The Central Idea
A scalar field is a rule that assigns one scalar value to each point in a spatial domain.
In two dimensions, we may write
\[ \boxed{f(x,y)=\text{one number at the point }(x,y)}. \]In three dimensions,
\[ \boxed{f(x,y,z)=\text{one number at the point }(x,y,z)}. \]The point tells us where we are. The scalar tells us what numerical quantity is present there.
What Does Scalar Mean?
A scalar is a quantity described by magnitude alone. It does not require a direction.
Examples include:
- temperature;
- mass density;
- pressure;
- altitude;
- concentration;
- electric potential;
- gravitational potential.
A temperature of \(22^\circ\text{C}\) is a scalar value. It tells us how warm a location is, but it does not point north, south, upward, or in any other direction.
A Field Is a Quantity Spread across Space
Earlier functions often assigned outputs to numerical inputs. A scalar field uses the same function idea, but its inputs represent positions in space.
For example,
\[ T(x,y)=20+2x-y \]may represent temperature across a flat region.
Every point \((x,y)\) receives exactly one temperature value.
Location and Value Play Different Roles
In the field
\[ T(x,y)=20+2x-y, \]the ordered pair \((x,y)\) identifies a location, while \(T(x,y)\) is the scalar value assigned to that location.
At
\[ (x,y)=(3,4), \]the field value is
\[ T(3,4)=20+2(3)-4=22. \]So the location \((3,4)\) carries the scalar value \(22\).
Scalar Fields in One, Two, and Three Dimensions
The basic idea does not depend on the number of spatial coordinates.
Along a line:
\[ f(x). \]Across a plane:
\[ f(x,y). \]Through three-dimensional space:
\[ f(x,y,z). \]In each case, the output is still one scalar value.
A One-Dimensional Example
Suppose the temperature along a thin metal rod is modeled by
\[ T(x)=30-2x, \]where \(x\) is measured in centimeters and \(T\) in degrees Celsius.
At \(x=4\),
\[ T(4)=30-2(4)=22. \]The field assigns the temperature \(22^\circ\text{C}\) to the point located \(4\) centimeters from the chosen origin.
A Two-Dimensional Example: Elevation
A landscape may be represented by an elevation field
\[ h(x,y)=100+4x-3y. \]Here, \((x,y)\) gives horizontal position and \(h(x,y)\) gives elevation.
At \((2,5)\),
\[ h(2,5)=100+4(2)-3(5)=93. \]The point \((2,5)\) therefore has elevation \(93\) in the chosen height units.
A Three-Dimensional Example: Density
Inside a three-dimensional object, density may vary from point to point. One simple model is
\[ \rho(x,y,z)=5+x^2+y^2+z^2. \]At \((1,2,2)\),
\[ \rho(1,2,2)=5+1^2+2^2+2^2=14. \]The field assigns the density value \(14\) to that location.
The Domain of a Scalar Field
The domain is the set of spatial points where the field is defined.
A temperature field for a room may be defined only inside the room. An elevation field may be defined only over a surveyed region. A density field for a solid object may be defined only inside the material.
Symbolically, a two-dimensional scalar field may be written
\[ f:D\subseteq\mathbb{R}^2\to\mathbb{R}. \]The set \(D\) is the spatial domain, and the output is a real number.
The Codomain Is Scalar
The defining feature of a scalar field is that its output is one number.
\[ f:\mathbb{R}^n\to\mathbb{R}. \]The input may need several coordinates, but the output remains one-dimensional.
This is the key distinction between a scalar field and the vector fields introduced in the next lesson.
Scalar Fields Need Not Fill All of Space
A field can be restricted to any meaningful spatial region.
For example,
\[ f(x,y)=x+y \]might be studied only on the circular disk
\[ x^2+y^2\leq 25. \]The formula tells us how to compute values. The domain tells us where the model applies.
Constant Scalar Fields
The simplest scalar field has the same value everywhere.
\[ f(x,y)=7. \]No matter which point is chosen,
\[ f(x,y)=7. \]A perfectly uniform temperature, pressure, density, or concentration would be represented by a constant scalar field.
