Solve the Problem $g ( x , y , z )$

question 268

Essay

Solve the problem.
-For some inexact differential forms df, a function $g ( x , y , z )$ can be found such that dh $= g ( x , y , z ) d f$ is exact. When it exists, the function $g ( x , y , z )$ is called an "integrating factor". Show that $g ( x , y , z ) = \frac { y z } { x }$ is an integrating factor for the inexact differential $\mathrm { df } = - \frac { 1 } { \mathrm { x } } \mathrm { dx } + \frac { 1 } { \mathrm { y } } \mathrm { dy } + \frac { 1 } { \mathrm { z } } \mathrm { dz }$ .

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Definitions:

Overloading

The act of exceeding the recommended weight capacity of a vehicle, structure, or system, which can lead to various safety issues.

Sidesway

A phenomenon where a structure or vehicle exhibits lateral movement or sway, often due to uneven loads or external forces.

Section Modulus

A geometric property of a cross-section used in the design and analysis of beams, indicating the section's resistance to bending.

I-Beam Frame Rail

A structural component in the chassis of vehicles and equipment characterized by its I-shaped cross-section, offering strength and rigidity.