Elements in Mathematics, Economics, Basic Engineering Sciences, and ME Laws

Comprehensive review flashcards featuring essential terminology, mathematical axioms, scientific laws, and economic definitions covering Algebra, Trigonometry, Geometry, Calculus, Physics, Mechanics, and Engineering Economics.

Even symmetry

The type of symmetry exhibited by a function when $f(t) = f(-t)$.

Corollary

A statement of truth which follows with little or no proof from a theorem.

Ogive

The graphical presentation of a cumulative frequency distribution plotted at class marks and connected by straight lines.

Trivial solution

The classification given to roots of an equation when they are all zero.

Symmetric axiom

The algebraic axiom stating that if $a = b$, then $b = a$.

Dyadic number system

Another name for the binary number system, which uses place values with base $2$.

Argand diagram

A rectangular coordinates diagram used to represent the complex plane.

Postulate

A proposition concerning geometric constructions and objects whose possibility is admitted without proof.

Lemma

A proved proposition or ancillary theorem that is useful mainly as a preliminary step in proving another theorem.

Conjecture

A mathematical statement which has neither been proved nor denied by counterexamples.

Constant of variation

The ratio or product of two expressions in direct or inverse relation with each other.

Literal numbers

Numbers or known quantities represented by letters.

Unit fraction

A common fraction with unity ($1$) as the numerator and a positive integer as the denominator ($\frac{1}{n}$).

Square-free integer

A positive integer that contains no perfect-square factor greater than $1$.

Perfect number

An integer that is equal to the sum of all its proper divisors excluding the number itself (the smallest being $6$).

Amicable numbers

Also called friendly numbers; a pair of integers where each number is equal to the sum of all proper divisors of the other (smallest pair is $220$ and $284$).

Goldbach conjecture

The mathematical conjecture asserting that every even integer greater than $2$ can be expressed as the sum of two prime numbers.

Fundamental theorem of arithmetic

The theorem stating that every positive integer greater than $1$ is a prime or can be expressed as a unique product of prime powers.

Antecedent

The first term of a ratio.

Consequent

The second term of a ratio.

Cramer's rule

A method of simultaneously solving a system of linear equations with several unknowns by using determinants.

Surd

An irrational root of a positive integer or fraction, equivalent to a non-terminating, non-repeating decimal.

Pure quadratic equation

A quadratic equation of the form $ax^2 + c = 0$, lacking a first-degree term.

Discriminant

The value $B^2 - 4AC$ in a quadratic equation that determines the nature of its roots.

Harmonic progression

A sequence of numbers whose reciprocals form an arithmetic progression.

Pascal's triangle

A triangular array of numbers where each entry is the sum of the two numbers directly above it, providing the coefficients for a binomial expansion.

Permutation

An arrangement of objects in a specific order.

Combination

An arrangement or selection of objects regardless of order.

Evolution

The algebraic operation of root extraction.

Involution

The algebraic operation of raising a quantity to an integral power.

Diophantine equations

Equations that are restricted exclusively to having integer solutions.

Coaltitude

The angular distance of a point on the terrestrial sphere measured from the North Pole.

Centroid

The point of concurrency where the three medians of a triangle intersect.

Orthocenter

The point of concurrency where the three altitudes of a triangle intersect.

Circumcenter

The point of concurrency of the perpendicular bisectors of the sides of a triangle.

Incenter

The point of concurrency of the angle bisectors of a triangle, which is also the center of its inscribed circle.

Cologarithm

The logarithm of the reciprocal of a number, which equals the negative of the logarithm of that number.

Pedal triangle

A triangle inscribed in a given triangle whose vertices are the feet of the three perpendiculars dropped to the sides from an interior point; it possesses the minimum perimeter for an inscribed triangle.

Primitive triangle

A right triangle whose side lengths can be expressed as a ratio of integral units.

Colure

One of two great circles intersecting at right angles at the celestial poles that divide the equinoctial points and ecliptic into four parts.

Zenith

The point where a straight line from the center of the Earth through the observer's position intersects the celestial sphere.

Nadir

The celestial point diametrically opposite the zenith.

Annulus

A plane figure bounded by two concentric circles.

Apothem

The perpendicular line from the center of a regular polygon to any of its sides, equal to the radius of its inscribed circle.

Dihedral angle

The angle formed by the divergence between two intersecting planes.

Gon

Also called a grad or centesimal degree; an angular unit equal to $\frac{1}{400}$ of a complete revolution.

Deflection angle

Another term for an exterior angle of a polygon, formed by prolonging one side and the adjacent side.