Linear Scalar Fields
A field such as
\[ f(x,y)=2x-3y+5 \]changes at a constant rate with respect to each coordinate.
Increasing \(x\) by \(1\) while keeping \(y\) fixed increases the field value by \(2\).
Increasing \(y\) by \(1\) while keeping \(x\) fixed decreases the field value by \(3\).
This gives us an early glimpse of how fields encode spatial change.
Quadratic Scalar Fields
Scalar fields may curve rather than change uniformly.
Consider
\[ f(x,y)=x^2+y^2. \]At the origin,
\[ f(0,0)=0. \]At \((3,4)\),
\[ f(3,4)=3^2+4^2=25. \]The value grows as the point moves farther from the origin.
A Radially Symmetric Field
The field
\[ f(x,y)=x^2+y^2 \]depends only on distance from the origin because
\[ x^2+y^2=r^2. \]Every point at the same distance from the origin has the same field value.
Such a field is called radially symmetric about the origin.
Distance as a Scalar Field
Distance from a fixed point can itself be treated as a scalar field.
Distance from the origin in the plane is
\[ d(x,y)=\sqrt{x^2+y^2}. \]At \((3,4)\),
\[ d(3,4)=5. \]Thus every point in the plane is assigned one scalar: its distance from the origin.
Distance from an Arbitrary Point
The distance from a fixed point \((a,b)\) is
\[ d(x,y)=\sqrt{(x-a)^2+(y-b)^2}. \]This field is smallest at \((a,b)\), where its value is \(0\).
Points equally far from \((a,b)\) receive the same value.
Field Values Can Represent Physical Measurements
A scalar field often models a measurable physical quantity.
Examples include:
- \(T(x,y,z)\): temperature;
- \(P(x,y,z)\): pressure;
- \(\rho(x,y,z)\): density;
- \(C(x,y,z)\): concentration;
- \(V(x,y,z)\): potential.
The mathematics of scalar fields is shared across many different applications even when the physical interpretation changes.
Units Matter
Coordinates and field values may use different units.
For example, if \(x\) and \(y\) are measured in kilometers and \(h(x,y)\) is elevation in meters, then
\[ h(2,3)=450 \]means that the location \((2,3)\) kilometers in the coordinate system has elevation \(450\) meters.
A field formula should therefore be interpreted together with its units and domain.
Values Can Be Positive, Negative, or Zero
A scalar field is not restricted to positive values.
For
\[ f(x,y)=x-y, \]we have
\[ f(5,2)=3, \] \[ f(2,5)=-3, \]and
\[ f(4,4)=0. \]The meaning of the sign depends on what the field represents.
Field Values Can Be Compared across Space
Suppose
\[ T(x,y)=18+x+2y. \]Then
\[ T(1,1)=21 \]while
\[ T(3,2)=25. \]The second location has the larger field value.
Comparing values at different points is the simplest way to study spatial variation.
Change along a Chosen Direction
Even before introducing formal calculus, we can compare nearby points to see how a field changes.
For
\[ f(x,y)=x^2+y, \]compare \((2,1)\) and \((3,1)\):
\[ f(2,1)=5, \] \[ f(3,1)=10. \]Moving one unit in the positive \(x\)-direction increased the field value by \(5\) over this step.
Later lessons formalize this idea using gradients and directional change.
Tables as Field Representations
A scalar field does not need to begin as a formula. It may be measured at a collection of locations.
| \((x,y)\) | Field value |
|---|---|
| \((0,0)\) | \(12\) |
| \((1,0)\) | \(14\) |
| \((0,1)\) | \(11\) |
| \((1,1)\) | \(13\) |
Such data may come from sensors, surveys, simulations, or experiments.
Grids of Measurements
In practice, fields are often sampled on a regular grid.
A weather station network, digital terrain map, medical scan, or computer simulation may store field values only at selected points rather than at every possible location.
The continuous field is then approximated from discrete samples.
Interpolation between Samples
If field values are known only at selected locations, intermediate values may be estimated by interpolation.
For example, if temperature is measured as \(20^\circ\text{C}\) at one point and \(24^\circ\text{C}\) at a nearby point, a simple model might estimate values between them.
The quality of such an estimate depends on the field, the spacing of the measurements, and the interpolation method.
Visualizing a Scalar Field as Height
A two-dimensional scalar field
\[ z=f(x,y) \]can often be visualized as a surface in three dimensions.
The horizontal coordinates \((x,y)\) locate a point in the domain, while the vertical coordinate \(z\) represents the field value.
This converts numerical values into visible height.
The Graph Is Not the Same as the Physical Field
A height graph is a visualization technique, not necessarily the physical shape of the object being modeled.
If \(T(x,y)\) represents temperature across a flat floor, plotting
\[ z=T(x,y) \]creates a three-dimensional surface. That surface represents temperature values; it does not mean the floor is physically warped upward and downward.
Color Maps
Another common visualization assigns colors or shades to ranges of scalar values.
Weather maps may use different colors for different temperatures. Topographic maps may shade elevations. Medical images may assign brightness to tissue density or signal intensity.
The color is a visual encoding of a scalar number.
Level Sets
A level set contains all points where a scalar field has the same value.
For the field
\[ f(x,y)=x^2+y^2, \]the level set \(f(x,y)=9\) satisfies
\[ x^2+y^2=9. \]This is a circle of radius \(3\).
Level sets and contour maps receive their own detailed treatment later in this section.
Contours Connect Equal Values
On a two-dimensional map, curves joining locations with equal field values are often called contours.
Familiar examples include:
- elevation contours on a topographic map;
- isobars joining equal atmospheric pressure;
- isotherms joining equal temperature.
Each contour is a geometric trace of one selected scalar value.
A Scalar Field Can Have Peaks and Valleys
Consider
\[ f(x,y)=10-x^2-y^2. \]The largest value occurs at the origin:
\[ f(0,0)=10. \]Moving away from the origin decreases the value.
In a height interpretation, the field resembles a hill. In a temperature interpretation, the origin could represent a hot spot.
A Scalar Field Can Have Basins
For
\[ f(x,y)=x^2+y^2, \]the smallest value occurs at the origin:
\[ f(0,0)=0. \]Field values increase in every direction away from that point.
When viewed as height, the graph resembles a bowl.
Several Peaks or Valleys Are Possible
Real fields may contain many local regions of high and low values.
A temperature map may contain several hot spots. A terrain model may contain many mountains and valleys. A pressure field may contain multiple high-pressure and low-pressure systems.
Scalar fields can therefore encode complicated spatial structure even though each individual point receives only one number.
Fields Can Be Smooth
A field is informally called smooth when nearby points tend to have nearby values and the field changes without abrupt jumps.
The field
\[ f(x,y)=x^2+y^2 \]is smooth everywhere.
Smoothness allows local changes to be studied systematically with calculus.
Fields Can Also Have Discontinuities
Not every scalar field changes smoothly.
For example, a simplified material model might assign
\[ \rho(x)= \begin{cases} 2, & x<0,\\ 7, & x\geq 0. \end{cases} \]The density jumps at \(x=0\).
Such abrupt boundaries can represent interfaces between different materials or regions.
Fields Can Have Undefined Points
Consider
\[ f(x,y)=\frac{1}{\sqrt{x^2+y^2}}. \]At the origin,
\[ \sqrt{x^2+y^2}=0, \]so the formula would require division by zero.
Therefore, the origin is not in the domain of this field.
Singular Behavior
Near the origin, the field
\[ f(x,y)=\frac{1}{\sqrt{x^2+y^2}} \]becomes very large.
A location where a field is undefined or behaves without an ordinary finite value is often associated with a singularity.
Later mathematics treats singular behavior with more precision.
Symmetry Can Simplify a Field
If
\[ f(x,y)=x^2+y^2, \]then changing \(x\) to \(-x\) does not change the value:
\[ f(-x,y)=(-x)^2+y^2=x^2+y^2. \]The same is true for \(y\).
Symmetry helps us understand the entire field from a smaller amount of information.
