Colorado, ASEN 5227

Excerpt: ... Vector Calculus & General Coordinate Systems Derivatives of a General Unitary Basis We have examined the computation of derivatives in an orthonormal curvilinear system. With the exception of the Cartesian system, these are the easiest to compute. Now we will develop more general procedures and formulae for an arbitrary curvilinear system with a unitary basis. Although the development is straightforward, we will introduce symbols and nomenclature that is probably unfamiliar. You must pay close attention to the definitions. Computational mechanics (fluid, structures, etc.) when the physical domain is mapped onto a curvilinear grid is an example where the basis vectors change with respect to location in the grid, thus partial derivatives such as ei / q j must be evaluated. 175 Vector Calculus & General Coordinate Systems Christoffel symbols and contravariant derivatives Recall, for an arbitrary vector a, a = (a ei )ei If a = ei / q j, we can write ei ei k = j e ek . j q q (11) For a more compact not ...

Chester, PHY 430

Excerpt: ... Electricity and Magnetism PHY 430 Exam #1 - Study Guide Note: The back and front inside covers (4 pages) of your text will be available to you on the exam. This exam will assess the following learning outcomes: 1. Student has a grasp of the following terms: position vector, displacement vector, source point, field point, and separation vector. 2. Student holds a physical understanding of the following differential vector operations: gradient, divergence, and curl. 3. Student can calculate the following vector integrals: line integral, surface integral, and a volume integral. 4. Student can apply the fundamental theorem of calculus to o path integrals involving gradients, o surface integrals involving curls, and o volume integrals involving divergences. 5. Student can use the product rules with the appropriate fundamental theorem of calculus to perform integration by parts. 6. Student can define a position in space using spherical and cylindrical coordinates. 7. Student can use differential vector calculus and ...

UCSD, MATH 20

Excerpt: ... MATH 20E VECTOR CALCULUS Lecture B Spring 2007 Textbook: Vector Calculus , fifth edition, by Jerrold E. Marsden and Anthony J. Tromba, published by W. H. Freeman and Company, 2003. Prerequisite: Math 20C or equivalent with a grade of C- or better. Catalog Description: We shall study the gradient, divergence, and curl of function of several variables, line integrals, volume and surface integrals, and the major theorems of vector calculus : Green's theorem, Stoke's Theorem, and the Divergence Theorem. These topics are covered in chapters 2, 4, 5, 6, 7, and 8 of the text. Lecture: B00 MWF 9:00-9:50am WLH 2005 Attending the lecture is a fundamental part of the course; you are responsible for material presented in the lecture whether or not it is discussed in the textbook. You should expect questions on the exams that will test your understanding of concepts discussed in the lecture. Reading: Reading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assi ...

UC Irvine, EECS 180

Excerpt: ... EECS 180 ENGINEERING ELECTROMAGNETICS (Required for EE; Elective for MSE) Catalog Data: EECS 180 Engineering Electromagnetics (Credit Units: 4) Electromagnetic fields and solutions to problems in engineering applications; Maxwell's equations and plane wave propagation, reflection, and transmission. Corequisites: Mathematics 2D and 3D. Prerequisite: Physics 7E. Formerly ECE170. (Design units: 1) Schwarz, S. E., Electromagnetics for Engineers, Saunders College Publishing, holt, Rinehart and Winston, Inc., 1990. Cheng, David K., Field and Wave Electromagnetics, 2nd edition, PrenticeHall, 1989. Chen Tsai and Franco DeFlaviis To provide the students a firm foundation in Engineering Electromagnetics. To help the students review vector calculus and establish competence in their applications for the study of Engineering Electromagnetics. To instill in the students an appreciation of the new and flourishing applications of Engineering Electromagnetics such as planar microwave technology, wireless and fiber optic commu ...

Colorado, ASEN 5227

Excerpt: ... Vector Calculus & General Coordinate Systems Homework #4, Prob 2: 1 x1 = 1 2 cos 3 , x2 = 1 2 sin 3 , x3 = ( 1 ) 2 - ( 2 ) 2 2 126 Vector Calculus & General Coordinate Systems 127 Vector Calculus & General Coordinate Systems Curvilinear Systems: Spherical Coordinates The curvilinear spherical coordinate system is probably familiar to all of you. In engineering and physics this coordinate system is used to take advantage of spherical symmetry. Let's examine this coordinate system in detail. The curvilinear transformations and inverse transformations that define the spherical system are given by, q1 = r = x 2 + y 2 + z 2 z -1 2 q = = cos x2 + y 2 + z 2 y q 3 = = tan -1 x x = r sin cos y = r sin sin z = r cos 128 Vector Calculus & General Coordinate Systems and scale factors and fundamental metric components are x y z h1 = hr = + + r r r 2 2 2 2 1/ 2 = (sin cos ) + (sin cos ) + cos h2 = h = r 2 2 1/ 2 =1 h3 = h = r sin g11 = h12 = 1 g 22 = h ...

Idaho, ECE 339

Excerpt: ... ckground in vector calculus is necessary in EE330. The prerequisites specified in the catalog are: Math 275 (Analytic Geometry and Calculus III), Math 310 (Ordinary Differential Equations) and Physics 212 (Engineering Physics II). Grading University of Idaho Engineering Outreach ECE339: The course is graded Pass/Fail based on the results of a 3 hour cumulative final examination. The final exam is open book, open note. The passing grade on the final exam is 70%. You are welcome to take and submit midterm examinations for grading. Although homeworks, quizzes, and midterm exams are not graded and will not affect your final course grade, they may be good practice tools. ...

ASU, PHY 501

Excerpt: ... PHY 501, John Shumway Lec. 10: Forms and Traditional Vector Calculus Sep. 17, 2003 Lecture 10: Forms and Traditional Vector Calculus 1 Differential Forms Forms are antisymmetric, covarient tensors. A p-form has p subscripted indicies, i1 i2 .ip A zero form is defined to be a scalar, = f . A one form is the usual differential form we talked about before, such as the differential of a scalar function, = df . To generate other forms, we need a wedge product with the following properties, 1. = - . 2. (a + b) = a + b . 3. = 0. Then two-forms can be any linear combination of wedge products of two one-forms. Similarly, p-forms are linear combinations of wedge products of p one-forms. Note that an in an n-dimensional all p-forms with p > n are necessarily zero. Also, all p-forms with p = n are linear combinations of each other (this p-form is the n-dimensional volume element. 2 Turning forms into vectors in three dimensions In three dimensional Euclidean space we have several structures tha ...

Evergreen, HW 0607

Excerpt: ... Vector calculus HW #6 due Tues.6.March: Ch.1.6 # 49, 52, 53, 56 ...

Colorado, ASEN 5227

Excerpt: ... orresponding to a set of distinct eigenvalues form a linearly independent set. Thus, these eigenvectors form a basis. If an nn matrix A has a basis of eigenvectors, then D = X 1 AX is diagonal with eigenvalues of A as the entries on the main diagonal. 86 Vector Calculus & General Coordinate Systems The vector algebra included operations involving sums and products of vectors. The definitions and operations defined in the linear algebra provide the basis for linear transformations and matrix operations useful in tensor analysis. The vector calculus allows us to apply the methods of differential and integral calculus in the general tensor analysis. We begin with the usual basic definitions and operations. Derivative of a Vector Function of a Scalar es a a(t) s a(t + t) da a(t + t ) a(t ) = lim dt t 0 t s =| a | da a s ds = lim = es t 0 s t dt dt 87 Vector Calculus & General Coordinate Systems Product Rules d da db b + a (a b) = dt dt dt d da ...

Evergreen, ACADEMIC 0607

Excerpt: ... This is an empty Template Requested by Faculty to use to provide evaluations of students to Program Secretaries. Examples of completed templates can be found at http:/www.evergreen.edu/deans/newevaluationprocess.htm. Student Last and First Name: Program, Course or Contract Title: Physical Systems Quarter and Academic Year: Winter Spring 2007 DESCRIPTION: We studied Electromagnetism with vector calculus , Modern Physics, and Quantum Mechanics with linear algebra (in bra-ket, integral, and matrix form). We studied most of Introduction to Electrodynamics by Griffiths (3rd Ed.) except fields in matter; Ch.1-7 of Modern Physics by Tipler and Llewellyn (4th Ed.); and Ch.1-16 of Shankar's Principles of Quantum Mechanics. We seminared on articles from Physics Today and Science News, and chapters from Women in Mathematics (Osen, 1974, MIT), and Out of the Shadows: Contributions of twentieth-century women to physics (Byers and Williams, 2006, Cambridge), in Winter quarter. In Spring we seminared on The View from th ...

Sveriges lantbruksuniversitet, MATH 252

Excerpt: ... MATH 252-3 Vector Calculus Homework Set 7 Spring 2005 Due Wednesday, 9 March 2005 Course Web Site: http:/www.math.sfu.ca/ralfw/math252/ Textbook: Davis and Snider Introduction to Vector Analysis Reading: Sections 3.103.11, 4.1 Some of these questions were originally assigned for Homework Set 6 Problems to study (for practice; you do not need to hand these in): Section 3.10 Section 3.11 Section 4.1 (pp.169170): (pp.180182): (pp.190192): 6, 7, 11, 14 4, 6, 8, 10 2, 4, 7, 8, 20 Problems to hand in: Section 3.10 (pp.169170): Section 3.11 Section 4.1 (pp.180182): (pp.190192): 9, 10, 12, 13 3, 7, 9, 11, 12, 13, 14 3, 6, 14 Notes: 1. There is a typographical error in problem 12 of Section 3.10: the last term should be r cos e . 2. Compare the ease of calculating the divergence of the inverse-square force eld F(R) = R/R3 in spherical coordinates (problem 13 of Section 3.10, for n = 2) with the same calculation in Cartesian coordinates (problem 4 o ...

Cornell, ECE 3030

Excerpt: ... School of Electrical and Computer Engineering, Cornell University ECE 303: Electromagnetic Fields and Waves Fall 2005 Homework 1 Due on Sep. 02, 2005 by 5:00 PM Reading Assignments: i) Review the material on cartesian, cylindrical, and spherical co-ordinate systems from your favorite freshman calculus book. Make sure you are comfortable in using these co-ordinate systems. ii) Relevant sections of the online Haus and Melcher book for this week are 2.0-2.6, 3.2, 3.3. Note that the book contains more material than you are responsible for in this course. Determine relevance by what is covered in the lectures and the recitations. Problem 1.1: (basic vector calculus review) Consider a scalar quantity given by the expression, ( x, y , z ) = a x 2 + y 2 + z 2 Where a is some constant. ( ) r a) Find the gradient of the scalar in the Cartesian co-ordinate system. b) Express the scalar given above in variables of the cylindrical co-ordinate system and then find the r gradient in the cylinderical co-ordinate ...