# VECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Prof. S.M. Tobias Jan 2009 VECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors A scalar is a physical quantity with magnitude only A vector is a physical quantity with magnitude and direction A unit vector has magnitude one. In Cartesian coordinates a = a e + a 2 e 2 + a 3 e 3 = (a, a 2, a 3 ) Magnitude: a = a 2 + a a 2 3 The position vector r = (x, y, z) The dot product (scalar product) is a scalar a b = a b cos θ = a b + a 2 b 2 + a 3 b 3 The cross product (vector product) a b is a vector with magnitude a b cos θ and a direction perpendicular to both a and b in a right-handed sense. a b = (a 2 b 3 a 3 b 2 )e + (a 3 b a b 3 )e 2 + (a b 2 a 2 b )e 3 The scalar triple product [a, b, c] is a scalar [a, b, c] = a b c = a b c = b c a The vector triple product is a vector a (b c) = (a c)b (a b)c

2 Vector Algebra and Suffix Notation The rules of suffix notation: () Any suffix may appear once or twice in any term in an equation (2) A suffix that appears just once is called a free suffix. The free suffices must be the same on both sides of the equation. Free suffices take the values, 2 and 3 (3) A suffix that appears twice is called a dummy suffix. Summation Convention: Dummy Suffices are summed over from to 3 The name of a dummy suffix is not important. a b = a j b j = a l b l = a p b p = a b + a 2 b 2 + a 3 b 3 (4) The Kronecker Delta: or δ ij = { if i = j, 0 if i j 0 0 δ ij = The Kronecker Delta is symmetric δ ij = δ ji and δ ij a j = a i (5) The Alternating Tensor: 0 if any of i, j or k are equal, ɛ ijk = if (i, j, k) = (, 2, 3), (2, 3, ) or (3,, 2) if (i, j, k) = (, 3, 2), (3, 2, ) or (2,, 3) The Alternating Tensor is antisymmetric: ɛ ijk = ɛ jik The Alternating Tensor is invariant under cyclic permutations of the indices: ɛ ijk = ɛ jki = ɛkij (6) The vector product: (7) The relation between δ ij and ɛ ijk : (a b) i = ɛ ijk a j b k ɛ ijk ɛ klm = δ il δ jm δ im δ jl

3 Vector Differentiation In all of the below formulae we are considering the vector F = (F, F 2, F 3 ) Basic Vector Differentiation () If F = F (t) then ( df dt = df dt, df 2 dt, df ) 3 dt (2) The unit tangent to the curve x = ψ(t) is given by dx/dt dx/dt Grad, Div and Curl (3) The gradient of a scalar field f(x, y, z) (= f(x, x 2, x 3 )) is given by ( f gradf = f = x, f y, f ) ( f =, f, f ) z x x 2 x 3 (4) f is the vector field with a direction perpendicular to the isosurfaces of f with a magnitude equal to the rate of change of f in that direction. (5) The directional derivative of f in the direction of a unit vector û is ( f) û (6) pronounced del or nabla is a vector differential operator. It is possible to study the algebra of. (7) The divergence of a vector field F is given by div F = F = F x + F 2 y + F 3 z = F x + F 2 x 2 + F 3 x 3 (8) A vector field F is solenoidal if F = 0 everywhere. (9) The curl of a vector field F is given by ( F3 curl F = F = y F ) ( 2 F e + z z F ) ( 3 F2 e 2 + x x F ) y ( F3 = F ) ( 2 F e + F ) ( 3 F2 e 2 + F ) x 2 x 3 x 3 x x x 2 e e 2 e 3 e e 2 e 3 = x y z = x x 2 x 3 F F 2 F 3 F F 2 F 3 e 3 e 3

4 (0) A vector field F is irrotational if F = 0 everywhere. () (F ) is a vector differential operator which can act on a scalar or a vector f (F ) f = F x + F f 2 y + F f 3 z (F )G = ((F ) G, (F ) G 2, (F ) G 3 ) (2) The Laplacian operator 2 = 2 x y + 2 can act on a scalar or a vector. 2 z2 Grad, Div and Curl and suffix notation In suffix notation Note: Here you cannot move the r = (x, y, z) = x i gradf = ( f) i = f x i ( ) i = x i div F = F = F j (curl F ) i = ( F ) i = ɛ ijk F k (F ) = F j around as it acts on everything that follows it. These can all be used to prove the vector differential identities.

