5.3.1 Scalar Field

5.3.1. Scalar Representation of Urban Land Value

The mathematical formalization of the Urban Land Value Field (ULVF) begins by representing urban land value as a scalar field. In mathematics and physics, a scalar field assigns a single numerical value to every point in space. Typical examples include temperature, atmospheric pressure, and gravitational potential, all of which vary continuously across space while remaining uniquely defined at each location.

Urban land value exhibits similar characteristics. Every location within a city possesses a measurable level of value determined by the combined influence of accessibility, transportation infrastructure, land use, economic activity, public investment, environmental quality, institutional conditions, and other urban factors. Although these influences originate from different sources, they collectively produce a single quantitative measure representing the urban land value at each location.

Within the ULVF framework, urban land value is therefore modeled as a scalar field defined over geographic space and evolving through time.

The Urban Land Value Field is formally expressed as

or equivalently,

where

• denotes the spatial domain of the urban area;

• represent geographic coordinates;

• denotes time;

• is the scalar value describing the urban land value at every location and every moment.

Unlike traditional parcel-based valuation, this formulation assigns a value to every point within the urban space rather than only to observed land parcels or market transactions. Consequently, the city is represented as a continuous value surface capable of supporting interpolation, spatial analysis, simulation, and prediction.

Representing urban land value as a scalar field establishes the mathematical foundation of ULVF. It enables the application of differential geometry, spatial statistics, Geographic Information Systems (GIS), numerical modeling, and artificial intelligence to analyze how urban value is distributed, evolves, and responds to changes in the urban environment.

From this perspective, individual land prices are interpreted as discrete observations sampled from an underlying continuous scalar field rather than isolated economic measurements. The primary objective of ULVF is therefore not only to estimate land prices but also to reconstruct and analyze the continuous urban value field that governs value formation throughout the city.

Key Characteristics of the Scalar Field

Property

Description

Continuous

Value exists at every spatial location.

Scalar

Each location has a single urban value magnitude.

Spatially Variable

Value changes continuously across space.

Time-dependent

Value evolves through time.

Multi-factor Driven

Generated by interactions among multiple urban variables.

Figure đề xuất

Figure 5.8 Urban Land Value as a Scalar Field

                 V(x,y,t)

                 ▲

                 │

       High Value│

                 │

       /\    │

     / \      │

______/______\____________

    Urban Space (x,y)

Continuous scalar representation of urban land value across geographic space.

Key References

The discussions on scalar fields and continuous spatial representation presented in this section are primarily based on the following references:

[29] Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists.

[30] Longley, P. A., Goodchild, M. F., Maguire, D. J., & Rhind, D. W. Geographic Information Systems and Science.