Translation of a Field
The field
\[ f(x,y)=x^2+y^2 \]is centered at the origin.
Replacing \(x\) by \(x-a\) and \(y\) by \(y-b\) gives
\[ g(x,y)=(x-a)^2+(y-b)^2. \]The same pattern is now centered at \((a,b)\).
Coordinate transformations therefore shift spatial field patterns just as they shift ordinary graphs.
Scaling a Field
Multiplying a scalar field by a constant rescales its values.
If
\[ g(x,y)=3f(x,y), \]then every field value is three times as large:
\[ g(a,b)=3f(a,b). \]The spatial locations are unchanged; only the assigned scalar values are rescaled.
Adding Scalar Fields
When two scalar fields are defined on the same domain and their quantities are compatible, they can be combined point by point.
If
\[ f(x,y)=x+y \]and
\[ g(x,y)=x-y, \]then
\[ (f+g)(x,y)=2x. \]Field operations are ordinary function operations performed at every location.
Subtracting Scalar Fields
The difference between two fields may itself be informative.
If \(T_1(x,y)\) and \(T_2(x,y)\) represent temperature at two different times, then
\[ \Delta T(x,y)=T_2(x,y)-T_1(x,y) \]represents the temperature change at each location.
One field can therefore describe how another field has changed.
Scalar Multiplication of a Field
If \(f(x,y)\) is a scalar field and \(c\) is a constant, then
\[ cf(x,y) \]is another scalar field.
Every location keeps the same position, while its assigned value is multiplied by \(c\).
Composing with a Scalar Function
A scalar field can be transformed by applying an ordinary one-variable function to its values.
If
\[ f(x,y)=x^2+y^2 \]and
\[ g(u)=\sqrt{u}, \]then
\[ g(f(x,y))=\sqrt{x^2+y^2}, \]which is the distance field from the origin.
Scalar Fields and Time
A field may also change with time.
For example,
\[ T(x,y,t) \]may represent temperature at position \((x,y)\) and time \(t\).
For each fixed time, \(T(x,y,t)\) gives a scalar field across space. As \(t\) changes, the entire spatial pattern may evolve.
Space Variables and Time Variables
It is useful to distinguish the roles of variables.
In
\[ T(x,y,z,t), \]the coordinates \(x,y,z\) locate a point in space, while \(t\) identifies time.
The output \(T\) remains a scalar temperature.
Temperature as a Scalar Field
Temperature is one of the most intuitive scalar fields.
A room can contain warmer and cooler regions. A weather system can contain temperature differences across hundreds of kilometers. A metal object can heat unevenly.
In each case,
\[ T(\text{position})=\text{temperature at that position}. \]Pressure as a Scalar Field
Pressure also assigns one scalar value to every location in a fluid or gas.
An atmospheric pressure field may be written
\[ P(x,y,z). \]Weather maps often display selected pressure values through curves of equal pressure.
Density as a Scalar Field
Density describes how much mass is associated with a region of space.
A nonuniform material may be modeled by
\[ \rho(x,y,z). \]Different locations can have different density values even though they belong to the same object.
Potential as a Scalar Field
Many physical theories use scalar fields called potentials.
Examples include gravitational potential and electric potential.
A potential field assigns a scalar value to each position. Later lessons will show how directional field behavior can be related to how a potential changes across space.
Scalar Fields and Vector Fields Are Different
A scalar field gives one number at each point:
\[ f(x,y)=\text{scalar}. \]A vector field gives a vector at each point:
\[ \mathbf{F}(x,y)=\text{vector}. \]Temperature is naturally scalar. Wind velocity is naturally vector because velocity has both magnitude and direction.
A Scalar Field Does Not Point
Suppose the temperature at a point is
\[ T(x,y)=25. \]The number \(25\) does not itself tell us which direction is warmer or cooler.
Directional information must be extracted by comparing how field values change from one location to another.
The gradient will later provide a systematic way to do this.
