The Central Idea
Local behavior describes what a field does in a neighborhood of a particular point.
Global behavior describes the field across an entire domain or across a substantial connected region.
The distinction can be summarized as
\[ \boxed{ \text{local}=\text{nearby behavior}, \qquad \text{global}=\text{whole-domain behavior} }. \]Why Scale Matters
A field may look simple when examined near one point but complicated across a larger region.
Conversely, a globally simple pattern may contain local features that require careful analysis.
Field analysis therefore moves repeatedly between two questions:
- What is happening here?
- How does that fit into the structure of the whole field?
A Local Neighborhood
To study a point
\[ \mathbf{p}, \]we examine points sufficiently close to it.
In the plane, a small neighborhood may be represented by
\[ \|\mathbf{x}-\mathbf{p}\|<\varepsilon, \]where \(\varepsilon>0\) is small.
Local statements concern behavior within some such neighborhood.
Global Domain
Global behavior depends on the full domain under consideration.
A field defined on all of
\[ \mathbb{R}^2 \]may behave differently from the same formula restricted to a disk, rectangle, punctured plane, or disconnected region.
The domain is part of the field, not merely a background detail.
Local Scalar-Field Behavior
Near a point in a scalar field, we may ask whether values:
- increase in some directions;
- decrease in some directions;
- remain nearly constant;
- form a local peak;
- form a local basin;
- change rapidly or slowly.
Derivatives and gradients provide systematic tools for these questions.
Global Scalar-Field Behavior
Across the entire domain, we may ask:
- What is the range of field values?
- Are there absolute maxima or minima?
- What symmetries are present?
- How are level sets organized?
- Does the field remain bounded?
- What happens far from the origin?
Local Vector-Field Behavior
Near one point in a vector field, we may examine:
- the field vector itself;
- changes in magnitude;
- changes in direction;
- local spreading or convergence;
- local rotational tendency;
- behavior near an equilibrium.
Global Vector-Field Behavior
Across the domain, we may ask:
- where trajectories go;
- whether closed orbits exist;
- whether equilibria organize the flow;
- whether the field has singularities;
- whether the field is conservative;
- how the field behaves near boundaries or at large distances.
A Simple Local Approximation
Smooth fields often look nearly linear when viewed closely enough.
For a scalar field \(f(x,y)\) near \((a,b)\),
\[ f(x,y) \approx f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b). \]This approximation captures local behavior even when the global field is nonlinear.
Local Linearity
Differentiability means, in part, that sufficiently small pieces of a field behave approximately like linear objects.
Curved level sets look nearly straight when magnified around a regular point.
Smooth vector fields look nearly constant plus a linear variation over a sufficiently small neighborhood.
A Globally Curved Field Can Be Locally Simple
Consider
\[ f(x,y)=x^2+y^2. \]Globally, the field has circular level curves and grows without bound.
Near a nonzero point \((a,b)\), however, its change is approximately described by the linear expression
\[ 2a(x-a)+2b(y-b). \]The local picture is simpler than the global one.
Local Maxima
A point \(\mathbf{p}\) is a local maximum of a scalar field if nearby points have values no larger than
\[ f(\mathbf{p}). \]Informally,
\[ f(\mathbf{x})\leq f(\mathbf{p}) \]for all sufficiently nearby \(\mathbf{x}\).
Local Minima
A point \(\mathbf{p}\) is a local minimum if nearby points have values no smaller than
\[ f(\mathbf{p}). \]Informally,
\[ f(\mathbf{x})\geq f(\mathbf{p}) \]for all sufficiently nearby \(\mathbf{x}\).
Global Maxima and Minima
A global, or absolute, maximum is larger than or equal to every other field value in the entire domain.
A global minimum is smaller than or equal to every other value in the entire domain.
Global extrema make statements about the whole domain, not merely a small neighborhood.
Local Does Not Mean Global
Consider a field with several hills of different heights.
The top of a smaller hill may be a local maximum because nearby points are lower.
But it is not a global maximum if another hill elsewhere is higher.
Local optimality does not guarantee global optimality.
A Local Minimum That Is Also Global
For
\[ f(x,y)=x^2+y^2, \]the origin has
\[ f(0,0)=0. \]Since
\[ x^2+y^2\geq0 \]everywhere, the origin is both a local minimum and a global minimum.
A Field with No Global Maximum
Again consider
\[ f(x,y)=x^2+y^2. \]As
\[ \sqrt{x^2+y^2}\to\infty, \]the field also tends to infinity.
Therefore, on all of \(\mathbb{R}^2\), the field has no global maximum.
Restriction Can Change Global Behavior
The same field
\[ f(x,y)=x^2+y^2 \]restricted to the disk
\[ x^2+y^2\leq25 \]does have a global maximum:
\[ \boxed{25}. \]Global conclusions depend on the domain.
Boundary Behavior
When a domain has a boundary, global extrema may occur there even if no interior critical point identifies them.
For example, on the disk
\[ x^2+y^2\leq25, \]the maximum of
\[ f(x,y)=x^2+y^2 \]occurs on the boundary circle.
Critical Points Are Local Candidates
For a differentiable scalar field, points satisfying
\[ \nabla f=\mathbf{0} \]are critical points.
These points are candidates for important local behavior, but the gradient equation alone does not determine whether they are maxima, minima, or saddles.
Saddle Behavior
Consider
\[ f(x,y)=x^2-y^2. \]At the origin,
\[ \nabla f(0,0)=\mathbf{0}. \]Along the \(x\)-axis,
\[ f(x,0)=x^2\geq0. \]Along the \(y\)-axis,
\[ f(0,y)=-y^2\leq0. \]The point is neither a local maximum nor a local minimum.
Local Behavior Can Depend on Direction
The saddle example shows that a field can rise in some directions and fall in others.
Looking along only one path may therefore give a misleading picture of a multivariable field.
Local analysis must account for all nearby directions.
Local Contour Geometry
Near a regular point where
\[ \nabla f\neq\mathbf{0}, \]a level curve is locally smooth and has a tangent direction perpendicular to the gradient.
This is a local geometric statement.
The full contour may curve, close, split into separate components, or extend across the domain.
Global Contour Structure
Globally, level sets may form:
- closed loops;
- open curves;
- multiple disconnected components;
- nested families;
- surfaces with several pieces;
- sets changed by singular or critical points.
Local tangent information alone does not reveal the entire level-set topology.
Local Vector Magnitude
At a point \(\mathbf{p}\), the magnitude
\[ \|\mathbf{F}(\mathbf{p})\| \]describes the local strength of the vector field.
Nearby magnitudes reveal whether the field is strengthening, weakening, or remaining nearly uniform around that point.
Global Magnitude Patterns
Across a large domain, vector magnitude may:
- remain constant;
- grow with distance;
- decay with distance;
- peak near selected sources;
- vanish at equilibria;
- become unbounded near singularities.
Example: Magnitude Growing with Distance
For
\[ \mathbf{F}(x,y)=\langle x,y\rangle, \]the magnitude is
\[ \|\mathbf{F}\| = \sqrt{x^2+y^2}. \]The field becomes stronger as distance from the origin increases.
This is a global radial pattern.
Example: Magnitude Decaying with Distance
Consider the radial field
\[ \mathbf{F}(x,y) = \frac{\langle x,y\rangle} {(x^2+y^2)^{3/2}}, \]defined away from the origin.
Its magnitude is
\[ \|\mathbf{F}\| = \frac{1}{x^2+y^2}. \]Thus the field weakens as distance from the origin increases and becomes singular at the origin.
Local Singular Behavior
A singularity is a point or set where a field fails to be defined or loses ordinary smooth behavior.
Near a singularity, field magnitude may become very large or direction may vary sharply.
Singular points often organize nearby field geometry.
Global Effect of a Singularity
A single excluded point can alter the topology of the entire domain.
For example,
\[ \mathbb{R}^2\setminus\{(0,0)\} \]is the plane with one point removed.
That small local exclusion can affect global questions about closed loops, circulation, and the existence of potentials.
Local versus Global Conservativeness
A field may satisfy a local derivative condition suggesting conservative behavior in every small neighborhood.
Yet a globally defined single-valued potential may fail to exist if the domain has a topological obstruction such as a hole.
This is a classic example of local information not automatically producing a global conclusion.
Simply Connected Domains
A simply connected domain has no holes that obstruct shrinking closed loops continuously to a point inside the domain.
On such domains, local differential tests often extend cleanly to global statements about conservative fields under the usual smoothness assumptions.
Connected versus Disconnected Domains
A domain may contain several disconnected pieces.
Behavior on one component does not automatically determine behavior on another.
A field could be positive everywhere on one component and negative everywhere on another without contradiction.
Local Equilibrium Behavior
An equilibrium point of an autonomous vector field satisfies
\[ \mathbf{F}(\mathbf{p})=\mathbf{0}. \]Local analysis asks how nearby trajectories behave:
- Do they approach the equilibrium?
- Do they move away?
- Do they circulate nearby?
- Do different directions behave differently?
Global Trajectory Behavior
Global trajectory questions include:
- Where does a trajectory go as \(t\to\infty\)?
- Does it remain bounded?
- Does it approach an equilibrium?
- Does it form a closed orbit?
- Does it escape to infinity?
- Does it encounter a boundary or singularity?
An Attracting Local Pattern
For
\[ \mathbf{F}(x,y)=\langle -x,-y\rangle, \]the origin is an equilibrium.
Nearby vectors point inward, and trajectories satisfy
\[ \mathbf{r}(t) = \langle C_1e^{-t},C_2e^{-t}\rangle. \]Every trajectory approaches the origin as \(t\to\infty\).
A Repelling Local Pattern
For
\[ \mathbf{F}(x,y)=\langle x,y\rangle, \]the origin is again an equilibrium.
But nearby trajectories move outward:
\[ \mathbf{r}(t) = \langle C_1e^t,C_2e^t\rangle. \]The same local zero value can belong to very different surrounding behavior.
Saddle-Type Vector Behavior
Consider
\[ \mathbf{F}(x,y)=\langle x,-y\rangle. \]Along the \(x\)-direction, trajectories move away from the origin.
Along the \(y\)-direction, they move toward the origin.
The equilibrium combines attraction and repulsion in different directions.
Rotational Local Behavior
For
\[ \mathbf{F}(x,y)=\langle -y,x\rangle, \]vectors are tangent to circles centered at the origin.
Nearby trajectories circulate rather than approach or leave the origin radially.
This local rotational structure extends globally into closed circular trajectories.
Local Behavior Need Not Persist Globally
A field may look inward-pointing near one equilibrium but change direction farther away.
It may behave nearly rotationally in one region and nearly radial in another.
Local classifications describe neighborhoods, not necessarily the entire field.
Global Symmetry
Symmetry is often a global organizing feature.
A field may be invariant under:
- reflection;
- rotation;
- translation;
- coordinate interchange;
- sign reversal combined with spatial transformation.
Recognizing symmetry can reduce the amount of analysis required.
Radial Symmetry
A scalar field of the form
\[ f(x,y)=g(x^2+y^2) \]depends only on distance from the origin.
Its level curves are typically concentric circles.
One radial slice can therefore reveal information about the entire field.
Translational Symmetry
A field such as
\[ f(x,y)=x \]is unchanged when \(y\) changes.
Its level curves are vertical lines:
\[ x=c. \]The field has a global invariance in the \(y\)-direction.
Periodic Global Behavior
Some fields repeat spatially.
For example,
\[ f(x,y)=\sin x+\cos y \]satisfies
\[ f(x+2\pi,y)=f(x,y) \]and
\[ f(x,y+2\pi)=f(x,y). \]A finite repeating cell can encode the global pattern.
Behavior at Large Distances
A global study often asks what happens as position moves far from a chosen origin.
We may examine
\[ \|\mathbf{x}\|\to\infty. \]A field might grow, decay, oscillate, approach a constant, or fail to have one limiting pattern.
Asymptotic Behavior
Asymptotic behavior describes how a field behaves in some limiting regime.
For example,
\[ f(x,y) = \frac{1}{1+x^2+y^2} \]satisfies
\[ f(x,y)\to0 \]as \(\sqrt{x^2+y^2}\to\infty\).
Bounded Fields
A scalar field is bounded if its values remain within finite limits.
For
\[ f(x,y)=\frac{1}{1+x^2+y^2}, \]we have
\[ 0Unbounded Fields
The field
\[ f(x,y)=x^2+y^2 \]is unbounded above on \(\mathbb{R}^2\).
No matter how large a number \(M\) is chosen, points sufficiently far from the origin satisfy
\[ f(x,y)>M. \]Global Range
The range of a scalar field is the set of values actually achieved across its domain.
For
\[ f(x,y)=x^2+y^2 \]on all of \(\mathbb{R}^2\),
\[ \boxed{ \operatorname{range}(f)=[0,\infty) }. \]Range is fundamentally a global property.
Local Range near a Point
Near a selected point, the values achieved may occupy only a small portion of the global range.
Near \((0,0)\) for
\[ f(x,y)=x^2+y^2, \]the values are small and nonnegative even though the global field reaches arbitrarily large values.
Local Continuity
Continuity is tested locally.
A field is continuous at a point when field values near that point remain close to the field value at the point.
A field can be continuous at one point and discontinuous at another.
Global Continuity
Saying a field is continuous on a domain means it is continuous at every point of that domain.
This is a global statement assembled from local continuity at all points.
Many global properties arise from local properties holding everywhere plus suitable domain conditions.
Local Smoothness
Smoothness concerns the existence and continuity of derivatives near a point or throughout a region.
A field may be smooth on most of its domain but fail at isolated singularities or interfaces.
Piecewise Global Structure
Some fields are naturally divided into regions with different formulas.
For example,
\[ f(x)= \begin{cases} x, & x<0,\\ x^2, & x\geq0. \end{cases} \]Local analysis near \(x=0\) must reconcile the two pieces, while global analysis considers how both regions fit together.
Interfaces
An interface is a boundary separating regions with different field behavior.
Across an interface, a scalar field or vector field may:
- remain continuous;
- jump in value;
- change derivative;
- change physical interpretation.
Local behavior near the interface can dominate the analysis.
Global Conservation Statements
Some of the most powerful field theorems connect local derivative behavior to global integrals over boundaries or regions.
These ideas link:
- local divergence to total outward flux;
- local curl to circulation around boundaries;
- gradients to endpoint differences.
Such theorems show how local structure can accumulate into global behavior.
Local Contributions Can Accumulate
A tiny amount of field change in one small region may seem insignificant.
But integrating similar contributions across a large domain can produce a substantial global effect.
This local-to-global accumulation is a recurring theme throughout calculus and field theory.
Cancellation Can Hide Local Activity
Strong local behavior does not always produce a large global total.
Positive and negative contributions from different regions may cancel.
For example, local directional effects can sum to zero over a symmetric domain even though the field is far from zero at individual points.
Global Totals Can Hide Local Variation
Two fields can have the same total integral over a region while being distributed very differently inside it.
A global total therefore does not uniquely determine the local field.
Local and global descriptions provide different kinds of information.
Sampling and Scale
Measured fields are usually sampled at finitely many locations.
A coarse sampling grid may reveal global trends but miss sharp local features.
A dense local sampling may reveal fine structure but cover only a small region.
Resolution affects what kind of field behavior can be detected.
Local Features Can Be Missed by Coarse Data
A narrow temperature spike, pressure drop, or vortex may lie between widely spaced measurement points.
If the sampling scale is too coarse, the global map may smooth over the feature entirely.
Numerical field representations must therefore be interpreted at the scale of their data.
Global Trends Can Be Missed by Narrow Data
A detailed measurement over one small neighborhood may reveal excellent local information.
But it cannot by itself establish the behavior of a field thousands of units away.
Extrapolating local behavior globally requires justification.
Local Models
A local model is designed to approximate a field accurately over a limited region.
Linearization is one common example.
Local models are useful precisely because they do not attempt to reproduce every global feature.
Global Models
A global model seeks to describe the field across a larger domain.
It must account for:
- boundaries;
- singularities;
- multiple regions;
- long-range behavior;
- global constraints;
- possible changes of regime.
Exact and Approximate Global Structure
Some global patterns follow exactly from a formula.
Others are inferred numerically from measured or simulated data.
A global contour map or vector-field plot may therefore be exact in a mathematical model but approximate in an empirical setting.
Local Symmetry
A small neighborhood may appear approximately symmetric even when the entire field is not.
For example, a smooth curved surface can look nearly planar under strong magnification.
Approximate local symmetry should not automatically be promoted to an exact global symmetry.
Global Constraints Can Shape Local Behavior
Boundaries and conservation laws can influence what local patterns are possible.
A field inside a closed container may behave differently from the same local equation posed on an unbounded domain.
Global conditions can therefore restrict local solutions.
Local Direction versus Global Destination
A vector at one point tells us only the instantaneous direction there.
It does not by itself determine the eventual destination of a trajectory.
The path depends on the sequence of field vectors encountered as the position changes.
Example: Curving Trajectory from Changing Local Directions
In the rotational field
\[ \mathbf{F}(x,y)=\langle -y,x\rangle, \]every individual vector is straight.
Yet the direction changes continuously from point to point.
Following those local directions produces a globally curved circular trajectory.
Local Potential Information
If
\[ \mathbf{F}=\nabla\phi, \]then locally the field points toward increasing \(\phi\).
Nearby equipotential sets are perpendicular to the field.
These are local geometric relationships.
Global Potential Information
A globally defined potential can provide:
- path-independent line integrals;
- zero closed-loop work;
- global equipotential families;
- endpoint formulas across connected regions.
These conclusions require more than information at a single point.
Local and Global Questions Are Complementary
Neither scale is inherently more important.
Local analysis reveals mechanisms and immediate behavior.
Global analysis reveals organization, constraints, destinations, and cumulative effects.
Mature field analysis uses both.
Common Error: Assuming a Local Maximum Is Global
A local maximum only compares the field value with sufficiently nearby points.
A larger value may exist elsewhere in the domain.
Global claims require global comparison.
Common Error: Assuming Zero Gradient Means Global Minimum
The equation
\[ \nabla f=\mathbf{0} \]identifies a critical point.
The point may be a minimum, maximum, saddle, or another stationary type.
Additional local and global analysis is required.
Common Error: Ignoring the Boundary
Global extrema on bounded domains often occur on the boundary.
Examining only interior critical points can therefore miss the true global maximum or minimum.
Common Error: Extending Local Behavior Too Far
A field that looks approximately linear near one point need not remain linear over the entire domain.
Local approximations have a limited region of reliability.
Common Error: Ignoring Domain Topology
Removing even one point can change global properties of a field.
Statements about potentials, closed loops, and connectedness may depend on whether the domain contains holes or separate components.
Common Error: Treating a Global Plot as Complete Information
A field plot displays only selected samples, arrows, contours, or numerical approximations.
Fine local structure can be hidden by plotting resolution.
Visualization is evidence about the field, not a replacement for the mathematical definition.
Common Error: Assuming Similar Local Pictures Imply the Same Global Field
Two fields can look almost identical near one point yet behave very differently far away.
Local agreement does not imply global equality.
Global structure requires information over the full domain.
A Reliable Procedure for Local Field Analysis
- Choose the point or small region of interest.
- Evaluate the field at the point.
- Inspect nearby values or vectors.
- Use derivatives, gradients, or local field operators when available.
- Identify critical points, equilibria, singularities, or interfaces.
- Describe how the field changes within the neighborhood.
- Avoid making whole-domain claims from local evidence alone.
A Reliable Procedure for Global Field Analysis
- Identify the complete domain.
- Locate boundaries, holes, singularities, and disconnected components.
- Determine symmetries and invariant structures.
- Study extrema, range, boundedness, and asymptotic behavior.
- Examine trajectory or level-set organization across the domain.
- Use integral and conservation relationships when appropriate.
- Check whether local conclusions remain valid globally.
Example: Local and Global Analysis of a Scalar Field
Consider
\[ f(x,y)=x^2+y^2. \]Locally at the origin,
\[ \nabla f(0,0)=\mathbf{0}, \]and nearby values are positive, so the origin is a local minimum.
Globally,
\[ f(x,y)\geq0 \]everywhere and grows without bound, so the origin is also the global minimum and there is no global maximum on \(\mathbb{R}^2\).
Example: Domain Changes the Global Conclusion
Restrict
\[ f(x,y)=x^2+y^2 \]to
\[ x^2+y^2\leq9. \]The local behavior at the origin is unchanged.
But globally, the field now has both:
\[ \boxed{\min f=0} \]and
\[ \boxed{\max f=9}. \]Example: Local and Global Vector Behavior
Consider
\[ \mathbf{F}(x,y)=\langle -x,-y\rangle. \]Locally near the origin, vectors point toward the equilibrium.
Globally, every trajectory has the form
\[ \langle C_1e^{-t},C_2e^{-t}\rangle \]and approaches the origin as \(t\to\infty\).
Here the local attracting pattern extends to the entire plane.
Example: Local Derivative Information and Global Topology
Suppose a planar vector field satisfies the local cross-partial condition
\[ P_y=Q_x \]throughout its domain.
On a simply connected domain, this strongly supports a global potential under the usual smoothness assumptions.
On a domain with a hole, more analysis is required.
The same local derivative information can therefore lead to different global conclusions depending on the domain.
Practice 1: Local or Global?
