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Linear Approximations

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

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Related Experiment Video

Updated: Jul 17, 2026

Composition and Distribution Analysis of Bioaerosols Under Different Environmental Conditions

Composition and Distribution Analysis of Bioaerosols Under Different Environmental Conditions

Published on: January 7, 2019

Monitoring, Mapping, and Modeling Spatial-Temporal Patterns of PM_2.5 for Improved Understanding of Air Pollution

Ronan Hart¹, Lu Liang¹, Pinliang Dong¹

¹Department of Geography and the Environment, University of North Texas, 1155 Union Circle, Denton, TX 76203, USA.

International Journal of Environmental Research and Public Health

|July 12, 2020

Summary

Fine particulate matter (PM2.5) levels vary significantly, peaking during evening commutes. A new wind-aware system improved modeling of PM2.5 dynamics, highlighting landscape and built environment impacts on air quality.

Keywords:

GIS Lidar landscape pattern mobile monitoring particulate pollution

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Last Updated: Jul 17, 2026

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Published on: January 7, 2019

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Area of Science:

Environmental Science
Atmospheric Science
Urban Planning

Background:

Fine particulate matter (PM2.5) exhibits high spatial and temporal variability.
Understanding local-scale air quality dynamics is crucial for public health and urban environmental management.

Purpose of the Study:

To map PM2.5 concentrations at high spatio-temporal resolutions.
To identify key environmental determinants of local PM2.5 dynamics.
To evaluate a novel wind wedge-based system for air pollution modeling.

Main Methods:

Bicycle-based mobile measurements for high-resolution PM2.5 mapping.
Integration of GIS, airborne imagery, and LiDAR data for predictor variables.
Panel data analysis incorporating a wind wedge-based system to quantify influencing factors.

Main Results:

Significant diurnal and daily variations in PM2.5 were observed, with evening peaks.
Eight natural and built environment variables were identified as significant determinants.
The wind wedge system demonstrated higher variable significance compared to conventional methods.

Conclusions:

Local-scale air quality is influenced by a combination of meteorological, landscape, and urban factors.
Incorporating wind effects and source-receptor relationships is vital for accurate air pollution modeling.
The study provides a framework for understanding and managing PM2.5 in urban environments.