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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
NASA NTRS2000

Convergent properties of vestibular-related brain stem neurons in the gerbil

Three classes of vestibular-related neurons were found in and near the prepositus and medial vestibular nuclei of alert or decerebrate gerbils, those responding to: horizontal translational motion, horizontal head rotation, or both. Their distribution ratios were 1:2:2, respectively. Many cells responsive to translational motion exhibited spatiotemporal characteristics with both response gain and phase varying as a function of the stimulus vector angle. Rotationally sensitive neurons were distributed as Type I, II, or III responses (sensitive to ipsilateral, contralateral, or both directions, respectively) in the ratios of 4:6:1. Four tested factors shaped the response dynamics of the sampled neurons: canal-otolith convergence, oculomotor-related activity, rotational Type (I or II), and the phase of the maximum response. Type I nonconvergent cells displayed increasing gains with increasing rotational stimulus frequency (0.1-2.0 Hz, 60 degrees /s), whereas Type II neurons with convergent inputs had response gains that markedly decreased with increasing translational stimulus frequency (0.25-2.0 Hz, +/-0.1 g). Type I convergent and Type II nonconvergent neurons exhibited essentially flat gains across the stimulus frequency range. Oculomotor-related activity was noted in 30% of the cells across all functional types, appearing as burst/pause discharge patterns related to the fast phase of nystagmus during head rotation. Oculomotor-related activity was correlated with enhanced dynamic range compared with the same category that had no oculomotor-related response. Finally, responses that were in-phase with head velocity during rotation exhibited greater gains with stimulus frequency increments than neurons with out-of-phase responses. In contrast, for translational motion, neurons out of phase with head acceleration exhibited low-pass characteristics, whereas in-phase neurons did not. Data from decerebrate preparations revealed that although similar response types could be detected, the sampled cells generally had lower background discharge rates, on average one-third lower response gains, and convergent properties that differed from those found in the alert animals. On the basis of the dynamic response of identified cell types, we propose a pair of models in which inhibitory input from vestibular-related neurons converges on oculomotor neurons with excitatory inputs from the vestibular nuclei. Simple signal convergence and combinations of different types of vestibular labyrinth information can enrich the dynamic characteristics of the rotational and translational vestibuloocular responses.

Non-NASA Center↗
NASA NTRS2017

Microphysical Properties of Snow and Their Link to Ze-S Relations During BAECC 2014

This study uses snow events from the Biogenic Aerosols-Effects on Clouds and Climate (BAECC) 2014 campaign to investigate the connection between properties of snow and radar observations. The general hydrodynamic theory is applied to video-disdrometer measurements to retrieve masses of falling ice particles. Errors associated with the observation geometry and the measured particle size distribution (PSD) are addressed by devising a simple correction procedure. The value of the correction factor is determined by comparison of the retrieved precipitation accumulation with weighing-gauge measurements. Derived mass-dimensional relations are represented in the power-law form m = amDbm. It is shown that the retrieved prefactor am and exponent bm react to changes in prevailing microphysical processes. From the derived microphysical properties, event-specific relations between the equivalent reflectivity factor Ze and snowfall precipitation rate S (Ze = 5 azsSbzs ) are determined. For the studied events, the prefactor of the Ze-S relation varied between 53 and 782 and the exponent was in the range of 1.19-1.61. The dependence of the factors azs and bzs on the m(D) relation and PSD are investigated. The exponent of the Ze-S relation mainly depends on the exponent of the m(D) relation, whereas the prefactor azs depends on both the intercept parameterN0 of the PSDand the prefactors of them(D) andy(D) relations. Changes in azs for a given N0 are shown to be linked to changes in liquid water path, which can be considered to be a proxy for degree of riming.

von Lerber, Annakaisa↗
NASA NTRS2017

A Trilateration Scheme for Relative Positioning

We introduce a trilateration scheme that evaluates the 3-dimensional (3-D) relative position between a reference spacecraft and a target spacecraft using raw-range measurements from a distance baseline of known locations, which we call “anchors”. The anchors can be antennas of a ground-based network (e.g., Deep Space Network (DSN) or Near Earth Network (NEN) stations), or satellites of a spacebased network (e.g., global positioning system (GPS) or tracking and data relay satellite (TDRS)). We define raw-range as the range that includes all the systematic errors that occur during range measurements. A unique feature of this approach is that accurate relative position is derived from a “differencing function” of raw-range measurements of the reference spacecraft and target spacecraft, thereby eliminating most of the systematic errors, such as media effects, ephemeris errors, instrument delays, clock bias, etc. There can be an arbitrary number of target spacecraft, and relative positioning of target spacecraft with respect to the reference spacecraft can be done simultaneously. In this paper, we first assume an idealized system in which clocks on the reference and target spacecraft are synchronized, with clocks of the anchors synchronized as well. We develop a novel iterative algorithm that computes the relative position of the target spacecraft with respect to the reference spacecraft. We illustrate the relative positioning method using the scenario of a network of three ground stations (i.e., the anchors) at Goldstone, California, USA, Madrid, Spain, and Marlargue, Argentina tracking two spacecraft at geosynchronous orbit distance. We demonstrate that the algorithm converges to submeter accuracy in estimating the relative position, in the presence of random errors and systematic errors in raw-range measurements, and in the presence of angular errors in estimating the pointing vectors between the anchors and the reference spacecraft. Next, we relax the requirement of perfect time synchronization between spacecraft, and show that by using an additional anchor, one can estimate and remove the clock biases between the reference and target spacecraft. We add a ground station at Kourou to the above example of three ground stations of Goldstone, Madrid, and Marlargue, and demonstrate that the updated algorithm also converges to meter-level accuracy (submeter in some cases) in the presence of clock biases in addition to the random errors, systematic errors, and angular errors as shown in the above case. We compare this scheme with a similar trilateration scheme for relative positioning scheme first proposed by Montenbruck in 2002.

Cheung, Kar-Ming↗
DOE OSTI2024

Precise relative magnitude measurement improves fracture characterization during hydraulic fracturing

SUMMARY Microseismic monitoring is an important technique to obtain detailed knowledge of in-situ fracture size and orientation during stimulation to maximize fluid flow throughout the rock volume and optimize production. Furthermore, considering that the frequency of earthquake magnitudes empirically follows a power law (i.e. Gutenberg–Richter), the accuracy of microseismic event magnitude distributions is potentially crucial for seismic risk management. In this study, we analyse microseismicity observed during four hydraulic fracture treatments of the legacy Cotton Valley experiment in 1997 at the Carthage gas field of East Texas, where fractures were activated at the base of the sand-shale Upper Cotton Valley formation. We perform waveform cross-correlation to detect similar event clusters, measure relative amplitude from aligned waveform pairs with a principal component analysis, then measure precise relative magnitudes. The new magnitudes significantly reduce the deviations between magnitude differences and relative amplitudes of event pairs. This subsequently reduces the magnitude differences between clusters located at different depths. Reduction in magnitude differences between clusters suggests that some attenuation-related biases could be effectively mitigated with relative magnitude measurements. The maximum likelihood method is applied to understand the magnitude frequency distributions and quantify the seismogenic index of the clusters. Statistical analyses with new magnitudes suggest that fractures that are more favourably oriented for shear failure have lower b-value and higher seismogenic index, suggesting higher potential for relatively larger earthquakes, rather than fractures subparallel to maximum horizontal principal stress orientation.

58 GEOSCIENCES↗
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 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)↗
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.↗