T

T

Tan : Abbreviation of tangent.

Tangent

A line that meets a smooth curve at a single point and does not cut across the curve. ii) In Trigonometry, the tangent is function defined as tan x = sinx / cosx. In a right angle triangle, the tangent of an angle θ is the ratio of the length of the sides opposite and adjacent to the angle. Tan θ = Perpendicular / Base. Value of Tan varies from  0 to infinity for the angles from zero to 90⁰

Tangent Line : The tangent line is represented a the limit of secant line.

Tangent Plane : A Tangent plane is a plane that touches a surface defined in 3dimensional coordinate system at a given point p. The normal vector of the tangent plane is perpendicular to the surface at the point p. the surface should be smooth near the  surface at the point p. the tangent plane contains all the lines tangent to a specific point on a surface.  

Tanh : Hyperbolic of tan.

Tautology : A statement of proportional logic which can be inferred from any proposition whatsoever  that is true because of its logical structure.

Taylor’s Series :  According to Taylor’s series or theorem, a polynomial f(x) can be expanded as given below around a point ‘a’ valid for an integer n .

F(x) = f(a) + f’(a) 9 x –a) + f”(a) (x – a)² / 2! + f”’(a) (x – a)³ / 3! + ………+ fⁿ(a) (x – a)ⁿ / n! + ……..

When a = 0 , the series is called Maclaurin series .

Tensor :  A set of quantities that obey certain transformation laws relating the bases in one generalized coordinate system to those of another and involving partial derivative sums . vectors are simple tensors.

Tera : Symbol T. A prefix having meaning 10¹².

Term : A mathematical expression of the form rxⁿ, pxⁿy^r, where n,p,r are numbers and x and y are variable.

Terminating :To come to an end. E.g., 8.5 is a terminating decimal, while 1.733333.. is non terminating decimal.

Terminating continued fraction : A continuous fraction having finite number of terms.

Terminating decimal : A decimal that terminates is a terminating decimal. All Terminating decimal are Rational numbers but not  all rational numbers are terminating decimal expression .

Ternary number system : A number system whose base is 3.

Terrestrial triangle : The Terrestrial triangle is formed on the earth by the great circles connecting the pole and two palces on the earth.

Tessellation : A Tessellation is a repeated geometric design that covers a plane without gaps or overlaps.

Tetrahedron : A polyhedron with four faces.

Tetromino : A four-square polyomino.

THEOREM : A Theorem is a statement which is stated in natural language and can be proved on the basis explicitly states or precisely known confusions.

Theorem of Parallel axis : The moment of inertia of any object about an axis through its centre of mass is the minimum moment of inertia for an axis in that direction in space. The moment of inertia about any axis parallel to the axis is given by

I_pa  =  I_cm  + Md²

Where I_pa  is moment of inertia of the body about an axis passing through centre of mass, M is the body of mass and d is the distance between the two axes.

Theoretical Probability : The chances of events happening as determined by calculating results that would occur under ideal circumstances.

Theory of Equations : Theory of equations includes the study of equations, polynomials , algebraic expressions, algebraic equations, separation of roots, approximation of roots and the application of matrices and determinants for solving the equations etc.

Theory of Games : The study of ways in which strategic interactions among rational players produce outcomes with respect of the preferences of the players, none of which might have been introduced by any one of them.

Theory of Indices : Theory of indices describes the laws for multiplication and division of quantities having same base as given below.

 1.a^m aⁿ  = a^{m + n}                   2^.  a^m /aⁿ  = a^{m -  n}            3. (a^m )ⁿ  = a^{m n}                4. (ab)ⁿ  = aⁿ b ⁿ

5. (a/b)ⁿ  = aⁿ /b ⁿ              6. a⁰ =1                       7.   a^-m =1/a^m                                  8. a^m/n =  ^m

Theory of Probability : A theory of Probability is a way of understanding probability statements. A theory of probability connects the mathematics of probability, which is the set of consequences of axioms of probability, with real world of observation and experimentation.

Therm : A unit of energy equal to 10⁵.

Time series : Times series analysis accounts for the fact that data points taken over time may have an internal structure ( such as autocorrelation, correlation, trend or seasonal variation ) that should be accounted for the value of the variable is generated successively in time. A continuous graph trace is an example of continuous time series.

