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PSCI 702. Preliminaries September 7, 2005. Table of Content. Algorithms Sets and Groups Scalars and Vectors Matrices Coordinate Systems Coordinate Transformations Operators. Algorithm.
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PSCI 702 Preliminaries September 7, 2005
Table of Content • Algorithms • Sets and Groups • Scalars and Vectors • Matrices • Coordinate Systems • Coordinate Transformations • Operators
Algorithm • “An algorithm is simply a set of finite mathematical operations which, when performed in sequence, lead to the numerical answer to some specific problem.”
Properties of Algorithms: • Finiteness: Algorithm must complete after a finite number of instructions been executed. • Clarity: Each step must be clearly defined, having only one interpretation. • Sequential: Each step has a unique preceding and succeeding step. • Feasibility: All instructions must be able to be performed. Illegal operations are not allowed • Input: 0 or more data values. • Output: 1 or more results.
Sets and Groups • A set is a collection of elements. • Elements are related by some “Law” (‡). • If a, b and c a “set”, where a‡b=c => the set is closed with respect to ‡. • A set with the following properties is called a group: • a‡i=a (unit element) • a-1‡a=I (inverse) • a‡(b‡c)=(a‡b) ‡c (associativity) • ‡ is communitative if a a‡b=b‡a. • ‡ and ^ are distributive if a‡(b^c)=(a‡b)^(a‡c). • Subsets that form a group under addition and scalar multiplication are called fields.
Scalars and Vectors • Scalars are quantities with magnitude. • Vectors are quantities that require more than one number for its specification. Vectors have magnitude and direction. • The number of components that are required for the vector’s specification is called the dimensionality of the vector.
Scalar and Vector Products • Scalar product: • a * b = c • Vector products: • Scalar product or dot product: A.B=c or • Cross product: AxB=C or
Matrices • Matrices are two dimensional vectors with m columns and n rows. • Matrix product is defined as: • AB=C where • Unit Matrix:
Matrices • Inverse: AA-1=1 • Transpose: AT=Aji • Trace of a Matrix: TrA=∑I Aii • Symmetric Matrix: Aij=Aji • Conjugate Transpose: A† • Unitary Matrix: A-1=A† • Normal Matrix: AA†=A†A
Matrices • Determinant is a single parameter that can be used to characterize the matrix.
Coordinate Systems • If the vectors that define the space are locally perpendicular, they are said to form an Orthogonal coordinate frame. • Cartesian (x,y,z) • Cylindrical (ρ,ξ,z) • Polar (r,θ,φ)
Operators • An Operator is a set of instructions represented by a symbol. • Scalar operators such as [d/dx]. • Vector operators such as • Divergence: • Gradient: • Curl: