The Central Idea
Earlier mathematics often associates one output with one input. A field extends that idea across space.
At every point, we ask:
What quantity or directed quantity is present here?
A scalar field answers with one number:
\[ f(x,y,z). \]A vector field answers with a vector:
\[ \mathbf{F}(x,y,z). \]The distinction between these two kinds of output is the foundation of vector calculus.
What Is a Field?
A field is a rule that assigns a mathematical object to every point in some region of space.
In two dimensions, the input point is commonly written
\[ (x,y). \]In three dimensions, it is commonly written
\[ (x,y,z). \]The assigned object might be a number, a vector, a matrix, or something more advanced. In this lesson, we focus on scalar and vector fields.
Scalar Fields
A scalar field assigns one scalar value to each point in a region.
In two dimensions:
\[ f:\mathbb{R}^2\to\mathbb{R}. \]In three dimensions:
\[ f:\mathbb{R}^3\to\mathbb{R}. \]The output may represent temperature, pressure, density, concentration, altitude, electric potential, or any other quantity described by one numerical value at each point.
Example: A Temperature Field
Suppose the temperature on a flat metal plate is modeled by
\[ T(x,y)=100-x^2-2y^2. \]At the point \((2,3)\),
\[ T(2,3)=100-2^2-2(3^2). \]Therefore,
\[ T(2,3)=100-4-18=78. \]The field assigns the scalar value \(78\) to that location.
One Field, Many Points
A field is not just one measurement. It describes how a quantity is distributed throughout an entire region.
For
\[ T(x,y)=100-x^2-2y^2, \]we have
\[ T(0,0)=100, \] \[ T(1,0)=99, \] \[ T(0,1)=98, \]and
\[ T(2,3)=78. \]The formula gives a systematic description of the temperature at every point where the model applies.
Scalar Fields as Surfaces
A two-variable scalar field
\[ z=f(x,y) \]can often be visualized as a surface in three-dimensional space.
The point \((x,y)\) determines the location in the horizontal plane, while the scalar value \(f(x,y)\) determines the height.
For example,
\[ f(x,y)=x^2+y^2 \]produces an upward-opening paraboloid.
Scalar Fields as Maps
A scalar field does not have to be viewed as a three-dimensional surface. It can also be displayed over its original spatial region.
Common representations include:
- color maps;
- heat maps;
- contour maps;
- level curves;
- level surfaces.
A weather map showing temperature across a state is a familiar example of a scalar-field visualization.
Level Curves
A level curve of a scalar field \(f(x,y)\) is a curve on which the field has a constant value.
We write
\[ f(x,y)=c, \]where \(c\) is constant.
For
\[ f(x,y)=x^2+y^2, \]the level curve \(f(x,y)=9\) is
\[ x^2+y^2=9, \]a circle of radius \(3\).
Level Surfaces
For a scalar field of three variables,
\[ f(x,y,z), \]the equation
\[ f(x,y,z)=c \]generally describes a level surface.
For example, if
\[ f(x,y,z)=x^2+y^2+z^2, \]then
\[ f(x,y,z)=25 \]gives
\[ x^2+y^2+z^2=25, \]a sphere of radius \(5\).
Scalar Fields in Physical Models
Many measurable physical quantities are naturally modeled as scalar fields.
- Temperature: \(T(x,y,z)\)
- Pressure: \(P(x,y,z)\)
- Mass density: \(\rho(x,y,z)\)
- Electric potential: \(V(x,y,z)\)
- Concentration: \(C(x,y,z)\)
- Gravitational potential: \(\Phi(x,y,z)\)
The field value changes as position changes.
The Domain of a Scalar Field
The domain is the set of points where the field is defined.
Consider
\[ f(x,y)=\frac{1}{x^2+y^2}. \]The denominator cannot equal zero. Therefore,
\[ x^2+y^2\ne0. \]The only excluded point is
\[ (0,0). \]So the field is defined everywhere in the plane except the origin.
Vector Fields
A vector field assigns a vector to every point in a region.
