The Central Idea
A vector field assigns one vector to each point in a spatial domain.
In two dimensions, we may write
\[ \boxed{\mathbf{F}(x,y)=\langle P(x,y),Q(x,y)\rangle}. \]In three dimensions,
\[ \boxed{\mathbf{F}(x,y,z)=\langle P(x,y,z),Q(x,y,z),R(x,y,z)\rangle}. \]The coordinates tell us where we are. The vector tells us what magnitude and direction are assigned there.
From Scalar Fields to Vector Fields
A scalar field has the form
\[ f:\mathbb{R}^n\to\mathbb{R}. \]A vector field instead has the form
\[ \mathbf{F}:\mathbb{R}^n\to\mathbb{R}^m. \]In the most common spatial cases, the output vector has the same number of components as the space has coordinate directions.
For example, a planar vector field typically assigns a two-component vector to each point in the plane.
What Does a Vector Add?
A vector provides directional information that a scalar alone does not.
For example, the scalar
\[ 20 \]might describe a speed of \(20\) meters per second.
The vector
\[ \langle 12,16\rangle \]gives both direction and magnitude. Its magnitude is
\[ \sqrt{12^2+16^2}=20. \]The components tell us how that magnitude is distributed across coordinate directions.
A Simple Constant Vector Field
Consider
\[ \mathbf{F}(x,y)=\langle 2,1\rangle. \]Every point in the plane receives exactly the same vector.
Such a field represents uniform direction and uniform magnitude throughout the domain.
A simplified model of steady wind blowing the same way everywhere could have this form.
A Position-Dependent Vector Field
Now consider
\[ \mathbf{F}(x,y)=\langle x,y\rangle. \]At \((1,2)\),
\[ \mathbf{F}(1,2)=\langle 1,2\rangle. \]At \((-3,1)\),
\[ \mathbf{F}(-3,1)=\langle -3,1\rangle. \]The assigned vector changes as the location changes.
Field Components
A two-dimensional vector field is often written
\[ \mathbf{F}(x,y)=\langle P(x,y),Q(x,y)\rangle. \]The function \(P\) gives the horizontal component, while \(Q\) gives the vertical component.
Each component is itself a scalar field.
Vector fields can therefore be understood as several coordinated scalar fields packaged into one vector-valued function.
Three-Dimensional Components
In three dimensions,
\[ \mathbf{F}(x,y,z) = \langle P(x,y,z),Q(x,y,z),R(x,y,z)\rangle. \]The three components describe how much of the vector points in the \(x\)-, \(y\)-, and \(z\)-directions.
Equivalently, we may write
\[ \mathbf{F} = P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}. \]Evaluating a Vector Field
Suppose
\[ \mathbf{F}(x,y)=\langle 2x-y,x+3y\rangle. \]At \((2,1)\),
\[ \mathbf{F}(2,1) = \langle 2(2)-1,2+3(1)\rangle. \]Therefore,
\[ \boxed{\mathbf{F}(2,1)=\langle 3,5\rangle}. \]Both component functions are evaluated at the same spatial point.
The Magnitude of a Field Vector
If
\[ \mathbf{F}(x,y)=\langle P(x,y),Q(x,y)\rangle, \]then the magnitude of the vector at \((x,y)\) is
\[ \boxed{ \|\mathbf{F}(x,y)\| = \sqrt{P(x,y)^2+Q(x,y)^2} }. \]Magnitude tells us the strength, speed, or size of the vector quantity at that location, depending on the interpretation.
Example: Magnitude at a Point
Let
\[ \mathbf{F}(x,y)=\langle x,2y\rangle. \]At \((3,2)\),
\[ \mathbf{F}(3,2)=\langle 3,4\rangle. \]Its magnitude is
\[ \|\mathbf{F}(3,2)\| = \sqrt{3^2+4^2} = 5. \]So the field vector has magnitude \(5\) at that point.
Direction at a Point
A nonzero vector field value points in a definite direction.
If
\[ \mathbf{F}(x,y)=\langle 3,4\rangle, \]then the corresponding unit direction is
\[ \frac{\mathbf{F}}{\|\mathbf{F}\|} = \left\langle \frac35,\frac45\right\rangle. \]The magnitude and direction can therefore be separated when useful.
