GATE XH key terms

A state function measuring how energy disperses among microscopic arrangements at a given temperature. In an isolated system it never decreases, bounding the efficiency of heat engines and chemical reactions.

A thermodynamic potential combining enthalpy and entropy, G = H − TS. Its decrease at constant temperature and pressure gives the maximum non-expansion work a process can deliver, and its sign predicts spontaneity.

The maximum fraction of heat convertible to work by any engine operating between two reservoirs, equal to 1 − Tc/Th with absolute temperatures. It sets an unattainable upper bound for real cycles.

The ratio of product to reactant activities at equilibrium, each raised to its stoichiometric coefficient. Its value at a given temperature follows from the standard Gibbs free energy change of reaction.

A rule stating that a system at equilibrium shifts to oppose an imposed change in concentration, pressure, or temperature. It qualitatively predicts the direction, though not the magnitude, of the resulting adjustment.

Energy conservation applied to thermodynamic systems: the heat added to a system equals the work it performs plus the change in its internal energy. It defines internal energy as a state function.

A state function defined as H = U + PV, useful because at constant pressure its change equals the heat exchanged. It is the natural energy measure for open processes such as combustion and phase change.

For a square matrix, a nonzero vector that the matrix merely scales, with the scaling factor called the eigenvalue. They reveal principal directions, stability, and vibrational modes of linear systems.

A factorisation of any rectangular matrix into rotation, diagonal scaling, and rotation. The diagonal entries are non-negative singular values whose squares give the eigenvalues of the matrix times its transpose.

The dimension of the vector space spanned by a matrix's columns, equal to the number of independent rows. It determines whether a linear system has a unique, infinite, or no solution.

An algorithm that reduces a linear system to row echelon form using elementary row operations. It yields solutions, rank, and determinant sign, and is the basis of LU decomposition.

The polynomial equation det(A − λI) = 0 whose roots are the eigenvalues of a square matrix. Its coefficients relate to trace and determinant through symmetric functions.

A result stating that sums or averages of many independent, identically distributed random variables approach a normal distribution as sample size grows, regardless of the underlying distribution's shape.

The standard deviation of a sample statistic's sampling distribution, commonly estimated as s/√n for a mean. It quantifies how much an estimate would vary across repeated samples.

A range of plausible values for an unknown parameter, constructed so that a stated proportion of such intervals would contain the true value under repeated sampling. Width shrinks with larger samples.

A procedure comparing observed data against a null model by computing a test statistic and a p-value. A small p-value indicates data inconsistent with the null hypothesis.

A Type I error rejects a true null hypothesis, while a Type II error fails to reject a false one. Their probabilities trade off against each other for a fixed sample size.

The probability that a test correctly rejects a false null hypothesis, equal to one minus the Type II error rate. It increases with larger effect size, larger samples, and lower noise.

An integral operation converting a function of time or space into its frequency spectrum. It decomposes signals into sinusoidal components and underlies filtering, imaging, and spectral analysis.

A sampling rule requiring at least two samples per period of the highest frequency present, otherwise aliasing occurs. It also names a stability criterion for feedback loops in control theory.

An operation combining two functions to produce a third, representing how one modifies the other across a shift. In linear systems the output equals the input convolved with the impulse response.

The ratio of output to input in the Laplace or frequency domain for a linear, time-invariant system under zero initial conditions. Its poles and zeros determine stability and dynamic response.

A pair of graphs showing a system's magnitude and phase response against logarithmic frequency. Gain margin and phase margin read from it indicate closed-loop stability.

Stress is internal force per unit area within a material; strain is the resulting fractional deformation. Their relationship, linear in the elastic regime, defines a material's stiffness.

The slope of the linear elastic portion of a stress–strain curve in tension or compression. It measures a material's resistance to elastic axial deformation.

The stress at which a material begins permanent plastic deformation, often reported as the 0.2 percent offset proof stress. Beyond it, unloading leaves residual strain.

The negative ratio of transverse contraction to axial extension in a uniaxially loaded elastic material. Typical metals lie near 0.3; auxetic materials show negative values.

For a cross-section, the second moment of area about an axis that measures resistance to bending; for a rotating body, resistance to angular acceleration. Both depend on axis choice.

Normal stress in a beam computed as My/I, where M is bending moment, y distance from the neutral axis, and I the second moment of area. It varies linearly across the section depth.

A dimensionless ratio of inertial to viscous forces in fluid flow, equal to ρVD/μ. Low values indicate laminar flow, high values turbulent flow, with transition around a geometry-specific threshold.

A statement of energy conservation along a streamline for steady, inviscid, incompressible flow: pressure plus kinetic and potential energy terms remain constant. It links speed and pressure.

A conservation statement that mass entering a control volume equals mass leaving plus accumulation. For steady incompressible flow it reduces to constant area times velocity along a streamtube.

The thin region near a solid surface where velocity rises from zero at the wall to the free-stream value. Its thickness and separation govern drag and lift.

An ideal reversible cycle of two isothermal and two adiabatic processes operating between hot and cold reservoirs. It establishes the maximum possible thermal efficiency for given temperatures.

A notation describing an asymptotic upper bound on an algorithm's growth rate in time or space. It abstracts away constants and lower-order terms to compare scalability.

A characterisation of how an algorithm's resource use grows with input size, usually expressed in asymptotic notation for best, average, or worst cases. It guides algorithm selection at scale.

In relational databases, organising tables to remove redundancy and update anomalies by satisfying successive normal forms based on functional dependencies. It trades some query joins for integrity.

Four guarantees for database transactions: atomicity, consistency, isolation, and durability. Together they ensure that concurrent failures cannot leave partially applied or corrupted state.

A binary tree where each node's key exceeds all keys in its left subtree and is less than all in its right subtree. Balanced variants give logarithmic search, insertion, and deletion.