentiate and substitute back into the field to verify.
- Interpret the resulting path, direction, and speed.
Example: Full Flow-Line Workflow
Let
\[ \mathbf{F}(x,y)=\langle x,-y\rangle. \]The flow equations are
\[ x'=x, \qquad y'=-y. \]Solve:
\[ x(t)=C_1e^t, \] \[ y(t)=C_2e^{-t}. \]Multiply the two equations:
\[ x(t)y(t)=C_1C_2. \]Therefore, the non-axis trajectories satisfy
\[ \boxed{xy=C}. \]Geometry of the Saddle-Type Flow
The curves
\[ xy=C \]are hyperbolas when \(C\neq0\).
The coordinate axes also require separate attention because dividing by \(x\) or \(y\) during algebraic manipulation can exclude them.
This field therefore produces a characteristic saddle-type flow pattern.
Example: Apply an Initial Condition
For
\[ \mathbf{F}(x,y)=\langle x,-y\rangle, \]suppose
\[ \mathbf{r}(0)=\langle 2,3\rangle. \]Then
\[ C_1=2, \qquad C_2=3. \]Therefore,
\[ \boxed{ \mathbf{r}(t) = \langle 2e^t,3e^{-t}\rangle }. \]Its geometric path satisfies
\[ xy=6. \]Practice 1: Constant Field
Let
\[ \mathbf{F}(x,y)=\langle 3,-2\rangle. \]Write the differential equations for a trajectory.
Practice 2: Apply an Initial Condition
For
\[ \mathbf{F}(x,y)=\langle 3,-2\rangle, \]find the trajectory satisfying
\[ \mathbf{r}(0)=\langle 1,4\rangle. \]Practice 3: Radial Flow
For
\[ \mathbf{F}(x,y)=\langle x,y\rangle, \]determine whether trajectories move toward or away from the origin as \(t\) increases.
Practice 4: Inward Flow
For
\[ \mathbf{F}(x,y)=\langle -x,-y\rangle, \]describe the long-term behavior of a nonzero trajectory.
Practice 5: Rotational Flow
For
\[ \mathbf{F}(x,y)=\langle -y,x\rangle, \]identify a quantity that remains constant along each trajectory.
Practice 6: Equilibrium Point
Find the equilibrium point of
\[ \mathbf{F}(x,y)=\langle x-2,y+1\rangle. \]Practice 7: Flow-Line Differential Equation
For
\[ \mathbf{F}(x,y)=\langle y,x\rangle, \]write \(\dfrac{dy}{dx}\) where \(y\neq0\).
Practice 8: Three-Dimensional Flow
Let
\[ \mathbf{F}(x,y,z)=\langle 1,0,-2\rangle. \]Write a general parameterized trajectory.
Practice 9: Check a Proposed Trajectory
Determine whether
\[ \mathbf{r}(t)=\langle e^t,2e^t\rangle \]is a trajectory of
\[ \mathbf{F}(x,y)=\langle x,y\rangle. \]Practice 10: Interpret Speed
If a velocity field satisfies
\[ \mathbf{v}(2,1)=\langle 3,4\rangle, \]what is the instantaneous speed of a particle following the field at that point?
Practice Answers
Practice 1
\[ \boxed{ x'=3, \qquad y'=-2 }. \]Practice 2
Integrate:
\[ x(t)=3t+C_1, \qquad y(t)=-2t+C_2. \]The initial condition gives
\[ C_1=1, \qquad C_2=4. \]Therefore,
\[ \boxed{ \mathbf{r}(t) = \langle 1+3t,4-2t\rangle }. \]Practice 3
The solutions have the form
\[ \langle C_1e^t,C_2e^t\rangle. \]Therefore, every nonzero trajectory moves away from the origin as \(t\) increases.
Practice 4
The solutions have the form
\[ \langle C_1e^{-t},C_2e^{-t}\rangle. \]Therefore, every trajectory approaches the origin as \(t\to\infty\).
Practice 5
Along a trajectory,
\[ \boxed{x^2+y^2=\text{constant}}. \]Therefore, the nonzero trajectories lie on circles centered at the origin.
Practice 6
Set both components equal to zero:
\[ x-2=0, \qquad y+1=0. \]Therefore,
\[ \boxed{(2,-1)}. \]Practice 7
\[ \frac{dy}{dx} = \frac{y'}{x'} = \frac{x}{y}. \]Therefore,
\[ \boxed{ \frac{dy}{dx}=\frac{x}{y} }. \]Practice 8
Integrating
\[ x'=1, \qquad y'=0, \qquad z'=-2 \]gives
\[ \boxed{ \mathbf{r}(t) = \langle t+C_1,C_2,-2t+C_3\rangle }. \]Practice 9
Differentiate:
\[ \mathbf{r}'(t) = \langle e^t,2e^t\rangle. \]Evaluate the field along the curve:
\[ \mathbf{F}(\mathbf{r}(t)) = \langle e^t,2e^t\rangle. \]Since these are equal,
\[ \boxed{\mathbf{r}(t)\text{ is a trajectory of the field}}. \]Practice 10
\[ \|\mathbf{v}(2,1)\| = \sqrt{3^2+4^2} = \boxed{5}. \]The instantaneous speed is \(5\) in the velocity units of the model.
What to Remember
A trajectory follows a vector field when
\[ \boxed{ \mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t)) }. \]- the field vector gives the tangent direction of the path;
- for velocity fields, its magnitude gives instantaneous speed;
- initial conditions select particular trajectories;
- autonomous fields depend on position but not explicitly on time;
- time-dependent fields can assign different vectors to the same location at different times;
- equilibrium points occur where the field is zero;
- flow lines may be straight, circular, radial, hyperbolic, closed, open, bounded, or unbounded;
- numerical integration is often needed when exact solutions are unavailable.
The central geometric habit is to ask: if the field gives the local velocity, what curve remains tangent to those vectors as the position changes?
Where We Go Next
Flow lines show how vector fields organize motion through space. The next step is to identify two important kinds of local field behavior: spreading and rotation.
The next lesson, Introductory Divergence and Curl, introduces scalar and vector measures that describe local expansion, convergence, and rotational tendency in vector fields.