Zero Vectors in a Field
A vector field may assign the zero vector to some locations.
For
\[ \mathbf{F}(x,y)=\langle x,y\rangle, \]the origin has
\[ \mathbf{F}(0,0)=\langle 0,0\rangle. \]The field has no directional arrow at that point because the vector has zero magnitude.
A Radial Vector Field
The field
\[ \mathbf{F}(x,y)=\langle x,y\rangle \]points directly away from the origin.
At \((2,1)\), the vector is
\[ \langle 2,1\rangle, \]while at \((-2,-1)\), it is
\[ \langle -2,-1\rangle. \]In each case, the vector points outward from the center.
An Inward Radial Field
Reversing the signs gives
\[ \mathbf{F}(x,y)=\langle -x,-y\rangle. \]Every nonzero vector now points toward the origin.
This simple pattern resembles the directional structure of an attractive force, although realistic physical force laws may also change magnitude with distance in different ways.
A Rotational Vector Field
Consider
\[ \mathbf{F}(x,y)=\langle -y,x\rangle. \]At \((1,0)\),
\[ \mathbf{F}(1,0)=\langle 0,1\rangle. \]At \((0,1)\),
\[ \mathbf{F}(0,1)=\langle -1,0\rangle. \]The vectors are tangent to circles centered at the origin, creating a rotational pattern.
Why the Rotational Pattern Appears
The position vector at \((x,y)\) is
\[ \mathbf{r}=\langle x,y\rangle. \]The field vector is
\[ \mathbf{F}=\langle -y,x\rangle. \]Their dot product is
\[ \mathbf{r}\cdot\mathbf{F} = x(-y)+y(x)=0. \]Therefore, the field vector is perpendicular to the radial direction at every nonzero point.
Vector Fields as Arrow Diagrams
A common visual representation places an arrow at selected points in the domain.
Each arrow communicates two things:
- its orientation represents direction;
- its length may represent magnitude.
Because infinitely many arrows cannot usually be drawn, the field is sampled at a manageable set of locations.
Arrow Lengths May Be Rescaled
In a field plot, very large arrows can obscure nearby structure and very small arrows can become invisible.
For visualization, arrow lengths are therefore sometimes normalized or rescaled.
The displayed arrows may emphasize direction rather than preserve exact physical magnitude.
A field diagram should always be interpreted according to its plotting convention.
Vector Fields from Data
Like scalar fields, vector fields do not have to begin with formulas.
A set of measurements may record a vector at each sampled location:
| Location | Vector value |
|---|---|
| \((0,0)\) | \(\langle 1,2\rangle\) |
| \((1,0)\) | \(\langle 2,2\rangle\) |
| \((0,1)\) | \(\langle 1,3\rangle\) |
| \((1,1)\) | \(\langle 2,3\rangle\) |
Such measurements can come from instruments, simulations, or tracked motion.
Wind as a Vector Field
Wind velocity is naturally vector-valued.
At each location, a wind vector gives:
- how fast the air is moving;
- the direction in which it is moving.
A wind field may therefore be written
\[ \mathbf{v}(x,y,z,t), \]where position and time determine the local velocity vector.
Fluid Velocity as a Vector Field
The velocity of a flowing fluid may vary throughout a pipe, river, ocean, or atmosphere.
A velocity field
\[ \mathbf{v}(x,y,z) \]describes the instantaneous velocity assigned to each point in the fluid.
Later lessons connect these vectors to flow lines and trajectories.
Force as a Vector Field
A force field assigns a force vector to every position.
Symbolically,
\[ \mathbf{F}(x,y,z) \]may describe the force that a test object would experience if placed at \((x,y,z)\).
Gravity and electrostatic interactions are important physical examples.
Acceleration as a Vector Field
An acceleration field can assign an acceleration vector to every point in space.
For example, near the surface of Earth, a simplified model may treat gravitational acceleration as approximately constant:
\[ \mathbf{g} = \langle 0,0,-g\rangle. \]The downward direction is built directly into the vector.