Reading a Formula Geometrically
Consider
\[ f(x,y)=4-x^2-y^2. \]Several spatial facts are visible from the formula:
- the field is centered at the origin;
- the origin has value \(4\);
- values decrease with distance from the origin;
- points at equal distance from the origin have equal values.
Algebraic form can therefore reveal geometric structure before any graph is drawn.
Reading a Field from Data
Sometimes the formula is unknown.
If measurements show larger values toward one side of a region, repeated high values near one location, or similar values along a curve, those patterns provide information about the underlying field.
Field analysis therefore combines algebra, geometry, data, and visualization.
Local Information
Local behavior concerns what happens near a selected point.
We may ask:
- Is the field increasing nearby?
- Is it decreasing?
- Is the point near a peak or valley?
- Do nearby values change slowly or rapidly?
These questions lead naturally toward derivatives and gradients.
Global Information
Global behavior concerns the field across a larger domain.
We may ask:
- What are the highest and lowest values?
- Where are large regions of similar values?
- What symmetries are present?
- How do distant regions compare?
A later lesson will distinguish local and global field behavior in greater depth.
Common Error: Confusing the Point with the Field Value
If
\[ f(2,3)=7, \]then \((2,3)\) is the location and \(7\) is the field value.
They are different mathematical objects with different roles.
Common Error: Treating Every Field as a Height
A scalar field can be visualized as height, but its actual interpretation may be temperature, pressure, density, concentration, or another scalar quantity.
The vertical graph is a representation of the values, not necessarily a literal physical surface.
Common Error: Ignoring the Domain
A formula does not automatically describe the field everywhere.
If
\[ f(x,y)=\frac{1}{x^2+y^2}, \]then the origin must be excluded because the denominator is zero there.
Physical models may impose additional domain restrictions beyond those forced by algebra.
Common Error: Confusing Scalar and Vector Quantities
A speed field can assign one speed to every location and is therefore scalar.
A velocity field assigns both magnitude and direction and is therefore vector-valued.
The word field alone does not tell us whether the output is scalar or vector. We must inspect what each point receives.
Common Error: Assuming Equal Values Mean Equal Points
Different locations can have the same field value.
For
\[ f(x,y)=x^2+y^2, \]both \((3,4)\) and \((0,5)\) satisfy
\[ f=25. \]Scalar fields are generally many-to-one: multiple locations may share one value.
A Reliable Procedure for Evaluating a Scalar Field
- Identify the spatial coordinates of the point.
- Check that the point lies in the field's domain.
- Substitute each coordinate into the correct variable.
- Evaluate the expression carefully.
- Attach the appropriate physical or geometric units when relevant.
- Interpret the result as the scalar value at that location.
Example: Full Evaluation Workflow
Let
\[ C(x,y,z)=4+x^2+2y-z. \]Find the field value at
\[ (x,y,z)=(2,-1,3). \]Substitute:
\[ C(2,-1,3)=4+2^2+2(-1)-3. \]Simplify:
\[ C(2,-1,3)=4+4-2-3=3. \]Therefore,
\[ \boxed{C(2,-1,3)=3}. \]Example: Comparing Two Locations
Suppose
\[ T(x,y)=30-x^2-y^2. \]Compare the field at \((1,1)\) and \((3,0)\).
\[ T(1,1)=30-1-1=28, \] \[ T(3,0)=30-9=21. \]Therefore, the field value is larger at \((1,1)\).
Example: Solving for Locations with a Given Value
Let
\[ f(x,y)=x^2+y^2. \]Which points have field value \(16\)?
Set
\[ f(x,y)=16. \]Then
\[ x^2+y^2=16. \]The answer is the circle of radius \(4\) centered at the origin.
A question about field values has become a question about spatial geometry.
Example: A Restricted Domain
Suppose
\[ f(x,y)=\sqrt{9-x^2-y^2}. \]For real values, the expression under the square root must satisfy
\[ 9-x^2-y^2\geq 0. \]Therefore,
\[ x^2+y^2\leq 9. \]The field is defined on the disk of radius \(3\) centered at the origin.
Practice 1: Evaluate a Two-Dimensional Field
Let
\[ f(x,y)=3x-2y+4. \]Find \(f(2,5)\).