5 Vector Identities Here are some simple vector identities that can all be proved with suffix notation. If F and G are vector fields and ϕ and ψ are scalar fields then ( ϕ) = 2 ϕ ( F ) = 0 ( ϕ) = 0 (ϕψ) = ϕ ψ + ψ ϕ (ϕf ) = ϕ F + F ϕ (ϕf ) = ϕ F + ϕ F ( F ) = ( F ) 2 F (F G) = F ( G) + G ( F ) + (F )G + (G )F (F G) = G ( F ) F ( G) (F G) = F ( G) G( F ) + (G )F (F )G

6 Vector Integral Theorems Alternative Definitions of divergence and curl () An alternative definition of divergence is given by F = lim F n ds, δv 0 δv where δv is a small volume bounded by a surface δs which has outward-pointing normal n. (2) An alternative definition of curl is given by δs n F = lim δs 0 δs δc F dr, where δs is a small open surface bounded by a curve δc which is oriented in a right-handed sense. Physical Interpretation of divergence and curl (3) The divergence of a vector field gives a measure of how much expansion and contraction there is in the field. (4) The curl of a vector field gives a measure of how much rotation or twist there is in the field. The Divergence and Stokes Theorems (5) The divergence theorem states that V F = F n ds, S where S is the closed surface enclosing the volume V and n is the outward-pointing normal from the surface. (6) Stokes theorem states that F n ds = F dr, S C where C is the closed curve enclosing the open surface S and n is the normal from the surface.

7 Conservative Vector fields, line integrals and exact differentials (7) The following 5 statements are equivalent in a simply-connected domain: (i) F = 0 at each point in the domain. (ii) C F dr = 0 around every closed curve in the region. Q (iii) F dr is independent of the path of integration from P to Q. P (iv) F dr is an exact differential. (v) F = φ for some scalar φ which is single-valued in the region. (8) If F = 0 then F = A for some A. (This vector potential A is not unique.)

### Index notation in 3D. 1 Why index notation?

Index notation in 3D 1 Why index notation? Vectors are objects that have properties that are independent of the coordinate system that they are written in. Vector notation is advantageous since it is elegant

### A Primer on Index Notation

A Primer on John Crimaldi August 28, 2006 1. Index versus Index notation (a.k.a. Cartesian notation) is a powerful tool for manipulating multidimensional equations. However, there are times when the more

### Index Notation for Vector Calculus

Index Notation for Vector Calculus by Ilan Ben-Yaacov and Francesc Roig Copyright c 2006 Index notation, also commonly known as subscript notation or tensor notation, is an extremely useful tool for performing

### MthSc 206 Summer1 13 Goddard

16.1 Vector Fields A vector field is a function that assigns to each point a vector. A vector field might represent the wind velocity at each point. For example, F(x, y) = y i+x j represents spinning around

### 1 Scalars, Vectors and Tensors

DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

### Primer on Index Notation

Massachusetts Institute of Technology Department of Physics Physics 8.07 Fall 2004 Primer on Index Notation c 2003-2004 Edmund Bertschinger. All rights reserved. 1 Introduction Equations involving vector

### Differentiation of vectors

Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where

### Vector Calculus: a quick review

Appendi A Vector Calculus: a quick review Selected Reading H.M. Sche,. Div, Grad, Curl and all that: An informal Tet on Vector Calculus, W.W. Norton and Co., (1973). (Good phsical introduction to the subject)

### Review of Vector Calculus

Review of Vector Calculus S. H. (Harvey) Lam January 8th, 2004 5:40PM version Abstract This review is simply a summary of the major concepts, definitions, theorems and identities that we will be using

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

### Vectors and Index Notation

Vectors and Index Notation Stephen R. Addison January 12, 2004 1 Basic Vector Review 1.1 Unit Vectors We will denote a unit vector with a superscript caret, thus â denotes a unit vector. â â = 1 If x is

### MVE041 Flervariabelanalys

MVE041 Flervariabelanalys 2015-16 This document contains the learning goals for this course. The goals are organized by subject, with reference to the course textbook Calculus: A Complete Course 8th ed.

