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Vector analysis
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
In this chapter we review some ideas from multi-variable calculus. We particularly emphasize line and surface integrals and their interpretations as work and flux. We develop the ideas in terms of both vector fields and differential forms, including the major theorems of vector analysis. We discuss Maxwell’s equations from both points of view and our use of differential forms goes well beyond a typical multi-variable calculus course.
An inconsistence-free integral-based dynamic one- and two-parameter mixed model
Published in Journal of Turbulence, 2018
After applying the top-hat test-filter of width on the continuous filtered momentum Equation (6), one has being the integral-based exact sub-test scale tensor. Thus, by comparing (8) to the exact unresolved tensor (5), the integral-based resolved tensor, denoted by L in analogy with the classic Leonard tensor, is obtained Identity (9) represents the counterpart of the classical differential-based Germano identity, extended to the integral form of the equation. Equation (9) differs from the standard Germano identity for the presence of the diffusive flux as well as for the fact that the quadratic product is not acted upon by the test-filter. Indeed, the term present in the differential form is absent in the integral-based expression (9). This fact becomes relevant for determining the model coefficients in a consistent mathematical way.
Thermoelastic damping in rectangular microplate/nanoplate resonators based on modified nonlocal strain gradient theory and nonlocal heat conductive law
Published in Journal of Thermal Stresses, 2021
Xiao Ge, Pu Li, Yuming Fang, Longfei Yang
The integral nonlocal constitutive relations can be converted to the differential form as follows [21] where are Lame’s constants, Laplace operator and Poisson’s ratio. For thermoelastic coupling problems in MEMS resonators, the thermal stresses are negligible compared with the total stresses. Thus, the thermal stresses can be ignored, which helps simplify the derivation without reducing the accuracy of the calculation of strain energy and thermal dissipation.
Analysis of Coal Spontaneous Combustion by Thermodynamic Methods
Published in Combustion Science and Technology, 2021
Zhilin Xi, Ze Shan, Meitong Li, Xiaodong Wang
This method avoids the approximate solution of the temperature integral but introduces the differential form dα/dT. The α presents a smooth curve with the change of T. However, the extraction of dα/dt data presents great difficulties, because the experimental noise is amplified, as shown by the red line in Figure 4. Therefore, when differential data dα/dT are used, Fourier smoothing noise reduction is needed. On this basis, it is convenient for data extraction but the real data will change slightly after smoothing.