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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

The atomic gas sequence and mass–metallicity relation from dwarfs to massive galaxies

Galaxy scaling relations provide insights into the processes that drive galaxy evolution. The extension of these scaling relations into the dwarf galaxy regime is of particular interest. This is because dwarf galaxies represent a crucial stage in galaxy evolution, and understanding them could also shed light on their role in reionizing the early Universe. There is currently no consensus on the processes that dominate the evolution of dwarfs. In this work, we constrain the atomic gas sequence (stellar mass versus atomic gas fraction) and mass–metallicity relation (stellar mass versus gas-phase metallicity) from dwarf (⁠10 6.5 M ⊙ ⁠) to massive (⁠10 11.5 M ⊙ ) galaxies in the local Universe. The combined optical and 21-cm spectroscopic observations of the Dark Energy Spectroscopic Instrument and Arecibo Legacy Fast ALFA surveys allow us to constrain both scaling relations simultaneously. We find a slope change of the atomic gas sequence at a stellar mass of ~10 9 M ⊙ ⁠. We also find that the shape and scatter of the atomic gas sequence and mass–metallicity relation are strongly linked for both dwarfs and more massive galaxies. Consequently, the low-mass slope change of the atomic gas sequence is imprinted onto the mass–metallicity relation of dwarf galaxies. The mass scale of the measured slope change is consistent with a predicted escape velocity threshold below which low-mass galaxies experience significant supernova-driven gas loss, as well as with a reduction in cold gas accretion onto more massive galaxies.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Crossing Symmetric Dispersion Relations in Quantum Field Theories

For 2-2 scattering in quantum field theories, the usual fixed t dispersion relation exhibits only two-channel symmetry. This Letter considers a crossing symmetric dispersion relation, reviving certain old ideas from the 1970s. Rather than the fixed t dispersion relation, this needs a dispersion relation in a different variable z, which is related to the Mandelstam invariants s, t, u via a parametric cubic relation making the crossing symmetry in the complex z plane a geometric rotation. The resulting dispersion is manifestly three-channel crossing symmetric. We give simple derivations of certain known positivity conditions for effective field theories, including the null constraints, which lead to two sided bounds and derive a general set of new nonperturbative inequalities. We show how these inequalities enable us to locate the first massive string state from a low energy expansion of the four dilaton amplitude in type II string theory. We also show how a generalized (numerical) Froissart bound, valid for all energies, is obtained from this approach.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

RG-stable parameter relations of a scalar field theory in absence of a symmetry

Abstract The stability of tree-level relations among the parameters of a quantum field theory with respect to renormalization group (RG) running is typically explained by the existence of a symmetry. We examine a toy model of a quantum field theory of two real scalars in which a tree-level relation among the squared-mass parameters of the scalar potential appears to be RG-stable without the presence of an appropriate underlying symmetry. The stability of this relation with respect to renormalization group running can be explained by complexifying the original scalar field theory. It is then possible to exhibit a symmetry that guarantees the relations of relevant beta functions of squared-mass parameters of the complexified theory. Among these relations, we can identify equations that are algebraically identical to the corresponding equations that guarantee the stability of the relations among the squared-mass parameters of the original real scalar field theory where the symmetry of the complexified theory is no longer present.

Haber, Howard E. (ORCID:0000000173388104)↗
DOE OSTI2022

MULTISTEP AND CONTINUOUS PHYSICS-INFORMED NEURAL NETWORK METHODS FOR LEARNING GOVERNING EQUATIONS AND CONSTITUTIVE RELATIONS

In this work, we investigate the applicability and relative merit of discrete and continuous versions of physics-informed neural network (PINN) methods for learning unknown governing equations or constitutive relations in a nonlinear dynamical system. In the case of unknown dynamics, entire right-hand-side (RHS) equations of the ordinary differential equations are unknown. In the case of unknown constitutive relations, however, the RHS equations are known up to the specification of constitutive relations (that may depend on the state of the system). We use a deep neural network to model unknown governing equations or constitutive relations. The discrete PINN approach combines classical multistep discretization methods for dynamical systems with neural-network-based machine learning methods. On the other hand, the continuous versions utilize deep neural networks to minimize the residual function for the continuous governing equations. We use the case of a fedbatch bioreactor system to study the effectiveness of these approaches and discuss conditions for their applicability. Our results indicate that the accuracy of the trained neural network models is much higher for the cases where we only have to learn a constitutive relation instead of all dynamics. This finding corroborates the well-known fact from scientific computing that building as much structural information as is available into an algorithm can enhance its efficiency and/or accuracy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