### Prof Joshua Le-Wei Li,, EM Research Group 2 EE2011: Engineering Electromagnetics

Lecture 1: Review of Vector Calculus Part I asic Vector lgebra Vector ddition and Subtraction Position and Distance Vectors Vector Multiplications Scalar and Vector Triple Products Orthogonal Coordinate

### Physics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus

Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations

### APPENDIX D. VECTOR ANALYSIS 1. The following conventions are used in this appendix and throughout the book:

APPENDIX D. VECTOR ANALYSIS 1 Appendix D Vector Analysis The following conventions are used in this appendix and throughout the book: f, g, φ, ψ are scalar functions of x,t; A, B, C, D are vector functions

### Differentiation of vectors

Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where

### Review of Vector Analysis in Cartesian Coordinates

R. evicky, CBE 6333 Review of Vector Analysis in Cartesian Coordinates Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers.

### 16.5: CURL AND DIVERGENCE

16.5: URL AN IVERGENE KIAM HEONG KWA 1. url Let F = P i + Qj + Rk be a vector field on a solid region R 3. If all first-order partial derivatives of P, Q, and R exist, then the curl of F on is the vector

### Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:

Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1

### Math 21a Curl and Divergence Spring, 2009. 1 Define the operator (pronounced del ) by. = i

Math 21a url and ivergence Spring, 29 1 efine the operator (pronounced del by = i j y k z Notice that the gradient f (or also grad f is just applied to f (a We define the divergence of a vector field F,

### Introduction to vector and tensor analysis

Introduction to vector and tensor analysis Jesper Ferkinghoff-Borg September 6, 2007 Contents 1 Physical space 3 1.1 Coordinate systems.......................... 3 1.2 Distances...............................

### The Matrix Elements of a 3 3 Orthogonal Matrix Revisited

Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, Three-Dimensional Proper and Improper Rotation Matrices, I provided a derivation

### Brief Review of Tensors

Appendix A Brief Review of Tensors A1 Introductory Remarks In the study of particle mechanics and the mechanics of solid rigid bodies vector notation provides a convenient means for describing many physical

### Lectures on Vector Calculus

Lectures on Vector Calculus Paul Renteln Department of Physics California State University San Bernardino, CA 92407 March, 2009; Revised March, 2011 c Paul Renteln, 2009, 2011 ii Contents 1 Vector Algebra

### Gradient, Divergence and Curl in Curvilinear Coordinates

Gradient, Divergence and Curl in Curvilinear Coordinates Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems.

### DIVERGENCE AND CURL THEOREMS

This document is stored in Documents/4C/Gausstokes.tex. with LaTex. Compile it November 29, 2014 Hans P. Paar DIVERGENCE AND CURL THEOREM 1 Introduction We discuss the theorems of Gauss and tokes also

### dv div F = F da (1) D

Note I.6 1 4 October The Gauss Theorem The Gauss, or divergence, theorem states that, if is a connected three-dimensional region in R 3 whose boundary is a closed, piece-wise connected surface and F is

### Chapter 2. Parameterized Curves in R 3

Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### CBE 6333, R. Levicky 1 Differential Balance Equations

CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,

### * Magnetic Scalar Potential * Magnetic Vector Potential. PPT No. 19

* Magnetic Scalar Potential * Magnetic Vector Potential PPT No. 19 Magnetic Potentials The Magnetic Potential is a method of representing the Magnetic field by using a quantity called Potential instead

### Elasticity Theory Basics

G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold

### CBE 6333, R. Levicky 1. Tensor Notation.

CBE 6333, R. Levicky 1 Tensor Notation. Engineers and scientists find it useful to have a general terminology to indicate how many directions are associated with a physical quantity such as temperature

### Elements of Linear Algebra: Q&A

Elements of Linear Algebra: Q&A A matrix is a rectangular array of objects (elements that are numbers, functions, etc.) with its size indicated by the number of rows and columns, i.e., an m n matrix A

### * Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No.

* Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B PPT No. 17 Biot Savart s Law A straight infinitely long wire is carrying

### MATHEMATICS FOR ENGINEERS & SCIENTISTS 7

MATHEMATICS FOR ENGINEERS & SCIENTISTS 7 We stress that f(x, y, z) is a scalar-valued function and f is a vector-valued function. All of the above works in any number of dimensions. For instance, consider

### 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important

### Brief Introduction to Tensor Algebra

Brief Introduction to Tensor Algebra CONTENT I. Basic concepts 1. different coordinate systems 2. tensor algorithm II. Differentiation of tensors 1.The differentiation of base vectors 2. Covariant derivatives

### Scalars, Vectors and Tensors

Scalars, Vectors and Tensors A scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time. Examples are hydrostatic pressure and temperature. A vector

### Chapter 4 Parabolic Equations

161 Chapter 4 Parabolic Equations Partial differential equations occur in abundance in a variety of areas from engineering, mathematical biology and physics. In this chapter we will concentrate upon the

### 6 J - vector electric current density (A/m2 )

Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J - vector electric current density (A/m2 ) M - vector magnetic current density (V/m 2 ) Some problems

### A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

### Figure 1.1 Vector A and Vector F

CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

### AB2.2: Curves. Gradient of a Scalar Field

AB2.2: Curves. Gradient of a Scalar Field Parametric representation of a curve A a curve C in space can be represented by a vector function r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k This is called

### MA 105 D1 Lecture 20

MA 105 D1 Lecture 20 Ravi Raghunathan Department of Mathematics Autumn 2014, IIT Bombay, Mumbai Orientation and the Jordan curve theorem Green s Theorem Various examples Green s Theorem Other forms of

### The force on a moving charged particle. A particle with charge Q moving with velocity v in a magnetic field B is subject to a force F mag = Q v B

Magnetostatics The force on a moving charged particle A particle with charge Q moving with velocity v in a magnetic field B is subject to a force F mag = Q v B If there is also an electric field E present,

### Scalar Valued Functions of Several Variables; the Gradient Vector

Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector) valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x) = φ(x 1,

### 52. The Del Operator: Divergence and Curl

52. The Del Operator: Divergence and Curl Let F(x, y, z) = M(x, y, z), N(x, y, z), P(x, y, z) be a vector field in R 3. The del operator is represented by the symbol, and is written = x, y, z, or = x,

### Scalar and vector potentials, Helmholtz decomposition, and de Rham cohomology

Scalar and vector potentials,, and de Rham cohomology Alberto Valli Department of Mathematics, University of Trento, Italy A. Valli Potentials,, de Rham cohomology Outline 1 Introduction 2 3 4 5 A. Valli

### 1 Vectors: Geometric Approach

c F. Waleffe, 2008/09/01 Vectors These are compact lecture notes for Math 321 at UW-Madison. Read them carefully, ideally before the lecture, and complete with your own class notes and pictures. Skipping

### PHY 301: Mathematical Methods I Curvilinear Coordinate System (10-12 Lectures)

PHY 301: Mathematical Methods I Curvilinear Coordinate System (10-12 Lectures) Dr. Alok Kumar Department of Physical Sciences IISER, Bhopal Abstract The Curvilinear co-ordinates are the common name of

### Vector Calculus 1. v t. v t. h A. v t. V = Ah = A(v t) cos α

Vector Calculus 1 The first rule in understanding vector calculus is draw lots of pictures. This subject can become rather abstract if you let it, but try to visualize all the manipulations. Try a lot

### How is a vector rotated?

How is a vector rotated? V. Balakrishnan Department of Physics, Indian Institute of Technology, Madras 600 036 Appeared in Resonance, Vol. 4, No. 10, pp. 61-68 (1999) Introduction In an earlier series

### Introduction to Tensor Calculus

Introduction to Tensor Calculus arxiv:1603.01660v3 [math.ho] 23 May 2016 Taha Sochi May 25, 2016 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT. Email: t.sochi@ucl.ac.uk.