Gnomon

The geometric figure remaining after a parallelogram has been removed from one corner of a larger similar polygon.

Platonic solids

The five regular polyhedra whose faces are all identical regular polygons (tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron).

Cavalieri's principle

The principle stating that if two solids have equal cross-sectional areas for every plane parallel to a given reference plane, they have equal volumes.

Golden ratio

The division of a segment into parts $a$ and $b$ ($a > b$) such that $\frac{a+b}{a} = \frac{a}{b}$.

Abscissa

The $x$-coordinate of a point representing its distance from the $y$-axis.

Ordinate

The $y$-coordinate of a point representing its distance from the $x$-axis.

Latus rectum

A chord passing through the focus of a conic section perpendicular to its principal axis.

Asymptote

A straight line which a curve approaches closer and closer to but never touches except at infinity.

Helix

A space curve generated by a point moving with uniform circular motion about an axis while advancing at constant speed parallel to the axis.

Catenary

The curve formed naturally by a uniform, perfectly flexible chain hanging freely between two supports (described by $y = \cosh(x)$).

Evolute

A curve that is the locus of the centers of curvature of another curve, forming the envelope of all its normals.

Envelope

A curve or surface that is tangential to each member of a family of curves or surfaces.

Point of inflection

A point on a curve where the second derivative is zero ($f''(x) = 0$) and the curve changes concavity from upward to downward or vice versa.

Extrema

The collective mathematical term for relative or absolute maxima and minima of a function.

First proposition of Pappus

A theorem stating that the area of a surface of revolution equals the product of the generating curve's length and the distance traveled by its centroid.

Second proposition of Pappus

A theorem stating that the volume of a solid of revolution equals the generating plane area multiplied by the circumference traced by its centroid.

Plasticity

The property or quality of a material that allows it to undergo permanent, irreversible elongation without fracture.

Elasticity

The property by virtue of which a deformed body returns to its original dimensions when the deforming forces are removed.

Coefficient of restitution

The negative ratio of the relative velocity of separation after impact to the relative velocity of approach before impact ($e$).

Pascal's principle

The principle stating that pressure applied to an enclosed, confined fluid is transmitted undiminished in all directions to every point of the fluid.

Archimedes' principle

The physical law stating that any body submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced.

Terminal speed

The constant downward velocity reached by a falling object when the upward force of air resistance balances its downward weight.

Modulus of elasticity

Also called Young's modulus; the ratio of tensile or compressive stress to corresponding axial strain in the linearly elastic region.

Modulus of rigidity

Also called shear modulus; the ratio of shear stress to shear strain within the elastic limit.

Poisson's ratio

The ratio of lateral unit strain to longitudinal unit strain under uniaxial stress within the proportional limit.

Slenderness ratio

The ratio of a column's unsupported length to its least radius of gyration.

Modulus of resilience

A mechanical property quantifying the capacity of a material to absorb and store strain energy elastically without permanent damage.

Monopsony

A market situation characterized by having only one buyer for a product or service with no viable substitutes.

Ordinary annuity

A series of equal periodic payments where each payment is made at the end of every payment period.

Deferred annuity

An annuity where the series of equal periodic payments begins only after several periods have passed.

Perpetuity

A series of uniform periodic payments that continue indefinitely over an infinite period of time ($\text{Present worth} = \frac{A}{i}$).

Book value

The recorded net worth of an asset on an enterprise's accounting records, equal to original cost minus accumulated depreciation.

Salvage value

The estimated value of a property or asset at the conclusion of its useful economic life.

Sunk cost

An unrecoverable past expenditure representing the difference between the recorded book value and actual resale value.

Acid-test ratio

An index of short-term debt-paying ability calculated as the ratio of current assets (excluding inventory) to total current liabilities.

Working capital

The liquid funds and short-term capital required to finance ongoing operations and make an enterprise a going concern.

Debenture bond

A type of corporate bond that carries no specific collateral lien other than the general credit and promise to pay of the issuing company.

Collateral trust bond

A bond backed by securities, such as stocks or bonds of subsidiaries, owned and pledged by the issuing corporation.

Economic Order Quantity (EOQ)

The optimal order quantity that minimizes the total inventory holding and ordering costs per unit time without shortages.

Tort

A civil wrong committed by an individual that damages another person's property, person, or reputation.

Condonation

A mode of extinguishing obligations that occurs when a creditor abandons or forgives the right to collect a debt.

R.A. 8495

The Philippine Mechanical Engineering Act of 1998 approved on February 12, 1998, regulating the practice of mechanical engineering.