In two dimensions, we often write
\[ \mathbf{F}(x,y) = \langle P(x,y),Q(x,y)\rangle. \]In three dimensions,
\[ \mathbf{F}(x,y,z) = \langle P(x,y,z),Q(x,y,z),R(x,y,z)\rangle. \]Each component is itself a scalar field.
Evaluating a Vector Field
Consider
\[ \mathbf{F}(x,y)=\langle 2x-y,\;x+3y\rangle. \]At the point \((2,-1)\),
\[ \mathbf{F}(2,-1) = \langle 2(2)-(-1),\;2+3(-1)\rangle. \]Therefore,
\[ \boxed{\mathbf{F}(2,-1)=\langle5,-1\rangle}. \]The vector field assigns the vector \(\langle5,-1\rangle\) to the point \((2,-1)\).
Magnitude and Direction
Because a vector field assigns vectors, every field value has both magnitude and direction.
If
\[ \mathbf{F}(x,y)=\langle P(x,y),Q(x,y)\rangle, \]then its magnitude is
\[ \|\mathbf{F}(x,y)\| = \sqrt{P(x,y)^2+Q(x,y)^2}. \]The components determine the direction of the vector at that point.
Example: Magnitude of a Field Vector
From the previous example,
\[ \mathbf{F}(2,-1)=\langle5,-1\rangle. \]Its magnitude is
\[ \|\mathbf{F}(2,-1)\| = \sqrt{5^2+(-1)^2} = \sqrt{26}. \]The field therefore has magnitude \(\sqrt{26}\) at the point \((2,-1)\).
How Vector Fields Are Drawn
A vector field is commonly visualized by drawing arrows at selected points.
At each point:
- the arrow points in the direction of the field vector;
- the arrow length represents the vector magnitude;
- the collection of arrows reveals the overall flow pattern.
In practical diagrams, arrow lengths are sometimes rescaled so that very large vectors do not overwhelm the picture.
A Constant Vector Field
Consider
\[ \mathbf{F}(x,y)=\langle3,1\rangle. \]The vector does not depend on \(x\) or \(y\).
Therefore, every point receives the same vector.
A plot of this field consists of parallel arrows with the same magnitude and direction everywhere.
A Radial Vector Field
Consider
\[ \mathbf{F}(x,y)=\langle x,y\rangle. \]At the point \((x,y)\), the field vector points in the same direction as the position vector from the origin.
Its magnitude is
\[ \|\mathbf{F}(x,y)\| = \sqrt{x^2+y^2}. \]The farther a point is from the origin, the larger the vector becomes.
An Inward Radial Field
If we reverse the previous field,
\[ \mathbf{F}(x,y)=\langle -x,-y\rangle, \]then every vector points toward the origin.
The magnitude is still
\[ \sqrt{x^2+y^2}, \]but the direction is reversed.
This simple comparison shows that magnitude alone does not determine a vector field.
A Rotational Vector Field
Consider
\[ \mathbf{F}(x,y)=\langle -y,x\rangle. \]At \((1,0)\),
\[ \mathbf{F}(1,0)=\langle0,1\rangle. \]At \((0,1)\),
\[ \mathbf{F}(0,1)=\langle-1,0\rangle. \]At \((-1,0)\),
\[ \mathbf{F}(-1,0)=\langle0,-1\rangle. \]The vectors circulate counterclockwise around the origin.
Vector Fields and Motion
A velocity field assigns a velocity vector to each point in a moving fluid.
For example,
\[ \mathbf{v}(x,y)=\langle -y,x\rangle \]can model a rotating flow.
A small particle placed in the fluid tends to move in the direction of the local velocity vector.
Vector Fields as Local Instructions
One useful way to interpret a vector field is as a collection of local instructions.
At every point, the vector says:
If you are here, this is the locally assigned direction and strength.
This interpretation is central to differential equations, fluid mechanics, dynamical systems, electromagnetism, and many optimization models.
Component Functions
A vector field can be analyzed component by component.
If
\[ \mathbf{F}(x,y,z) = \langle P(x,y,z),Q(x,y,z),R(x,y,z)\rangle, \]then \(P\), \(Q\), and \(R\) are scalar fields.
The vector field packages these component fields into one directed object.
Unit-Vector Notation
Instead of angle-bracket notation, a three-dimensional vector field may be written with the standard basis vectors
\[ \mathbf{i},\qquad \mathbf{j},\qquad \mathbf{k}. \]Thus,
\[ \mathbf{F} = \langle P,Q,R\rangle \]can also be written
\[ \boxed{\mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}}. \]Position-Vector Notation
The position vector in three-dimensional Cartesian coordinates is
\[ \mathbf{r} = \langle x,y,z\rangle. \]A radial field may therefore be written
\[ \mathbf{F}(\mathbf{r})=\mathbf{r}. \]More complicated radial fields may depend on the distance
\[ r=\|\mathbf{r}\|. \]This notation becomes especially useful in physical field models.
Example: A Three-Dimensional Vector Field
Let
\[ \mathbf{F}(x,y,z) = \langle yz,\;xz,\;xy\rangle. \]At the point \((1,2,3)\),
\[ \mathbf{F}(1,2,3) = \langle 2\cdot3,\;1\cdot3,\;1\cdot2\rangle. \]Therefore,
\[ \boxed{\mathbf{F}(1,2,3)=\langle6,3,2\rangle}. \]The Domain of a Vector Field
A vector field is defined only where all of its components are defined.
Consider
\[ \mathbf{F}(x,y) = \left\langle \frac{x}{x^2+y^2}, \frac{y}{x^2+y^2} \right\rangle. \]Both components fail when
\[ x^2+y^2=0. \]Therefore, the origin is excluded from the domain.
Scalar Multiplication of a Vector Field
A scalar field can multiply a vector field.
If
\[ f(x,y) \]is a scalar field and
\[ \mathbf{F}(x,y) \]is a vector field, then
\[ f\mathbf{F} \]is another vector field.
The scalar changes the magnitude, and a negative scalar can also reverse direction.
Example: Scaling a Vector Field
Let
\[ f(x,y)=x^2+y^2 \]and
\[ \mathbf{F}(x,y)=\langle -y,x\rangle. \]Then
\[ f\mathbf{F} = (x^2+y^2)\langle -y,x\rangle. \]Equivalently,
\[ f\mathbf{F} = \langle -(x^2+y^2)y,\;(x^2+y^2)x\rangle. \]Adding Vector Fields
Vector fields may be added pointwise.
If
\[ \mathbf{F}=\langle P,Q\rangle \]and
\[ \mathbf{G}=\langle R,S\rangle, \]then
\[ \boxed{\mathbf{F}+\mathbf{G}=\langle P+R,Q+S\rangle}. \]At each point, the usual rules of vector addition apply.
Vector Fields Can Depend on Time
A field may vary with both position and time.
A time-dependent scalar field may be written
\[ f(x,y,z,t), \]and a time-dependent vector field may be written
\[ \mathbf{F}(x,y,z,t). \]Weather, fluid flow, electromagnetic fields, and many engineering systems are naturally modeled this way.
Steady and Unsteady Fields
A field that does not explicitly depend on time is often called steady or time-independent.
For example,
\[ \mathbf{v}(x,y)=\langle-y,x\rangle \]is steady.
A field such as
\[ \mathbf{v}(x,y,t) = \langle -ty,tx\rangle \]changes with time and is therefore unsteady.
Uniform and Nonuniform Fields
A constant field such as
\[ \mathbf{F}(x,y)=\langle2,-1\rangle \]is spatially uniform.
A field such as
\[ \mathbf{F}(x,y)=\langle x,y\rangle \]is nonuniform because its value changes from one point to another.
Scalar Versus Vector: Temperature and Wind
A weather map provides a useful distinction.
Temperature is modeled by a scalar field:
\[ T(x,y). \]Wind velocity is modeled by a vector field:
\[ \mathbf{v}(x,y) = \langle u(x,y),v(x,y)\rangle. \]Temperature tells us how much. Wind velocity tells us both how much and which way.