### vector calculus 2 Learning outcomes

29 ontents vector calculus 2 1. Line integrals involving vectors 2. Surface and volume integrals 3. Integral vector theorems Learning outcomes In this Workbook you will learn how to integrate functions

### Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product

Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.

### UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH College of Informatics and Electronics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2004-05 MODULE TITLE: Fluid mechanics DURATION OF EXAMINATION:

### Lista 1. Para entregar em 16/3 (T1): 1.8, 1.10, 1.13, 1.14, 1.16, 1.17, 1.20, 1.27, 1.28, 1.51 e 1.52

Lista 1 Para entregar em 16/3 (T1): 1.8, 1.10, 1.13, 1.14, 1.16, 1.17, 1.20, 1.27, 1.28, 1.51 e 1.52 12 Chapter 1 Vector Analysis In general, an nth-rank tensor has n indices and 3n components, and transforms

### v 1. v n R n we have for each 1 j n that v j v n max 1 j n v j. i=1

1. Limits and Continuity It is often the case that a non-linear function of n-variables x = (x 1,..., x n ) is not really defined on all of R n. For instance f(x 1, x 2 ) = x 1x 2 is not defined when x

### Unified Lecture # 4 Vectors

Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

### Boundary Conditions in Fluid Mechanics

Boundary Conditions in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University The governing equations for the velocity and pressure fields are partial

### The Derivative and the Tangent Line Problem. The Tangent Line Problem

The Derivative and the Tangent Line Problem Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century. 1. The tangent line problem 2. The velocity

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### Divergence and Curl. . Here we discuss some details of the divergence and curl. and the magnetic field B ( r,t)

Divergence and url Overview and Motivation: In the upcoming two lectures we will be discussing Maxwell's equations. These equations involve both the divergence and curl of two vector fields the electric

### Chapter 3. Forces, Momentum & Stress. 3.1 Newtonian mechanics: a very brief résumé

Chapter 3 Forces, Momentum & Stress 3.1 Newtonian mechanics: a very brief résumé In classical Newtonian particle mechanics, particles (lumps of matter) only experience acceleration when acted on by external

### Fundamental Theorems of Vector Calculus

Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. In the case of integrating over an interval on the real line, we were able to use

### A Some Basic Rules of Tensor Calculus

A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in continuum mechanics and the derivation of the governing equations for applied problems.

### LECTURE 1: DIFFERENTIAL FORMS. 1. 1-forms on R n

LECTURE 1: DIFFERENTIAL FORMS 1. 1-forms on R n In calculus, you may have seen the differential or exterior derivative df of a function f(x, y, z) defined to be df = f f f dx + dy + x y z dz. The expression

### Geometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi

Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in three-dimensional space, we also examine the

### 4B. Line Integrals in the Plane

4. Line Integrals in the Plane 4A. Plane Vector Fields 4A-1 Describe geometrically how the vector fields determined by each of the following vector functions looks. Tell for each what the largest region

### F = U. (1) We shall answer these questions by examining the dimensions n = 1,2,3 separately.

Lecture 24 Conservative forces in physics (cont d) Determining whether or not a force is conservative We have just examined some examples of conservative forces in R 2 and R 3. We now address the following

### Vectors SCALAR PRODUCT. Graham S McDonald. A Tutorial Module for learning about the scalar product of two vectors. Table of contents Begin Tutorial

Vectors SCALAR PRODUCT Graham S McDonald A Tutorial Module for learning about the scalar product of two vectors Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk 1. Theory 2. Exercises

### Vectors VECTOR PRODUCT. Graham S McDonald. A Tutorial Module for learning about the vector product of two vectors. Table of contents Begin Tutorial

Vectors VECTOR PRODUCT Graham S McDonald A Tutorial Module for learning about the vector product of two vectors Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk 1. Theory 2. Exercises

### (a) F (x, y, z) = (x + y, y + z, x + z) To give a better feel for the vector field, here is a plot of the vector field.