An Integrated Framework for Risk Assessment of Safety-related Digital Instrumentation and Control Systems in Nuclear Power Plants: Methodology Refinement and Exploration

This report documents activities performed by Idaho National Laboratory (INL) during Fiscal Year (FY) 2023 for the U.S. Department of Energy (DOE) Light Water Reactor Sustainability (LWRS) Program, Risk Informed Systems Analysis (RISA) Pathway, digital instrumentation and control (DI&C) risk assessment project. In FY 2019, the RISA Pathway initiated a project to develop a risk assessment strategy for delivering a technical basis to support effective, and secure DI&C technologies for digital upgrades/designs. A risk assessment-informed framework was proposed for this strategy, which aims to (1) provide a best-estimate, risk informed capability to quantitatively estimate the safety margin obtained from plant modernization, especially for safety-related DI&C systems, (2) support and supplement existing risk informed DI&C design guides by providing quantitative risk information and evidence, (3) offer a capability of design architecture evaluation of various DI&C systems, (4) assure the long-term safety and reliability of safety-related DI&C systems, and (5) reduce uncertainty in costs and support integration of DI&C systems in the plant. To achieve these technical goals, the LWRS-developed framework provides a means to address relevant technical issues by: (1) defining a risk informed analysis process for DI&C upgrade that integrates hazard analysis, reliability analysis, and consequence analysis, (2) applying risk informed tools to address common cause failures (CCFs) and quantify corresponding failure probabilities for DI&C technologies, particularly software CCFs, (3) evaluating the impact of digital failures at the component level, system level, and plant level, and (4) providing insights and suggestions on designs to manage the risks, thus to support the development and deployment of advanced DI&C technologies in nuclear power plants (NPPs). Adding diversity within a system or components is the primary means to eliminate and mitigate CCFs, but diversity also increases system complexity and may not address all sources of systematic failures. Optimization of diversity and redundancy applications for the safety-critical DI&C systems remains a challenge. To deal with the technical issues in addressing potential software CCFs in safety-related DI&C systems of NPPs and supporting relevant design optimization, the proposed framework provides: (a) A best-estimate, risk informed capability to address new technical digital issues quantitatively, focusing on software CCFs in safety-related DI&C systems of NPPs; (b) A common and a modularized platform for DI&C designers, software developers, cybersecurity analysts, and plant engineers to predict and prevent risk in the early design stage of DI&C systems; (c) Technical bases and risk informed insights to assist users address the risk informed alternatives for evaluation of CCFs in safety-related DI&C systems of NPPs; and (d) A risk informed tool that offers a capability of design architecture evaluation of various DI&C systems to support system design decisions in diversity and redundancy applications. The research and development efforts of this project in FY 2023 are focused on refining current methods on software CCF modeling and estimation and exploring additional innovative approaches to risk assessment of DI&C systems to enable a more comprehensive and complete assessment of various safety-related DI&C design architectures. The primary audience of this report are DI&C designers, engineers, and probabilistic risk assessment (PRA) practitioners. This includes stakeholders, such as the nuclear utilities and regulators who consider the deployment and upgrade of DI&C systems, DI&C software developers and reviewers, and cybersecurity specialists. It should be noted that all the analyses are performed for the demonstration of the methodology, not for the evaluation of an actual digital control system. Results are obtained based on limited design information and testing data.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2021

Zwicky Transient Facility and Globular Clusters: the Period–Luminosity and Period–Luminosity–Color Relations for Late-type Contact Binaries

