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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: Jun 18, 2026

Automated, High-resolution Mobile Collection System for the Nitrogen Isotopic Analysis of NOx

Automated, High-resolution Mobile Collection System for the Nitrogen Isotopic Analysis of NOx

Published on: December 20, 2016

High-resolution urban air pollution mapping.

Joshua S Apte^1,2, Chirag Manchanda¹

¹Department of Civil and Environmental Engineering, University of California, Berkeley, Berkeley, CA 94720, USA.

Science (New York, N.Y.)

|July 25, 2024

Summary

This summary is machine-generated.

Understanding urban air quality requires advanced monitoring. Mobile monitoring and dense sensor networks offer complementary insights into pollution dynamics, human health impacts, and environmental justice.

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Area of Science:

Environmental Science
Public Health
Atmospheric Chemistry

Background:

Urban air pollution is complex, influenced by spatial, temporal, and chemical factors.
These factors significantly impact population exposure, human health, and environmental justice.
Characterizing the multidimensional urban air quality system is crucial for effective interventions.

Purpose of the Study:

To review and highlight insights from mobile monitoring and dense sensor networks for urban air quality assessment.
To discuss the distinct but complementary strengths of these in situ measurement methods.
To explore their implications for understanding fine-scale processes and impacts on health and disparities.

Main Methods:

Review of in situ measurement techniques: mobile monitoring and dense sensor networks.
Analysis of strengths and limitations of each method for characterizing urban air quality.
Examination of how combined methods enhance understanding of spatial and temporal pollution patterns.

Main Results:

Mobile monitoring excels at measuring multiple pollutants at fine spatial scales, aiding process understanding and control strategies.
Dense sensor networks provide high temporal resolution data across numerous locations.
Integrated approaches using both methods reveal detailed spatial-temporal patterns affecting exposure and disparities.

Conclusions:

Combining mobile monitoring and dense sensor networks offers powerful insights into urban air quality dynamics.
These methods are vital for identifying pollution-related health risks and environmental injustices.
Sophisticated studies using these techniques provide mechanistic understanding for targeted interventions.