Practice 2: Evaluate a Three-Dimensional Field
Let
\[ \rho(x,y,z)=2+x^2+y^2+z^2. \]Find \(\rho(1,2,3)\).
Practice 3: Identify Location and Value
Suppose
\[ T(4,-2)=19. \]Identify the spatial location and the scalar field value.
Practice 4: Compare Two Points
Let
\[ f(x,y)=10-x^2-y^2. \]Which has the larger field value: \((1,1)\) or \((2,2)\)?
Practice 5: Find a Level Set
Let
\[ f(x,y)=x^2+y^2. \]Describe the set of points where \(f(x,y)=36\).
Practice 6: Determine the Domain
Determine the real-valued domain of
\[ f(x,y)=\sqrt{25-x^2-y^2}. \]Practice 7: Scalar or Vector?
Classify each quantity as naturally scalar or vector-valued:
- temperature at every point in a room;
- wind velocity at every point in the atmosphere;
- air pressure at every point in the atmosphere;
- gravitational acceleration at every point in space.
Practice 8: Translate a Centered Field
The field
\[ f(x,y)=x^2+y^2 \]is centered at the origin. Write a corresponding field centered at \((3,-2)\).
Practice 9: Combine Two Fields
Let
\[ f(x,y)=2x+y \]and
\[ g(x,y)=x-3y. \]Find \((f+g)(x,y)\).
Practice 10: Interpret a Time-Dependent Field
A model uses
\[ T(x,y,t). \]Explain the roles of \(x\), \(y\), \(t\), and \(T\).
Practice Answers
Practice 1
\[ f(2,5)=3(2)-2(5)+4=6-10+4=0. \]Therefore,
\[ \boxed{f(2,5)=0}. \]Practice 2
\[ \rho(1,2,3)=2+1^2+2^2+3^2=2+1+4+9=16. \]Therefore,
\[ \boxed{\rho(1,2,3)=16}. \]Practice 3
The location is
\[ \boxed{(4,-2)}, \]and the scalar field value there is
\[ \boxed{19}. \]Practice 4
\[ f(1,1)=10-1-1=8, \] \[ f(2,2)=10-4-4=2. \]Therefore, \((1,1)\) has the larger field value.
Practice 5
Set
\[ x^2+y^2=36. \]The level set is
\[ \boxed{\text{the circle of radius }6\text{ centered at the origin}}. \]Practice 6
Require
\[ 25-x^2-y^2\geq 0. \]Thus,
\[ \boxed{x^2+y^2\leq 25}. \]The domain is the closed disk of radius \(5\).
Practice 7
- temperature: scalar;
- wind velocity: vector;
- air pressure: scalar;
- gravitational acceleration: vector.
Practice 8
Replace \(x\) by \(x-3\) and \(y\) by \(y+2\):
\[ \boxed{g(x,y)=(x-3)^2+(y+2)^2}. \]Practice 9
\[ (f+g)(x,y)=(2x+y)+(x-3y). \]Therefore,
\[ \boxed{(f+g)(x,y)=3x-2y}. \]Practice 10
The variables \(x\) and \(y\) locate a point in space, \(t\) identifies time, and \(T(x,y,t)\) gives the scalar temperature at that location and time.
What to Remember
A scalar field assigns one scalar value to every point in a spatial domain:
\[ \boxed{f:\mathbb{R}^n\to\mathbb{R}}. \]- the input identifies a location;
- the output is one number;
- the domain tells us where the field is defined;
- different points may share the same value;
- field values may be represented by formulas, tables, surfaces, colors, or contours;
- scalar fields may model temperature, pressure, density, elevation, concentration, and potential;
- spatial change is studied by comparing how values vary from point to point.
The central habit is to keep two ideas separate: where the point is and what scalar value the field assigns there.
Where We Go Next
Scalar fields attach one number to each location. Many spatial phenomena, however, also require direction.
The next lesson, Vector Fields, assigns a vector to every point and begins the transition from scalar distributions to directional behavior such as motion, force, and flow.