In this work, we aimed to derive the gri-band period–luminosity (PL) and period–luminosity–color (PLC) relations for late-type contact binaries, for the first time, located in globular clusters, using the homogeneous light curves collected by the Zwicky Transient Factory (ZTF). We started with 79 contact binaries in 15 globular clusters, and retained 30 contact binaries in 10 globular clusters that have adequate numbers of data points in the ZTF light curves and are unaffected by blending. Magnitudes at mean and maximum light of these contact binaries were determined using a fourth-order Fourier expansion, while extinction corrections were done using the Bayerstar2019 3D reddening map together with adopting the homogeneous distances to their host globular clusters. After removing early-type and “anomaly” contact binaries, our derived gri-band PL and period–Wesenheit (PW) relations exhibited a much larger dispersion with large errors on the fitted coefficients. Nevertheless, the gr-band PL and PW relations based on this small sample of contact binaries in globular clusters were consistent with those based on a larger sample of nearby contact binaries. Good agreements of the PL and PW relations suggested both samples of contact binaries in the local Solar neighborhood and in the distant globular clusters can be combined and used to derive and calibrate the PL, PW, and PLC relations. The final derived gr-band PL, PW, and PLC relations were much improved over those based on the limited sample of contact binaries in the globular clusters.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1972

On the mass-luminosity relation.

The results of a least-squares study of the mass-luminosity relation for eclipsing and visual binary stars consisting of main sequence components are presented. Two methods are discussed. First, the values of the coefficients A and B in the relation log M = A + BM sub BOL are determined. Then a technique which permits the determination of alpha and beta in the relation M = alpha L beta, when only the sum of the masses, and not the individual masses of each component, is known. The results and a comparison of the two methods are discussed. It is found that the following mass-luminosity relation represents the observational data satisfactorily: log M = 0.504 - 0.103 M sub BOL, -8 less than or equal to M sub BOL less than or equal to +10.5. A discussion of the data and of the possibility that separate mass-luminosity relations may exist for visual and eclipsing binaries is given. The possibility that more than one mass-luminosity relation is required in the range -8 less than or equal to M sub BOL less than or equal to +13 is also discussed.

Mccluskey, G. E., Jr.↗
NASA NTRS1981

Relations for the thermodynamic and transport properties in the testing environment of the Langley hypersonic CF4 tunnel

Thermodynamic and transport properties of gaseous CF4 that can be used in flow field computer codes and theoretical analyses for comparison of results with experimental data from the Langley Hypersonic CF4 Tunnel are presented. The thermodynamic relations which are based on thermally perfect but calorifically imperfect gas are adequate for the testing environment of the CF4 tunnel and are simpler to use than the imperfect gas relations required to define the complete expansion from the tunnel reservoir. Relations for the transport properties are based on the kinetic theory of gases in which published experimental data are used in the derivation of the relations. Extensive experimental data were located for viscosity and the derived relation should provide values for viscosity with errors of less than 1 percent. The experimental data for thermal conductivity were limited with significant disagreement between the various sources. The derived relation will probably provide values for the thermal conductivity with errors of no more than 5 percent which is within the accuracy of the experimental data.

Sutton, K.↗
NASA NTRS1985

Correlated peak relative light intensity and peak current in triggered lightning subsequent return strokes

The correlation between peak relative light intensity L(R) and stroke peak current I(R) is examined for 39 subsequent return strokes in two triggered lightning flashes. One flash contained 19 strokes and the other 20 strokes for which direct measurements were available of the return stroke peak current at ground. Peak currents ranged from 1.6 to 21 kA. The measurements of peak relative light intensity were obtained from photographic streak recordings using calibrated film and microsecond resolution. Correlations, significant at better than the 0.1 percent level, were found for several functional relationships. Although a relation between L(R) and I(R) is evident in these data, none of the analytical relations considered is clearly favored. The correlation between L(R) and the maximum rate of current rise is also examined, but less correlation than between L(R) and I(R) is found. In addition, the peak relative intensity near ground is evaluated for 22 dart leaders, and a mean ratio of peak dart leader to peak return stroke relative light intensity was found to be 0.1 with a range of 0.02-0.23. Using two different methods, the peak current near ground in these dart leaders is estimated to range from 0.1 to 6 kA.

Idone, V. P.↗
NASA NTRS1986

Statistical relation between monthly mean precipitable water and surface-level humidity over global oceans

Monthly summaries of atmospheric soundings taken over 17 years from 49 midocean stations at small islands and weather ships distributed over major oceans are examined. Over tropical oceans, precipitable water is found to be a better predictor of surface-level humidity than surface-level air temperature. A statistical relation in the form of a polynomial is derived; from this relation, the monthly-mean, surface-level mixing ratio can be computed from monthly-mean precipitable water. The root-mean-square differences between the measured and derived values were found to be less than 8 x 10 to the -4th over most ocean areas. Such a relation is useful in deriving large-scale evaporation and latent heat flux data from the ocean, using spaceborne observations. The temporal and spatial variabilities of data deviations from this relation are examined. This relation is found to be applicable to all major ocean basins and can be used to monitor interannual variability. Boundary-layer thermodynamics of different air masses are suggested as an explanation of some characteristics of this relation.