Position Can Generate a Vector Field
The position vector itself defines a simple field:
\[ \mathbf{r}(x,y,z)=\langle x,y,z\rangle. \]At every point, the assigned vector points from the origin to that point.
Many important fields are constructed from the position vector, its magnitude, or normalized versions of it.
Distance and Vector Magnitude
For the position field
\[ \mathbf{r}(x,y,z)=\langle x,y,z\rangle, \]the magnitude is
\[ \|\mathbf{r}\| = \sqrt{x^2+y^2+z^2}. \]This is exactly the scalar distance from the origin.
A scalar field can therefore arise naturally from the magnitude of a vector field.
Unit Vector Fields
Away from the origin, the radial position vector can be normalized:
\[ \widehat{\mathbf{r}} = \frac{\langle x,y,z\rangle} {\sqrt{x^2+y^2+z^2}}. \]This field assigns a vector of magnitude \(1\) pointing directly outward from the origin.
At the origin, the formula is undefined because the denominator is zero.
The Domain of a Vector Field
The domain is the set of spatial points where every component of the field is defined.
Consider
\[ \mathbf{F}(x,y) = \left\langle \frac{x}{x^2+y^2}, \frac{y}{x^2+y^2} \right\rangle. \]The denominator vanishes at \((0,0)\), so the origin must be excluded from the domain.
A Vector Field Can Be Smooth
A field such as
\[ \mathbf{F}(x,y)=\langle x^2-y,xy\rangle \]changes smoothly as the point moves through the plane.
Nearby input points produce nearby component values.
Smooth vector fields can be studied locally using calculus.
A Vector Field Can Have Discontinuities
Vector fields do not have to vary smoothly.
A simplified model could assign
\[ \mathbf{F}(x,y) = \begin{cases} \langle 1,0\rangle, & x<0,\\ \langle 0,1\rangle, & x\geq 0. \end{cases} \]The field direction changes abruptly across the line \(x=0\).
Magnitude Can Change without Direction Changing
Consider
\[ \mathbf{F}(x,y)=\langle x,0\rangle. \]Along the positive \(x\)-axis, every nonzero vector points horizontally to the right, but its magnitude is \(|x|\).
The direction may therefore remain fixed while the strength varies from point to point.
Direction Can Change without Magnitude Changing
Consider the unit radial field
\[ \mathbf{F}(x,y) = \frac{\langle x,y\rangle} {\sqrt{x^2+y^2}}. \]Wherever it is defined,
\[ \|\mathbf{F}(x,y)\|=1. \]The magnitude stays constant while the direction changes around the plane.
Magnitude and Direction Can Both Change
In many realistic fields, both features vary simultaneously.
For example,
\[ \mathbf{F}(x,y)=\langle x^2-y,xy\rangle \]changes in both magnitude and orientation as \((x,y)\) changes.
Understanding a vector field therefore requires attention to both parts of each vector.
Adding Vector Fields
Vector fields on the same domain can be added point by point.
If
\[ \mathbf{F}(x,y)=\langle x,y\rangle \]and
\[ \mathbf{G}(x,y)=\langle 1,-1\rangle, \]then
\[ (\mathbf{F}+\mathbf{G})(x,y) = \langle x+1,y-1\rangle. \]This can model the superposition of independent vector effects.
Subtracting Vector Fields
The difference between vector fields is also computed point by point.
\[ (\mathbf{F}-\mathbf{G})(x,y) = \mathbf{F}(x,y)-\mathbf{G}(x,y). \]If two velocity fields represent different times or scenarios, their difference can describe how the local velocity has changed throughout the domain.
Multiplying a Vector Field by a Scalar
If \(c\) is a scalar constant, then
\[ c\mathbf{F}(x,y) \]multiplies every component by \(c\).
Positive \(c\) rescales magnitude without reversing direction. Negative \(c\) also reverses the direction.
Multiplying by a Scalar Field
A scalar field can also scale a vector field point by point.
If \(f(x,y)\) is scalar and \(\mathbf{F}(x,y)\) is vector-valued, then
\[ f(x,y)\mathbf{F}(x,y) \]is a vector field.
