Transformations In Polar Firn

Author : MANJU SHARMA
Publisher : Ram Prasad Publications(R.P.H.)
ISBN 13 :
Total Pages : 206 pages
Book Rating : 4./5 ( download)

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Download or read book LINEAR ALGEBRA written by MANJU SHARMA and published by Ram Prasad Publications(R.P.H.). This book was released on with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: Unit-1 1. Analytic Functions, Cauchy-Riemann Equations, Harmonic Functions 1-40 Complex Number System 1; Complex Numbers as Ordered Pairs 1; The Polar Form 1; Function of a Complex Variable 2; Single Valued Function(or Uniform Function) 2; Multiple-Valued Function(or Many-Valued Function) 3; Limit of a Function 3; Theorems on Limits 3; Continuity 3; Fundamental Operations as Applied to Continuous Function 4; Continuity in Terms of Real and Imaginary Parts of f(z) 4; Uniform Continuity 4; Differentiability of a Complex Function 5; Geometric Interpretation of the Derivative 5; Partial Derivative 6; Analytic Function 6; The Necessary Conditions for f(z) to be Analytic [(Cauchy-Riemann Equations (C-R Equations)] 6; The Sufficient Condition for f(z) to be Analytic 8; Polar Form of Cauchy-Riemann Equations 9; Derivative of w in Polar Form 11; Functions of a Function 12; Derivative of a Function of a Function 12; Inverse Function 12; Laplace Equation 13; Harmonic Function 13; Theorem 13; Conjugate Harmonic Functions 14; Theorem 14; Determination of the Conjugate Function 14; To Construct a Function f(z) when One Conjugate Function is Given 15; Orthogonal System 16; Theorem 16. 2. Mobius Transformation, Cross Ratio 41-68 Elementary Functions 41; Mapping or Transformation 43; Definition of Mapping 43; Mobius Transformation or Bilinear Transformation or Fractional Transformation 43; Inverse Transformation 44; Critical Points and Critical Mapping 44; Resultant or Product of Two Mobius Transformations (Group Property) 45; Some Theorems 46; Fixed Points (or Invariant Points) of Mobius Transformation 47; Theorem 48; Cross Ratio 48; Some Theorems 49; The Circle 54; Inverse Points with Respect to a Circle 55; To find the Relation between the Inverse Points with Respect to the Circle 55; Nature of Transformations (Elliptic, Hyperbolic and Parabolic Transformations) 56; Some Theorems 58. 3. Vector Space 69-123 Vector Space 69; Various Notations 70; General Properties (Elementary Properties) of Vector Spaces 70; Vector Subspace 77; Union and Intersection of Subspaces 83; Sum of Subspaces 85; Some Definitions 91; Basis of a Vector Space 91; Dimension of a Vector Space 92; Finite Dimensional Vector Space 102; Some Theorems on Finite Dimensional Spaces 102; Quotient Space 115. 4. Linear Transformation 124-158 Definition 124; Purpose 124; Image 124; Existence and Uniqueness 124; Types of Linear Transformation 127; Determining whether a Mapping is Linear Transformation or Not 127; Isomorphism of Vector Spaces 133; Theorems on Isomorphism 134; Kernel of Linear Transformation T or Kernel of a Homomorphism T 142; Theorem 142; Range of a Linear Transformation 143; Theorem 143; Lemma 144; Sylvester Law of Nullity [Rank-Nullity Theorem] 144; Fundamental Theorem of Vector Space Homomorphism 146. 5. Inner Product Spaces 159-200 Inner Product 159; Usual or Standard Inner Product 159; Inner Product Space 162; Theorems 162; Some Important Terms about Vectors 168; Norm or Length of a Vector a in an Inner Product Space 168; Theorems 169; Orthogonality 170; Orthogonal Complement 171; Orthogonal Basis and Orthonormal Basis 172; Gram-Schmidt Orthogonalization Process 172; Theorems on Orthogonal/ Orthonormal Bases 173; Cauchy-Schwarz’s Inequality or Schwarz’s Inequality 186; Bessel’s Inequality 188; Normed Vector Space or Normal Vector Space 194; Distance in an Inner Product Space 195.