Liu, W. T.↗
NASA NTRS1990

Relative effects on stratospheric ozone of halogenated methanes and ethanes of social and industrial interest

Four atmospheric modeling groups have calculated relative effects of several halocarbons (chlorofluorocarbons (CFC's)-11, 12, 113, 114, and 115; hydrochlorofluorocarbons (HCFC's) 22, 123, 124, 141b, and 142b; hydrofluorocarbons (HFC's) 125, 134a, 143a, and 152a, carbon tetrachloride; and methyl chloroform) on stratospheric ozone. Effects on stratospheric ozone were calculated for each compound and normalized relative to the effect of CFC-11. These models include the representations for homogeneous physical and chemical processes in the middle atmosphere but do no account for either heterogeneous chemistry or polar dynamics which are important in the spring time loss of ozone over Antarctica. Relative calculated effects using a range of models compare reasonably well. Within the limits of the uncertainties of these model results, compounds now under consideration as functional replacements for fully halogenated compounds have modeled stratospheric ozone reductions of 10 percent or less of that of CFC-11. Sensitivity analyses examined the sensitivity of relative calculated effects to levels of other trace gases, assumed transport in the models, and latitudinal and seasonal local dependencies. Relative effects on polar ozone are discussed in the context of evolving information on the special processes affecting ozone, especially during polar winter-springtime. Lastly, the time dependency of relative effects were calculated.

Fisher, Donald A.↗
NASA NTRS1991

Relative navigation requirements for automatic rendezvous and capture systems

This paper will discuss in detail the relative navigation system requirements and sensor trade-offs for Automatic Rendezvous and Capture. Rendezvous navigation filter development will be discussed in the context of navigation performance requirements for a 'Phase One' AR&C system capability. Navigation system architectures and the resulting relative navigation performance for both cooperative and uncooperative target vehicles will be assessed. Relative navigation performance using rendezvous radar, star tracker, radiometric, laser and GPS navigation sensors during appropriate phases of the trajectory will be presented. The effect of relative navigation performance on the Integrated AR&C system performance will be addressed. Linear covariance and deterministic simulation results will be used. Evaluation of relative navigation and IGN&C system performance for several representative relative approach profiles will be presented in order to demonstrate the full range of system capabilities. A summary of the sensor requirements and recommendations for AR&C system capabilities for several programs requiring AR&C will be presented.

Kachmar, Peter M.↗
NASA NTRS1993

Complex ridged terrain-related ridge belts on Venus: Global distribution and classification

Fifty-six features were identified in a global survey of complex ridged terrain (CRT)-related ridge belts. Nonrandom aerial distribution was observed with all of the CRT-related ridge belts located to the north of 20 deg S and increasing in number towards the north. In the equatorial highlands the only concentration of CRT-related ridge belts occurs along N and NE Ovda Regio and western Thetis Regio. Major areas of CRT devoid of related ridge belts (Beta, Phoebe, Alpha Regiones, and northern Lada Terra) are regions dominated by rifting, fractures, coronae, and volcanic features. A noticeable concentration of ridge belts is located within a region 20 deg S - 80 deg N, 0 deg - 150 deg E. Three classes of CRT-related ridge belts were defined: (1) ridge belts directly in contact with CRT margins, (2) ridge belts located apart from the CRT boundary, but whose shape and strike are affected by CRT, and (3) ridge belts terminating against a margin of CRT. There does not appear to be any relation between ridge belt class and type of CRT margin. Some of the class 2 and 3 belts of the 20 deg S - 80 deg N, 0 deg - 150 deg E region seem to be continuations of adjacent elongated blocks of CRT and could reflect the hypothesized basement of tessera-like material. Majority of class 1 and 2 ridge belts within this region parallel N of NE boundaries of large CRT plateaus of arc-like arrangements of tesserae. These relationships show that this region was dominated by compressional stresses oriented perpendicular to the CRT boundaries, in N-S/NE-SW direction.

Toermaenen, T.↗
NASA NTRS2000

Measurements and Modeling of Aerosol Absorption and Single Scattering Albedo at Ambient Relative Hum

Uncertainties in the aerosol single scattering albedo have been identified to be an important source of errors in current large-scale model estimates of the direct aerosol radiative forcing of climate. A number of investigators have obtained estimates of the single scattering albedo from a variety of remote sensing and in situ measurements during aerosol field experiments. During the Tropospheric Aerosol Radiative Forcing Observational Experiment (TARFOX, 1996) for example, estimates of the aerosol single scattering albedo were obtained (1) as a best-fit parameter in comparing radiative flux changes measured by airborne pyranometer to those computed from independently measured aerosol properties; (2) from estimates of the aerosol complex index of refraction derived using a combination of airborne sunphotometer, lidar backscatter and in situ size distribution measurements; and (3) from airborne measurements of aerosol scattering and absorption using nephelometers and absorption photometers. In this paper, we briefly compare the results of the latter two methods for two TARFOX case studies, since those techniques provide height-resolved information about the aerosol single scattering albedo. Estimates of the aerosol single scattering albedo from nephelometer and absorption photometer measurements require knowledge of the scattering and absorption humidification (i.e., the increase in these properties in response to an increase in ambient relative humidity), since both measurements are usually carried out at a relative humidity different from the ambient atmosphere. In principle, the scattering humidification factor can be measured, but there is currently no technique widely available to measure the absorption of an aerosol sample as a function of relative humidity. Frequently, for lack of better knowledge, the absorption humidification is assumed to be unity (meaning that there is no change in aerosol absorption due to an increase in ambient relative humidity). This assumption then enters the estimate of the single scattering albedo at ambient relative humidity. To investigate the validity of this assumption we have carried out modeling studies of the absorption humidification factor, assuming that the aerosols contain an insoluble soot core and a coating which determines its hygroscopic growth behavior. The aerosol optical properties are then computed on the basis of the shell/core particle morphology using a Mie-code for concentric shells. From basic physical principles, it is conceivable that aerosol absorption increases when an atmospheric aerosol particle collects a non-absorbing shell, since the soot core is then exposed to an increased (focused) electric field strength. Indeed, our preliminary modeling studies show that the absorption of an atmospheric aerosol particle composed of a soot core and an aqueous sulfuric acid shell may increase by a factor of 50% due to a change in ambient relative humidity from 30 to 95%. We will show how this increased absorption is a function of the initial particle size and soot mass fraction.

Redemann, J.↗
NASA NTRS2001

Evaluation of Relative Navigation Algorithms for Formation-Flying Satellites

Goddard Space Flight Center is currently developing advanced spacecraft systems to provide autonomous navigation and control of formation flyers. This paper discusses autonomous relative navigation performance for formations in eccentric, medium, and high-altitude Earth orbits using Global Positioning System (GPS) Standard Positioning Service (SPS) and intersatellite range measurements. The performance of several candidate relative navigation approaches is evaluated. These analyses indicate that the relative navigation accuracy is primarily a function of the frequency of acquisition and tracking of the GPS signals. A relative navigation position accuracy of 0.5 meters root-mean-square (RMS) can be achieved for formations in medium-attitude eccentric orbits that can continuously track at least one GPS signal. A relative navigation position accuracy of better than 75 meters RMS can be achieved for formations in high-altitude eccentric orbits that have sparse tracking of the GPS signals. The addition of round-trip intersatellite range measurements can significantly improve relative navigation accuracy for formations with sparse tracking of the GPS signals.

Kelbel, David↗
NASA NTRS2004

Identification and Control of Gravity Related Defect Formation During Melt Growth of Electro-Optic Single Crystals Bismuth Silicate(Bi12SiO20)