Scalar Versus Vector: Pressure and Fluid Velocity
Inside a fluid, pressure at a point is a scalar:
\[ P(x,y,z). \]Fluid velocity at a point is a vector:
\[ \mathbf{v}(x,y,z). \]These fields can interact. Spatial changes in pressure can contribute to forces that influence the velocity field.
Scalar Versus Vector: Potential and Force
Many physical theories use both scalar and vector fields.
A potential may be represented by a scalar field
\[ V(x,y,z), \]while an associated force field may be represented by
\[ \mathbf{F}(x,y,z). \]The next lesson on gradient fields will show one of the most important mathematical connections between such scalar and vector fields.
Fields and Coordinate Systems
A field is a geometric object, but its formula depends on how we describe position.
Cartesian coordinates use
\[ (x,y,z). \]Cylindrical coordinates use
\[ (r,\theta,z). \]Spherical coordinates use
\[ (\rho,\theta,\phi). \]Later in this section, coordinate systems will be treated systematically because the best coordinates often follow the symmetry of the field.
Why Symmetry Matters
Fields often have geometric symmetry.
For example, a field generated by an idealized isolated point source may depend only on distance from that source.
Such a field has radial symmetry, so a radial coordinate may describe it more naturally than separate \(x\), \(y\), and \(z\) coordinates.
Recognizing symmetry can simplify both formulas and calculations.
Field Lines
A vector field can also be visualized using field lines.
A field line is a curve whose tangent direction agrees with the vector field at each point along the curve.
Informally, if we move along a field line, we are always moving in the direction locally indicated by the field.
Field lines are especially common in fluid flow and electromagnetic diagrams.
Field Lines Are Not the Field Itself
A field-line diagram is a representation of a vector field, not the field itself.
The actual field assigns a vector to every point in its domain.
A field-line drawing selects only enough curves to make the overall structure visible.
Zeros of a Vector Field
A point where
\[ \mathbf{F}(x,y,z)=\mathbf{0} \]is a zero of the vector field.
For
\[ \mathbf{F}(x,y)=\langle x,y\rangle, \]the only zero is
\[ (0,0). \]At that point, the assigned vector has zero magnitude and no unique direction.
Zeros Can Be Structurally Important
In applications, zeros of vector fields may represent:
- equilibrium points;
- stagnation points in fluid flow;
- locations where force vanishes;
- critical points in dynamical systems.
Later analysis studies how nearby field vectors behave around such points.
Singularities
A singularity is a location where a field fails to be defined or behaves irregularly.
Consider the radial vector field
\[ \mathbf{F}(x,y) = \frac{\langle x,y\rangle}{x^2+y^2}. \]The origin is excluded because the denominator becomes zero there.
The field may become very large as a point approaches the singularity.
Distance-Dependent Radial Fields
Let
\[ r=\sqrt{x^2+y^2+z^2}. \]A three-dimensional radial field can often be written in the form
\[ \mathbf{F} = g(r)\,\hat{\mathbf{r}}, \]where \(g(r)\) controls magnitude and
\[ \hat{\mathbf{r}} = \frac{\mathbf{r}}{\|\mathbf{r}\|} \]is the outward radial unit vector.
Inverse-Square Pattern
A common radial pattern has magnitude proportional to
\[ \frac{1}{r^2}. \]An abstract inverse-square vector field may therefore be written
\[ \mathbf{F} = \frac{k}{r^2}\hat{\mathbf{r}}, \]where \(k\) is a constant.
This kind of spatial dependence appears in idealized gravitational and electrostatic models.
Units Matter
A field formula should be interpreted together with its units.
A temperature field may have units of degrees Celsius.
A velocity field may have units of meters per second.
A force field may have units of newtons.
The input coordinates also carry units, commonly meters or another unit of length.
Local Value Versus Global Pattern
Evaluating a field at one point gives local information.
Understanding the full field requires examining how those values vary over a region.
Vector calculus develops tools that convert local field behavior into broader conclusions about flow, circulation, flux, sources, sinks, and accumulated effects.