Ton : 1. Long Ton – unit of weight used in U.K., equal to 2240 lbs or 1016 kg. (2) short Ton  ; unit of weight used in U.S.A., equal to 2000 pounds or 907.18 kg. (3) Metric Ton  : usually referred to as tone equal to 1000 kg.

Topological space : A set with topology defined upon it is called a topological space .

Topology :  Generally Topology is the study of those proportions of space which are invariant under continuous deformations, that is , deformations which do not create’tears’ or ‘holes’. More specifically, given set X , a Topology on X is a collection of subsets of X, such that the empty set and X itself are included in the collection, and such that the collection is closed under the formation of finite intersection and arbitrary unions.

Torque : Symbol T, Torque about an axis is the product of force and the distance of its line of action from the axis

Toeplitz matrix : A matrix in which all the elements are the same along any diagonal that slopes from northwest to southeast.

Torus : A geometric solid in the shape of a donut.; A surface of revolution generated by revolution of a circle in three dimensional spaces about an axis passing through its centre and perpendicular to the plane of the circle and it do or not touch the circle .

Total Derivative : The rate of change of a function of two or more variables with respect to a paprameter on which these variables depend.

If Z  = f(x,y,t), x=  θ(t)   and y =  Ø(t), the total derivative of z with respect to t.

Total Differential : In analogy with the differential of a function of one variable, the total differential of a function Z = f( x,y) of more than one variable is

Dz =  dx  +   dy 

Totally ordered Set : A set with total order defined on it. An order relation on a set S is a total order exactly if, for any two elements x and y of S, either x< y , x = y or y< x , but no two of the three.

Trace : The trace of a matrix is the sum of the terms along the principal diagonal.

Transcendental Function : A function that cannot be expressed as algebraic function. E.g. , In x, sinx, e^x etc., and any function containing them are Transcendental functions.

Transcendental  number : A real or complex number that is not algebraic.

Transform of Element of Group : The transform of an element ‘K’ by an element ‘X’ is X^-1 is X^-1 KX.

Transform of Matrix : A matrix B related to a given matrix A by the expression B = Q^-1 AQ, Q being a non-singular matrix. Eigen values remain same.

Transformation : The changing of an algebraic expression or equation into an equivalent one with a different form.

Transformation of Axis : Let the coordinate of a point be (x,y,z) in a coordinate system of axes. The axis can be transformed in three ways. One , the origin is shifted keeping the axes parallel. Two , the origin is kept as it is , but the axes are rotated. Third , the origin shifting and rotation of axes take place simultaneously. The coordinate of the point with respect to new coordinate axes will be different.  

Transformation of Coordinate : A coordinate transformation is conversion from one system to another, to describe the same space. E.g. Cartesian coordinates may be converted to spherical coordinates or vice versa.

Transitive Property : The property that states that if a=b and b = c , then a =c .

Transitive relation : A binary relation R over a set X is Transitive if it holds for a,b,c in X, that is . if a is related to b and b is related to c, then a is related to c . e.g. a>b, and b>c , then a>c  ( > relation is transitive ).

Transitive Set : A set is called Transitive  if it is equal to its Transitive closure. That is, X is a Transitive set if whenever y is an element of X and z is an element of Y, then z is an element of X.

Translation :  A function which we obtain if we add the same constant to each value of the variable of the given function and move the graphs of the function by a constant distance.

Transpose: To move a term from one side of an equation to another side , with change of sign.

Transpose of Matrix : The Transpose of a matrix A is obtained by interchanging its rows and columns. It is usually denoted by A^T, if A is the original matrix.

Transversal : A line that intersects two or more other lines in the same plane.

Trapezium : A quadrilateral in which no sides are parallel.

Trapezoid : A quadrilateral in which two sides are parallel.

Tree : A tree is a graph with the property that there is a unique path from any vertex to any other vertex traveling along the edges.

Triangle : A geometric figure with three sides.

Triangular number : A number of the form n(n+1)/2.

Triangular prism : A polyhedron-a prism with two identical, triangular bases.

Trinomial : An algebraic expression consisting of 3 terms.

Tromino : A three-square polyomino.

Truncated pyramid : A section of a pyramid between its base and a plane parallel to the base.

Twin primes : Two prime numbers that differ by 2. For example, 11 and 13 are twin primes.