A research carried out under Award Number NAG8-1487 was aimed at to the design, conduct and analysis of experiments directed at the identification and control of gravitational effects on crystal growth, segregation and defect formation in the Sillenite system: bismuth silicate (Bi(12)SiO(20)). Correlation analyses was conducted in order to establish the influence of gravity related defects introduced during crystal growth on critical, application specific properties. Achievement of the states objective was conducted during the period from Feb. 01, 1998 to Dec. 31, 2003 with the following anticipated milestones: 1. Establishment of capabilities for (a) reproducible Czochralski and Bridgman-type growth of BSO single crystals and (b) for comprehensive analysis of crystalline and chemical defects as well as for selective property characterization of grown crystals (year 1). 2. Design and execution of critical space growth experiment(s) based on analyses of prefatory space results (experiments aimed at establishing the viability of planned approaches and procedures) and on unresolved issues related to growth, segregation and defect formation associated with conventional growth in Bridgman geometries. Comparative analysis of growth under conventional and under mu-g conditions; identification of gravity related defect formation during conventional Bridgman growth and formulation of approaches for their control (years 2 and 3). Development of charge confinement system which permits growth interface demarcation (in a mu-g environment) as well as minimization of confinement related stress and contamination during growth; design of complementary mu-g growth experiments aimed at quantitative mu-g growth and segregation analyses (year 4). 3. Conduct of quantitative mu-g growth experiments directed at: (a) identification and control of gravity related crystalline and chemical defect formation during single crystal growth of Bi(12)SiO(20) and at (b) defect engineering -the development of approaches to the controlled generation during crystal growth of specified point defects in homogeneous distribution (year 5). The proposed research places focus on a class of materials which have outstanding electrical and optical properties but have so far failed to reach their potential, primarily because of our inability to control adequately their stoichiometry and crystal defect formation as well as confinement related contamination and lattice stress.

Becia, Piotr↗
NASA NTRS2002

Effects of perfluorohexane vapor on relative blood flow distribution in an animal model of surfactant-depleted lung injury

OBJECTIVE: To test the hypothesis that treatment with vaporized perfluorocarbon affects the relative pulmonary blood flow distribution in an animal model of surfactant-depleted acute lung injury. DESIGN: Prospective, randomized, controlled trial. SETTING: A university research laboratory. SUBJECTS: Fourteen New Zealand White rabbits (weighing 3.0-4.5 kg). INTERVENTIONS: The animals were ventilated with an FIO(2) of 1.0 before induction of acute lung injury. Acute lung injury was induced by repeated saline lung lavages. Eight rabbits were randomized to 60 mins of treatment with an inspiratory perfluorohexane vapor concentration of 0.2 in oxygen. To compensate for the reduced FIO(2) during perfluorohexane treatment, FIO(2) was reduced to 0.8 in control animals. Change in relative pulmonary blood flow distribution was assessed by using fluorescent-labeled microspheres. MEASUREMENTS AND MAIN RESULTS: Microsphere data showed a redistribution of relative pulmonary blood flow attributable to depletion of surfactant. Relative pulmonary blood flow shifted from areas that were initially high-flow to areas that were initially low-flow. During the study period, relative pulmonary blood flow of high-flow areas decreased further in the control group, whereas it increased in the treatment group. This difference was statistically significant between the groups (p =.02) as well as in the treatment group compared with the initial injury (p =.03). Shunt increased in both groups over time (control group, 30% +/- 10% to 63% +/- 20%; treatment group, 37% +/- 20% to 49% +/- 23%), but the changes compared with injury were significantly less in the treatment group (p =.03). CONCLUSION: Short treatment with perfluorohexane vapor partially reversed the shift of relative pulmonary blood flow from high-flow to low-flow areas attributable to surfactant depletion.

NASA Discipline Cardiopulmonary↗
NASA NTRS2005

Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity

Tensor analysis is one of the more abstruse, even if one of the more useful, higher math subjects enjoined by students of physics and engineering. It is abstruse because of the intellectual gap that exists between where most physics and engineering mathematics leave off and where tensor analysis traditionally begins. It is useful because of its great generality, computational power, and compact, easy to use, notation. This paper bridges the intellectual gap. It is divided into three parts: algebra, calculus, and relativity. Algebra: In tensor analysis, coordinate independent quantities are sought for applications in physics and engineering. Coordinate independence means that the quantities have such coordinate transformations as to leave them invariant relative to a particular observer s coordinate system. Calculus: Non-zero base vector derivatives contribute terms to dynamical equations that correspond to pseudoaccelerations in accelerated coordinate systems and to curvature or gravity in relativity. These derivatives have a specific general form in tensor analysis. Relativity: Spacetime has an intrinsic geometry. Light is the tool for investigating that geometry. Since the observed geometry of spacetime cannot be made to match the classical geometry of Euclid, Einstein applied another more general geometry differential geometry. The merger of differential geometry and cosmology was accomplished in the theory of relativity. In relativity, gravity is equivalent to curvature.

Kolecki, Joseph C.↗