Fields Prepare Us for Differential Operators
Several important operations in vector calculus act on fields.
- The gradient converts a scalar field into a vector field.
- The divergence converts a vector field into a scalar field.
- The curl converts a three-dimensional vector field into another vector field.
These operations will be developed in the next lessons.
A Useful Type Checklist
Before calculating, identify the mathematical type of each object.
- \(f\): scalar field;
- \(\mathbf{F}\): vector field;
- \(f(x,y,z)\): scalar value;
- \(\mathbf{F}(x,y,z)\): vector value;
- \(\|\mathbf{F}(x,y,z)\|\): scalar magnitude.
Keeping these types straight prevents many later errors.
Common Error: Confusing a Scalar Field with Its Graph
The scalar field
\[ f(x,y) \]is the rule assigning a scalar to each point.
The surface
\[ z=f(x,y) \]is one way to visualize that rule.
The field and the graph are related, but they are not the same mathematical object.
Common Error: Treating a Vector Field as One Vector
A vector field is not a single vector.
For
\[ \mathbf{F}(x,y)=\langle x,-y\rangle, \]the assigned vector changes as the point changes.
For example,
\[ \mathbf{F}(1,2)=\langle1,-2\rangle, \]while
\[ \mathbf{F}(3,-1)=\langle3,1\rangle. \]Common Error: Ignoring the Domain
A field formula may not be valid everywhere.
If a denominator, logarithm, square root, or other operation imposes a restriction, that restriction is part of the field definition.
Domain analysis becomes especially important around singularities.
Common Error: Confusing Field Magnitude with a Component
If
\[ \mathbf{F}=\langle P,Q\rangle, \]then neither \(P\) nor \(Q\) alone is the magnitude.
The magnitude is
\[ \boxed{\|\mathbf{F}\|=\sqrt{P^2+Q^2}}. \]Common Error: Forgetting That Components Are Functions
In
\[ \mathbf{F}(x,y) = \langle P(x,y),Q(x,y)\rangle, \]the components generally change from point to point.
Evaluating the vector field means evaluating every component at the same input point.
A Reliable Procedure for Evaluating a Scalar Field
- Identify the input point.
- Check that the point lies in the field's domain.
- Substitute the coordinates into the field formula.
- Evaluate carefully.
- Interpret the resulting scalar with its units when applicable.
A Reliable Procedure for Evaluating a Vector Field
- Identify all component functions.
- Check that the point lies in the common domain of the components.
- Substitute the coordinates into every component.
- Simplify each component separately.
- Assemble the resulting components into a vector.
- Compute magnitude only if the problem asks for it.
Example: Full Scalar-Field Workflow
Let
\[ C(x,y,z)=\frac{50}{1+x^2+y^2+z^2}. \]Find the value at \((1,2,2)\).
Substitute:
\[ C(1,2,2) = \frac{50}{1+1^2+2^2+2^2}. \]Therefore,
\[ C(1,2,2) = \frac{50}{10} = \boxed{5}. \]Example: Full Vector-Field Workflow
Let
\[ \mathbf{F}(x,y,z) = \langle x+y,\;yz,\;z-x\rangle. \]Evaluate at \((2,-1,3)\).
\[ \mathbf{F}(2,-1,3) = \langle 2+(-1),\;(-1)(3),\;3-2\rangle. \]Hence,
\[ \boxed{\mathbf{F}(2,-1,3)=\langle1,-3,1\rangle}. \]Its magnitude is
\[ \sqrt{1^2+(-3)^2+1^2} = \boxed{\sqrt{11}}. \]Practice 1: Scalar Field Evaluation
Let
\[ f(x,y)=3x^2-2xy+y^2. \]Find \(f(2,-1)\).
Practice 2: Level Curve
For
\[ f(x,y)=x^2+4y^2, \]write the equation of the level curve corresponding to \(f=16\).
Practice 3: Vector Field Evaluation
Let
\[ \mathbf{F}(x,y) = \langle x-y,\;2x+y\rangle. \]Find \(\mathbf{F}(3,1)\).
Practice 4: Magnitude
Find the magnitude of the vector from Practice 3.
Practice 5: Field Type
Classify each quantity as a scalar field or vector field:
- temperature throughout a room;
- wind velocity over a region;
- air pressure throughout the atmosphere;
- gravitational force at points around a mass.
Practice 6: Domain
Determine the domain of
\[ \mathbf{F}(x,y) = \left\langle \frac{1}{x-y}, \frac{x}{x-y} \right\rangle. \]Practice 7: Radial Behavior
Describe the direction of the field
\[ \mathbf{F}(x,y)=\langle -x,-y\rangle. \]Is it inward radial, outward radial, rotational, or constant?
Practice 8: Rotational Behavior
Evaluate
\[ \mathbf{F}(x,y)=\langle-y,x\rangle \]at \((0,-2)\), and describe the direction of the resulting vector.
Practice 9: Three-Dimensional Field
Let
\[ \mathbf{F}(x,y,z)=\langle xy,\;yz,\;zx\rangle. \]Evaluate at \((2,3,-1)\).
Practice 10: Scalar or Vector Output
Let
\[ \mathbf{F}(x,y)=\langle 3x,4y\rangle. \]Determine whether each expression is scalar-valued or vector-valued:
- \(\mathbf{F}(x,y)\)
- \(\|\mathbf{F}(x,y)\|\)
- \(\mathbf{F}(1,2)\)
- \(\|\mathbf{F}(1,2)\|\)
Practice Answers
Practice 1
\[ f(2,-1) = 3(2^2)-2(2)(-1)+(-1)^2 = 12+4+1 = \boxed{17}. \]Practice 2
\[ \boxed{x^2+4y^2=16}. \]This is an ellipse.
Practice 3
\[ \mathbf{F}(3,1) = \langle3-1,\;2(3)+1\rangle = \boxed{\langle2,7\rangle}. \]Practice 4
\[ \|\mathbf{F}(3,1)\| = \sqrt{2^2+7^2} = \boxed{\sqrt{53}}. \]Practice 5
- temperature: scalar field;
- wind velocity: vector field;
- air pressure: scalar field;
- gravitational force: vector field.
Practice 6
The denominator requires
\[ x-y\ne0. \]Therefore, the field is defined at all points except those on the line
\[ \boxed{x=y}. \]Practice 7
The vector
\[ \langle-x,-y\rangle \]points opposite the position vector, so the field is inward radial.
Practice 8
\[ \mathbf{F}(0,-2) = \langle2,0\rangle. \]The vector points directly to the right.
Practice 9
\[ \mathbf{F}(2,3,-1) = \langle(2)(3),\;(3)(-1),\;(-1)(2)\rangle = \boxed{\langle6,-3,-2\rangle}. \]Practice 10
- \(\mathbf{F}(x,y)\): vector-valued;
- \(\|\mathbf{F}(x,y)\|\): scalar-valued;
- \(\mathbf{F}(1,2)\): vector-valued;
- \(\|\mathbf{F}(1,2)\|\): scalar-valued.
What to Remember
A scalar field assigns one number to each point:
\[ \boxed{f:\mathbb{R}^n\to\mathbb{R}}. \]A vector field assigns a vector to each point:
\[ \boxed{\mathbf{F}:\mathbb{R}^n\to\mathbb{R}^m}. \]In the most common two- and three-dimensional vector-calculus settings, the field dimension matches the spatial dimension.
- Scalar fields model quantities such as temperature, pressure, and potential.
- Vector fields model quantities such as velocity and force.
- A field must always be interpreted together with its domain.
- A vector field has component functions, magnitude, and direction.
- Field plots reveal global patterns from local assignments.
- Radial, rotational, constant, and singular behaviors are basic patterns worth recognizing.
Where We Go Next
Scalar and vector fields give us the objects that vector calculus studies. The next step is to connect them.
The next lesson, Gradient Fields, develops the gradient operator, explains how a scalar field produces a vector field of steepest increase, and connects level sets, directional change, potentials, and conservative behavior.