The scalar factor can strengthen, weaken, or reverse the vector differently at different locations.
Time-Dependent Vector Fields
A vector field may evolve with time:
\[ \mathbf{F}(x,y,z,t). \]At each fixed time \(t\), this expression defines a vector field across space.
As time changes, the vector assigned to a particular location may change in magnitude, direction, or both.
Steady and Unsteady Fields
A field that does not depend explicitly on time is often called steady.
\[ \mathbf{F}(x,y,z). \]A field that depends on time is time-dependent or unsteady:
\[ \mathbf{F}(x,y,z,t). \]This distinction is especially important in flow and motion problems.
Field Vectors Are Attached to Locations
A field vector should be interpreted together with the point at which it is evaluated.
If
\[ \mathbf{F}(1,2)=\langle 3,4\rangle, \]then \((1,2)\) is the spatial location and \(\langle 3,4\rangle\) is the vector assigned there.
The same vector may occur at many different locations.
Equal Vectors Can Occur at Different Points
Consider the constant field
\[ \mathbf{F}(x,y)=\langle 2,-1\rangle. \]Then
\[ \mathbf{F}(0,0) = \mathbf{F}(5,7) = \mathbf{F}(-3,10) = \langle 2,-1\rangle. \]Different locations can share exactly the same field vector.
A Field Is Not a Single Vector
The symbol \(\mathbf{F}\) represents the entire rule assigning vectors throughout the domain.
By contrast,
\[ \mathbf{F}(2,3) \]is one particular vector from that field.
Confusing the entire field with one sampled vector obscures the spatial structure.
Vector Fields and Paths
Suppose an object moves along a path
\[ \mathbf{r}(t)=\langle x(t),y(t)\rangle. \]A vector field can be evaluated along that path:
\[ \mathbf{F}(\mathbf{r}(t)) = \mathbf{F}(x(t),y(t)). \]This tells us which field vector is present at the object's current location.
Field Direction and Actual Motion
A vector field does not automatically mean that an object must move along its vectors.
The interpretation depends on the model.
If the field represents fluid velocity, a small passive particle may move approximately with the local field. If the field represents force, an object's acceleration is related to the force but is not generally equal to its velocity.
Flow Lines
For a velocity-type field, curves can be drawn so that their tangent direction agrees with the local field direction.
These curves help visualize how the field guides motion through space.
The next lessons develop flow lines and trajectories more carefully rather than treating them as merely decorative arrows.
Local Field Behavior
Near a point, we may ask:
- Are nearby vectors becoming longer or shorter?
- Are they turning?
- Are they spreading apart?
- Are they converging?
- Does the field resemble rotation?
These questions lead toward more advanced field descriptors.
Sources and Sinks as Visual Patterns
A field whose arrows point outward from a region can resemble a source.
A field whose arrows point inward can resemble a sink.
These words describe useful geometric patterns. Later, divergence provides a quantitative way to analyze local spreading and convergence.
Rotational Patterns
A field may circulate around a point or axis.
The example
\[ \mathbf{F}(x,y)=\langle -y,x\rangle \]has this rotational appearance.
Later, curl will provide a mathematical measure connected to local rotational tendency.
Do Not Confuse Appearance with Measurement
A field may look as though it spreads outward, rotates, or converges, but visual inspection alone is not a complete quantitative analysis.
Arrow spacing, plotting scale, and sampling choices can affect what a diagram looks like.
Formal field operators provide more reliable measurements of these behaviors.
Vector Fields from Scalar Fields
Some vector fields arise naturally from how a scalar field changes across space.
For a scalar field \(f(x,y)\), one important associated vector field is the gradient,
\[ \nabla f. \]The gradient receives its own lesson because it connects scalar values, direction, and rate of change in a precise way.
Potential Fields
In some situations, a vector field can be obtained from a scalar potential.
Such relationships are central in mechanics, gravitation, and electromagnetism.
The later lesson on potential and conservative fields develops this connection systematically.
Flux Will Measure Passage through a Boundary
A vector field can pass across a curve or surface.
Informally, we may ask how much of the field points through that boundary rather than along it.
This idea leads to flux, which is treated later in this section.
Common Error: Treating a Vector Field as a Scalar Field
If
\[ \mathbf{F}(x,y)=\langle x,y\rangle, \]then the output is not the single number \(x+y\).
The output is the ordered vector
\[ \langle x,y\rangle. \]Components must remain distinct unless a separate operation combines them.
Common Error: Confusing Magnitude with the Vector
If
\[ \mathbf{F}(3,4)=\langle 3,4\rangle, \]then the field vector is
\[ \langle 3,4\rangle, \]while its magnitude is
\[ 5. \]A vector and its magnitude are different mathematical objects.
Common Error: Forgetting to Evaluate Every Component
Suppose
\[ \mathbf{F}(x,y)=\langle x+y,x-y\rangle. \]At \((4,1)\), both components must be evaluated:
\[ \mathbf{F}(4,1) = \langle 4+1,4-1\rangle = \langle 5,3\rangle. \]Evaluating only one component does not determine the vector.
Common Error: Ignoring the Domain
For
\[ \mathbf{F}(x,y) = \left\langle \frac{x}{x^2+y^2}, \frac{y}{x^2+y^2} \right\rangle, \]the origin is excluded because both components contain the same zero denominator there.
A vector field is defined only where all required component expressions are defined.
Common Error: Reading Arrow Lengths Too Literally
Some diagrams rescale arrows for visibility.
A long displayed arrow may represent a larger magnitude, but only if the plotting convention preserves relative lengths.
Always distinguish the mathematical field from the choices used to display it.
A Reliable Procedure for Evaluating a Vector Field
- Identify the spatial point.
- Check that the point lies in the field's domain.
- Substitute the coordinates into every component function.
- Simplify each component separately.
- Reassemble the components in the correct order.
- Compute magnitude or direction only if requested.
- Interpret the vector according to the quantity being modeled.
Example: Full Evaluation Workflow
Let
\[ \mathbf{F}(x,y,z) = \langle x+y,y-z,2z-x\rangle. \]Evaluate at
\[ (x,y,z)=(2,-1,3). \]Substitute into each component:
\[ x+y=2+(-1)=1, \] \[ y-z=-1-3=-4, \] \[ 2z-x=2(3)-2=4. \]Therefore,
\[ \boxed{ \mathbf{F}(2,-1,3)=\langle 1,-4,4\rangle }. \]Example: Magnitude and Unit Direction
Suppose
\[ \mathbf{F}(1,2)=\langle 6,8\rangle. \]The magnitude is
\[ \|\mathbf{F}(1,2)\| = \sqrt{6^2+8^2} = 10. \]The unit direction is
\[ \frac{\mathbf{F}}{\|\mathbf{F}\|} = \left\langle \frac{3}{5},\frac{4}{5}\right\rangle. \]Example: Identify a Radial Pattern
Consider
\[ \mathbf{F}(x,y)=\langle 3x,3y\rangle. \]This can be written
\[ \mathbf{F}(x,y)=3\langle x,y\rangle. \]Therefore, each vector is parallel to the position vector and points outward from the origin, except at the origin where the field is zero.
Example: Identify a Rotational Pattern
Let
\[ \mathbf{F}(x,y)=\langle -2y,2x\rangle. \]This is
\[ 2\langle -y,x\rangle. \]The scalar factor \(2\) changes the magnitude but not the basic rotational pattern.
Practice 1: Evaluate a Planar Vector Field
Let
\[ \mathbf{F}(x,y)=\langle 2x+y,x-3y\rangle. \]Find \(\mathbf{F}(2,-1)\).
Practice 2: Find a Magnitude
Suppose
\[ \mathbf{F}(1,4)=\langle 5,12\rangle. \]Find \(\|\mathbf{F}(1,4)\|\).
Practice 3: Constant or Variable?
Determine whether each field is constant or position-dependent:
- \(\mathbf{F}(x,y)=\langle 4,-2\rangle\);
- \(\mathbf{G}(x,y)=\langle x,y\rangle\).
Practice 4: Radial or Rotational?
Classify the basic pattern of
\[ \mathbf{F}(x,y)=\langle -y,x\rangle. \]Practice 5: Find a Zero of the Field
Find all points where
\[ \mathbf{F}(x,y)=\langle x-2,y+3\rangle \]equals the zero vector.
Practice 6: Determine the Domain
Determine the domain of
\[ \mathbf{F}(x,y) = \left\langle \frac{1}{x-y}, \sqrt{y} \right\rangle. \]Practice 7: Add Two Vector Fields
Let
\[ \mathbf{F}(x,y)=\langle x,2y\rangle \]and
\[ \mathbf{G}(x,y)=\langle 3,-y\rangle. \]Find \((\mathbf{F}+\mathbf{G})(x,y)\).
Practice 8: Scalar Multiplication
Let
\[ \mathbf{F}(x,y)=\langle x-y,x+y\rangle. \]Find \(3\mathbf{F}(x,y)\).
Practice 9: Interpret a Physical Field
Suppose
\[ \mathbf{v}(x,y,z,t) \]represents wind velocity. Explain what its inputs and output represent.
Practice 10: Distinguish Scalar and Vector Fields
Classify each field as scalar-valued or vector-valued:
- temperature throughout a room;
- velocity throughout a river;
- air pressure throughout the atmosphere;
- force throughout a region of space.
Practice Answers
Practice 1
\[ \mathbf{F}(2,-1) = \langle 2(2)+(-1),2-3(-1)\rangle. \]Therefore,
\[ \boxed{\mathbf{F}(2,-1)=\langle 3,5\rangle}. \]Practice 2
\[ \|\mathbf{F}(1,4)\| = \sqrt{5^2+12^2} = \sqrt{169} = \boxed{13}. \]Practice 3
- \(\mathbf{F}(x,y)=\langle 4,-2\rangle\) is constant.
- \(\mathbf{G}(x,y)=\langle x,y\rangle\) is position-dependent.
Practice 4
The field
\[ \boxed{\mathbf{F}(x,y)=\langle -y,x\rangle} \]has a rotational pattern around the origin.
Practice 5
Set both components equal to zero:
\[ x-2=0, \qquad y+3=0. \]Therefore,
\[ \boxed{(x,y)=(2,-3)}. \]Practice 6
The first component requires
\[ x-y\neq 0, \]so \(x\neq y\). The second component requires
\[ y\geq 0. \]Therefore, the domain is
\[ \boxed{\{(x,y)\in\mathbb{R}^2:y\geq0,\ x\neq y\}}. \]Practice 7
\[ (\mathbf{F}+\mathbf{G})(x,y) = \langle x+3,2y-y\rangle. \]Therefore,
\[ \boxed{(\mathbf{F}+\mathbf{G})(x,y)=\langle x+3,y\rangle}. \]Practice 8
\[ 3\mathbf{F}(x,y) = 3\langle x-y,x+y\rangle. \]Therefore,
\[ \boxed{ 3\mathbf{F}(x,y) = \langle 3x-3y,3x+3y\rangle }. \]Practice 9
The variables \(x,y,z\) identify a location in three-dimensional space, \(t\) identifies time, and \(\mathbf{v}(x,y,z,t)\) gives the wind velocity vector at that location and time.
Practice 10
- temperature: scalar field;
- velocity: vector field;
- pressure: scalar field;
- force: vector field.
What to Remember
A vector field assigns a vector to every point in a spatial domain:
\[ \boxed{ \mathbf{F}(x,y) = \langle P(x,y),Q(x,y)\rangle }. \]- the input identifies a location;
- the output has magnitude and direction;
- each component is itself a scalar field;
- vector fields may be constant or position-dependent;
- they can form radial, inward, rotational, and more complicated patterns;
- they model velocity, force, acceleration, flow, and related spatial quantities;
- field diagrams use arrows to represent sampled vectors;
- magnitude, direction, and domain must be interpreted separately.
The central habit is to keep three things distinct: the location, the vector assigned there, and the magnitude of that vector.
Where We Go Next
Scalar fields assign numerical values to locations, while vector fields assign directional quantities.
The next lesson, Field Values and Direction, focuses on reading, comparing, and interpreting these assignments geometrically across space.