1. Compute curl of the following vector fields. Use the curl to decide whether each vector field is conservative. If the vector field is conservative, find the potential function. (a) F (x, y, z) = (x

### Section 12.6: Directional Derivatives and the Gradient Vector

Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate

### 1 The basic equations of fluid dynamics

1 The basic equations of fluid dynamics The main task in fluid dynamics is to find the velocity field describing the flow in a given domain. To do this, one uses the basic equations of fluid flow, which

### 1 Conservative vector fields

Math 241 - Calculus III Spring 2012, section CL1 16.5. Conservative vector fields in R 3 In these notes, we discuss conservative vector fields in 3 dimensions, and highlight the similarities and differences

### A vector is a directed line segment used to represent a vector quantity.

Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector

### Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### Vector algebra Christian Miller CS Fall 2011

Vector algebra Christian Miller CS 354 - Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority

### Lie derivatives, forms, densities, and integration. John L. Friedman Department of Physics, University of Wisconsin-Milwaukee

Lie derivatives, forms, densities, and integration John L. Friedman Department of Physics, University of Wisconsin-Milwaukee 1 1 Lie derivatives Lie derivatives arise naturally in the context of fluid

### M PROOF OF THE DIVERGENCE THEOREM AND STOKES THEOREM

68 Theor Supplement Section M M POOF OF THE DIEGENE THEOEM ND STOKES THEOEM In this section we give proofs of the Divergence Theorem Stokes Theorem using the definitions in artesian coordinates. Proof

### FINAL EXAM SOLUTIONS Math 21a, Spring 03

INAL EXAM SOLUIONS Math 21a, Spring 3 Name: Start by printing your name in the above box and check your section in the box to the left. MW1 Ken Chung MW1 Weiyang Qiu MW11 Oliver Knill h1 Mark Lucianovic

### Relativistic Electromagnetism

Chapter 8 Relativistic Electromagnetism In which it is shown that electricity and magnetism can no more be separated than space and time. 8.1 Magnetism from Electricity Our starting point is the electric

### Math 497C Sep 17, Curves and Surfaces Fall 2004, PSU

Math 497C Sep 17, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 4 1.9 Curves of Constant Curvature Here we show that the only curves in the plane with constant curvature are lines and circles.

### Linear Algebra Concepts

University of Central Florida School of Electrical Engineering Computer Science EGN-3420 - Engineering Analysis. Fall 2009 - dcm Linear Algebra Concepts Vector Spaces To define the concept of a vector

### (x) = lim. x 0 x. (2.1)

Differentiation. Derivative of function Let us fi an arbitrarily chosen point in the domain of the function y = f(). Increasing this fied value by we obtain the value of independent variable +. The value

### Faraday s Law & Maxwell s Equations (Griffiths Chapter 7: Sections 2-3) B t da = S

Dr. Alain Brizard Electromagnetic Theory I PY 3 Faraday s Law & Maxwell s Equations Griffiths Chapter 7: Sections -3 Electromagnetic Induction The flux rule states that a changing magnetic flux Φ B = S

### VII MAXWELL S EQUATIONS

VII MAXWELL S EQUATIONS 71 The story so far In this section we will summarise the understanding of electromagnetism which we have arrived at so far We know that there are two fields which must be considered,

### 6. In cylindrical coordinate system, the differential normal area along a z is calculated as: a) ds = ρ d dz b) ds = dρ dz c) ds = ρ d dρ d) ds = dρ d

Electrical Engineering Department Electromagnetics I (802323) G1 Dr. Mouaaz Nahas First Term (1436-1437), Second Exam Tuesday 07/02/1437 H االسم: الرقم الجامعي: Start from here Part A CLO 1: Students will

### ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE

i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear

### Teaching Electromagnetic Field Theory Using Differential Forms

IEEE TRANSACTIONS ON EDUCATION, VOL. 40, NO. 1, FEBRUARY 1997 53 Teaching Electromagnetic Field Theory Using Differential Forms Karl F. Warnick, Richard H. Selfridge, Member, IEEE, and David V. Arnold

### Problem set on Cross Product

1 Calculate the vector product of a and b given that a= 2i + j + k and b = i j k (Ans 3 j - 3 k ) 2 Calculate the vector product of i - j and i + j (Ans ) 3 Find the unit vectors that are perpendicular

### Chapter 5A. Torque. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapter 5A. Torque A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University 2007 Torque is a twist or turn that tends to produce rotation. * * * Applications