Classify each statement as primarily local or global:
- The gradient at \((2,3)\) is \(\langle 4,-1\rangle\).
- The field has no maximum anywhere on \(\mathbb{R}^2\).
- Nearby trajectories approach an equilibrium.
- The field is bounded throughout its entire domain.
Practice 2: Local versus Global Minimum
Let
\[ f(x,y)=x^2+y^2. \]Is the origin a local minimum, global minimum, or both?
Practice 3: No Global Maximum
For
\[ f(x,y)=x^2+y^2 \]on \(\mathbb{R}^2\), explain why no global maximum exists.
Practice 4: Boundary Maximum
Restrict
\[ f(x,y)=x^2+y^2 \]to
\[ x^2+y^2\leq16. \]Find the global maximum.
Practice 5: Saddle Point
For
\[ f(x,y)=x^2-y^2, \]explain why the origin is not a local maximum or minimum.
Practice 6: Attracting or Repelling?
Consider
\[ \mathbf{F}(x,y)=\langle x,y\rangle. \]Do nearby nonzero trajectories approach or move away from the origin as \(t\) increases?
Practice 7: Bounded or Unbounded?
Determine whether
\[ f(x,y)=\frac{1}{1+x^2+y^2} \]is globally bounded on \(\mathbb{R}^2\).
Practice 8: Asymptotic Behavior
Find the limiting behavior of
\[ f(x,y)=\frac{1}{1+x^2+y^2} \]as
\[ \sqrt{x^2+y^2}\to\infty. \]Practice 9: Domain Topology
Explain why removing the origin from the plane can matter for global field properties even though only one point has been removed.
Practice 10: Local Evidence
A field looks almost constant inside a very small neighborhood. Does that prove the field is constant globally? Explain.
Practice Answers
Practice 1
- local;
- global;
- local;
- global.
Practice 2
Since
\[ x^2+y^2\geq0 \]everywhere and equality occurs at the origin,
\[ \boxed{\text{the origin is both a local and global minimum}}. \]Practice 3
As distance from the origin grows,
\[ x^2+y^2\to\infty. \]Therefore, the field values are unbounded above, so no global maximum exists.
Practice 4
On the disk
\[ x^2+y^2\leq16, \]the largest possible field value is
\[ \boxed{16}, \]attained on the boundary circle.
Practice 5
Along the \(x\)-axis,
\[ f(x,0)=x^2>0 \]for nonzero nearby \(x\).
Along the \(y\)-axis,
\[ f(0,y)=-y^2<0 \]for nonzero nearby \(y\).
The field therefore increases in some directions and decreases in others, so the origin is a saddle point.
Practice 6
The solutions are
\[ \langle C_1e^t,C_2e^t\rangle. \]Therefore, nearby nonzero trajectories
\[ \boxed{\text{move away from the origin}}. \]Practice 7
Since
\[ 0< \frac{1}{1+x^2+y^2} \leq1, \]the field is globally bounded.
Practice 8
As
\[ x^2+y^2\to\infty, \]the denominator grows without bound. Therefore,
\[ \boxed{ f(x,y)\to0 }. \]Practice 9
Removing the origin creates a hole in the domain. That change can affect whether closed loops can be contracted to a point and can alter global conclusions about potentials, circulation, and path independence.
Practice 10
No. A field may be nearly constant in one small region and vary strongly elsewhere. Local evidence describes only a neighborhood unless additional information justifies extending the conclusion across the full domain.
What to Remember
Local and global analysis answer different but complementary questions:
\[ \boxed{ \text{local}=\text{what happens near a point} } \] \[ \boxed{ \text{global}=\text{how the whole field is organized} }. \]- gradients, derivatives, equilibria, and singularities often describe local structure;
- range, boundedness, global extrema, topology, and asymptotic behavior require whole-domain analysis;
- a local maximum or minimum need not be global;
- domain boundaries can determine global extrema;
- local derivative conditions do not always imply global field properties;
- holes and disconnected regions can change global conclusions;
- local contributions may accumulate or cancel in global integrals;
- numerical resolution determines which scales of behavior are visible.
The central discipline is: never promote a nearby pattern into a whole-domain conclusion without checking the global structure that supports it.
Where We Go Next
Local and global field analysis provides a framework for interpreting spatial systems at multiple scales.
The next lesson, Gravity, Electricity, Fluids, and Temperature, brings the field ideas together through major physical applications and shows how scalar and vector fields encode real spatial phenomena.