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Tensor Calculus Homework Answers

Tensor analysis is a branch of mathematics that is mainly concerned with laws and relations that are valid and remain regardless of the system that coordinates the specified quantities. The covariant relations are the major vector components for each coordinate system.  It involves the various inverted tensors manipulate the vectors formalizing the manipulation of the geometric entities of the mathematical manifolds.

From the various definitions given by tensor Analysis help service providers, they indicate the various aspects of the vector entity as having magnitude and direction, making a representation of a [parallelogram with similar entities.  The various laws of the vector components’ giving their entities as those with system coordinates, the vector change comes in place with a mathematical law with deductible from the parallelogram law. The various laws give the pictorial image from the objective entity containing the various components and key properties of the law.

Possible difficulties in this subject area?

There are many areas covered by the tensor sensors, but the metrical tensor and the curvature tensor become of major interest and at times pose a major challenge in the converting the vector components into the magnitude of vectors. It does not become easy to make the two-dimensional case making the simple perpendicular coordinates.  The application of the Pythagoras theorem in getting the specific square of the magnitude of the vector V. there is hidden metrical tensors in the various equations that consist outside the Os.  The metric tensor gives a possible construct of a complicated tensor, called the curvature tensor.

The application of the sensors in geometry through the special theory of relativity that space and time, the close interrelationships constitute an indivisible Quadra-dimensional space-time.

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The online writers take the least time in calculating the magnitudes of the vectors. They easily convert the vector components and apply the Pythagoras theorem to get into the specific square of the magnitude vector. In most chances, the students just have to give their details and the assignments that they desire to be done for and wait for answers.

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FALL 2009

CLASS: 10:30AM–12:00N T-TH, Nicholson 435

Prof. Jonathan P. Dowling

Office Hours: 12:00N–1:00PM T-Th

Grader: Kaushik Seshadreesan

Required Book:
Mathematical Methods in the Physical Sciences, Third Edition
Mary Boas
Published by Wiley, 2005
ISBN 0471198269, 9780471198260

Required Software: Acrobat Reader 9.1 & Mathematica 7.01 (Available in PAWS  Tigerware)

NRL Plasma Formulary

Dyadic Tensor Notation

100% HomeworkSYLLABUS

Chapter 10 Tensor Analysis
Chapter 11 Special Functions
Chapter 12 Legendre, Bessel, Hermite, and Laguerre functions
Chapter 13 Partial Differential Equations

HOMEWORK         ASSIGNED         DUE                 HINTS                        SOLUTIONS
HW01                       THU 27 AUG       THU 03 SEP             HW01HINTS                   HW01SOL
HW02                       THU 03 SEP        THU 10 SEP                                                         HW02SOL
HW03                       THU 10 SEP        THU 17 SEP                                                         HW03SOL
HW04                       THU 17 SEP       THU 24 SEP                                                          HW04SOL
HW05                       THU 24 SEP        THU 08 OCT                                                        HW05SOL
HW06                       THU 08 OCT       THU 15 OCT                                                        HW06SOL
HW07                       THU 15 OCT       THU 22 OCT                                                        HW07SOL
HW08                       THU 22 OCT        THU 29 OCT                                                       HW08SOL
HW09                       THU 29 OCT       THU 05 NOV                                                        HW09SOL
HW10                       THU 05 NOV       THU 12 NOV                                                        HW10SOL
HW11                       THU 12 NOV       THU 19 NOV                                                        HW11SOL
HW12                       THU 19 NOV       THU 10 DEC                                                        HW12SOL


Ch.10      Tensor Analysis
10.1           Introduction to Tensors
10.2           Definition of 3-D Cartesian Tensors
10.3           Einstein Summation Convention
10.4           Moment of Inertia Tensor / Mathematica Notebook: 10.4.nb
10.4.A       Appendix: Review of Eigenvalues and Eigenvectors
10.5           Kronecker Delta and Levi-Civita Symbols / Mathematica Notebook: 10.5.nb
10.6           Pseudo-Vectors and Pseudo-Tensors
10.7           More Applications: Moment of Inertia Revisited
10.8           Curvilinear Coordinates
10.9           Vector Operators — Grad, Div, Curl, and Laplacian — in Curvilinear Coordinates
10.10         Non-Orthogonal Curvilinear Coordinates: Covariant and Contravariant Tensors

Ch.11        Special Functions
11.1             Speed Integration by Parts
11.2             Integral form of the Factorial Function
11.3             Euler Gamma Function as Generalized Factorial Function and the Recursion Relation
11.4             Euler Gamma Function of Negative Numbers
11.5             Special Values of Euler Gamma and the Reflection Formula
11.6             The Euler Beta Function
11.7             Euler Beta in terms of Euler Gamma
11.8             Exact Solution to the Simple Pendulum with Beta Function
11.9             Gaussian Integrals and the Error Functions Erf, Erfc, and Erfi
11.10           Asymptotic Series Expansions with Applications to ArcTan and Erf
11.11           Asymptotic Series Expansion of the Euler Gamma Function and Sterling's Approximation for n!
11.9-11        Mathematica Notebook for Above Sections [NB, PDF]
11.12           Jacobi Elliptic Integrals and Functions [NB, PDF]
11.13           Euler-MacLaurin & Abel-Plana Summation Formulas [Dowling89d, Prellberg07]

Ch.12        Series Solutions to Differential Equations
12.1             Introduction
12.2             Legendre's Differential Equation and the Legendre Polynomials
12.3             Leibnitz Product Rule
12.4             Rodriques Formula for Legendre Polynomials
12.5             Generating Function for Legendre Polynomials [NB, PDF]
12.6             Complete Sets of Orthonormal Functions
12.7             Orthogonality of the Legendre Polynomials
12.8             Normalization of the Legendre Polynomials
12.9             Completeness of the Legendre Polynomials
12.10           Associated Legendre Functions
12.11           Generalized Power Series and the Method of Frobenius
12.12           Bessel's Differential Equation and the Bessel Function of the first kind [NB,PDF]
12.13           Bessel's Function of the Second Kind
12.14           Graphs and Zeros of the Bessel's Functions
12.15           Recursion Relations
12.16           General Differential Equations with Bessel Function Solutions
12.17           Other Kinds of Bessel Functions (Airy, Kelvin, Spherical,  Hankel, Modified-Hyperbolic)
12.18          The Bouncing Quantum Ball [NB, PDF]
12.19          The Orthogonality of Bessel's Functions
12.20          Asymptotic Forms of Bessel's Functions [NB, PDF]
12.21          Fuch's Theorem
12.22          Hermite and Laguerre Polynomials

Ch.13         Partial Differential Equations
13.1            Introduction
13.2            Laplace Equation in 2D Cartesian Coordinates: Temperature of a Semi-Infinite Plate
13.2            Mathematica Notebook [NB, PDF]
13.3           Time-Dependent Schrödinger Equation Solution to 2D Particle in a Box
13.3            Mathematica Notebook [NB, PDF]
13.4            Vibrating String
13.4            Mathematica Notebook [NB, PDF]
13.5            Laplace Equation in 3D Cylindrical Coordinates: Temperature of a Semi-Infinite Cylinder
13.6            Vibration of a Circular Membrane: The Drum-Head Problem
13.7            Laplace's Equation in 3D Spherical Coordinates: Temperature of a Solid Sphere



Entire Course:






Ch. 15: