Lecture 10 Divergence, Gauss Law in Differential Form

Size: px
Start display at page:

Download "Lecture 10 Divergence, Gauss Law in Differential Form"

Transcription

1 Lecture 10 Divergence, Gauss Law in Differential Form ections: 3.4, 3.5, 3.6 Homework: ee homework file

2 Properties of the Flux Integral: Recap flux is the net normal flow of the vector field F through a surface Ψ= F ds= F cosαds cosα = af an flux is positive when flow is outward and negative when flow is inward F is the density of the flux (in electricity, this is the D vector) LECTURE 10 slide

3 Flux Lines in Field Plots flux lines show the direction and density of the flux the number of flux lines per unit area corresponds to the flux density strength ds F a n high density low density LECTURE 10 slide 3

4 Total Flux: Example A 1 Compute the flux of the vector field D= axxz ayyz azxy through the surface of a cube centered at the origin and with sides equal to units each and parallel to the Cartesian axes. Ψ= D d s = D ( a ) dydz + D a dydz ( x= 1) x ( x= 1) x Dx ( x= 1) Dx ( x= 1) ( x= 1) ( x=+ 1) z x y + D ( a ) dxdz + D a dxdz ( y= 1) y ( y= 1) y Dy ( y= 1) Dy ( y= 1) ( y= 1) ( y=+ 1) + D ( a ) dxdy + D a dxdy ( z= 1) z ( z= 1) z D ( z= 1) D ( z= 1) ( z= 1) z ( z=+ 1) z units LECTURE 10 slide 4

5 Total Flux: Example A Ψ= zdydz + ( z ) dxdz + xydxdy + y= 1z= 1 x= 1z= 1 x= 1y= Ψ= = x= 1y= 1 ( xy) dxdy LECTURE 10 slide 5

6 Flux Through a mall Volume 1 Ψ= F d s = front back left right top bottom assume volume is a small cuboid centered at (x 0, y 0, z 0 ) with edges along the principal axes the cuboid edges are very short: Δx, Δy, Δz assume F is constant on each face of the cuboid (faces are very small) a x a z a y x ( x0, y0, z0) a z a x y a y z a x a z a y LECTURE 10 slide 6

7 a y y Fy ( x0, y0, z0) Flux Through a mall Volume determine which F component is normal to which face determine whether the normal F component at each face goes out of or into the volume x z Fz ( x0, y0, z0 ) y a x ( x0, y0, z0) a z a x x Fx ( x0 +, y0, z0) a z z Fz ( x0, y0, z0 + ) a y z a x a z x a Fx ( x0, y0, z0) y y Fy ( x0, y0 +, z0) LECTURE 10 slide 7

8 top Flux Through a mall Volume 3 x x + = Fx( x0 +, y0, z0) Fx( x0, y0, z0) y z bottom xfx = x y z x v x, y, z x F x left yfy + = x y z y front right + = z back zfz x y z v x, y, z v x, y, z xfx yfy zfz Ψ= F d s = + + x y z x y z v x, y, z LECTURE 10 slide 8

9 Divergence in RC 1 lim x, y, z 0 Fx Fy Fz F ds= + + dxdydz x y z dv div F x, y, z F [ ] divf = lim v v 0 v d s or divf = dψ dv the divergence of a vector shows how much flux per unit volume is generated or lost at a given point the divergence is a local measure of the source strength LECTURE 10 slide 9

10 What is the divergence in these examples???? F=>< Other examples: air in tire, ionization LECTURE 10 slide 10

11 Divergence in RC in rectangular coordinates divf Fx Fy F = F= + + x y z the div operator as a vector and a dot product div where = a + a + a x y z x y z F= ax + ay + ax ( Fx x Fy y Fz z) x y z a + a + a Fx Fy Fz = + + x y z z the del vector operator and the nabla symbol LECTURE 10 slide 11

12 Divergence Example 1 F= 1 a F= x field lines neither diverge nor converge locally Fx Fy F F = + + x y z LECTURE 10 slide 1 z

13 Divergence Example F= xa + ya F= x y field lines diverge in directions Fx Fy F F = + + x y z LECTURE 10 slide 13 z

14 Divergence Example 3 F= xa + ya + za F= x y z field lines diverge in all 3 directions Fx Fy F F = + + x y z z LECTURE 10 slide 14

15 Divergence Example 4 Fx Fy F F = + + x y z z F= yax + xay F= field lines neither diverge nor converge, they spin around the z axis LECTURE 10 slide 15

16 Divergence Example 5 Given: ( x, y, z) = xz x yz y xy z D a a a Find: (a) = f( xyz,, ) =? (b) D D at P(1,,3) =? LECTURE 10 slide 16

17 Divergence in Curvilinear C divergence in CC and C is obtained analogously to the expression in RC using the generic definition F= lim v 0 F [ v ] v d s CC: 1 1 Fφ Fz F = ( ρfρ ) + + ρ ρ ρ φ z C: F ( rf φ F = ) ( sin ) r + Fθ θ + r r rsinθ θ rsinθ φ the divergence of a vector is a physical property and it does not depend on the coordinate system you use LECTURE 10 slide 17

18 Divergence in CC and C Find the divergence of the field F = xa x + ya y using rectangular and cylindrical coordinate systems. LECTURE 10 slide 18

19 Gauss Law of Electrostatics Revisited D d s Q D = lim = lim = ρv C/m v 0 v v 0 v [ v] 3 Gauss law in differential form D = ρ v as opposed to Gauss law in integral form D d s = Q = ρvdv v D d s = ( D ) dv [ v ] v divergence theorem (aka Gauss Theorem) LECTURE 10 slide 19

20 Example B: Gauss (Divergence) Theorem in Vector Analysis Verify the divergence theorem by comparing the flux Ψ of the field D= a xz a yz a xy x y z through the surface of a cube centered at the origin and with sides equal to units each and parallel to the Cartesian axes with the volume integral over the same cube. I cube = Ddv cube LECTURE 10 slide 0

21 Gauss (Divergence) Theorem in Vector Analysis Example B, olution: From Example A, we know that Ψ= d s = D cube 8 3 We also found in Example 5 that D = z z Icube Ddv ( z z ) dxdydz = = cube = ( ) dx dy z z dz = x= 1 y= 1 z= I cube =Ψ LECTURE 10 slide 1

22 You have learned: what divergence is and how it is computed in RC, CC, and C the relation between the electric flux density D and the charge density ρ v (Gauss law in differential form) D = ρ v that the total flux is equivalent to the volume integral of the divergence (divergence or Gauss theorem) D d s = ( D ) dv [ v ] v LECTURE 10 slide

Lecture 04. Curl and Divergence

Lecture 04. Curl and Divergence Lecture 04 Curl and Divergence UCF Curl of Vector Field (1) F c d l F C Curl (or rotor) of a vector field a n curlf F d l lim c s s 0 F s a n C a n : normal direction of s follow right-hand rule UCF Curl

More information

ENERGY IN ELECTROSTATICS

ENERGY IN ELECTROSTATICS ENERGY IN ELECTROSTATICS We now turn to the question of energy in electrostatics. The first question to consider is whether or not the force is conservative. You will recall from last semester that a conservative

More information

AE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.

AE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept. AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says

More information

3: Mathematics Review

3: Mathematics Review 3: Mathematics Review B. Rouben McMaster University Course EP 4D03/6D03 Nuclear Reactor Analysis (Reactor Physics) 015 Sept.-Dec. 015 September 1 Review of: Table of Contents Co-ordinate systems (Cartesian,

More information

Lecture 9 Electric Flux and Its Density Gauss Law in Integral Form

Lecture 9 Electric Flux and Its Density Gauss Law in Integral Form Lecture 9 Electric Flux and Its Density Gauss Law in Integral Form ections: 3.1, 3.2, 3.3 Homework: ee homework file Faraday s Experiment (1837), Electric Flux ΨΨ charge transfer from inner to outer sphere

More information

29.3. Integral Vector Theorems. Introduction. Prerequisites. Learning Outcomes

29.3. Integral Vector Theorems. Introduction. Prerequisites. Learning Outcomes Integral ector Theorems 9. Introduction arious theorems exist relating integrals involving vectors. Those involving line, surface and volume integrals are introduced here. They are the multivariable calculus

More information

Notes 19 Gradient and Laplacian

Notes 19 Gradient and Laplacian ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can

More information

ES.182A Topic 45 Notes Jeremy Orloff

ES.182A Topic 45 Notes Jeremy Orloff E.8A Topic 45 Notes Jeremy Orloff 45 More surface integrals; divergence theorem Note: Much of these notes are taken directly from the upplementary Notes V0 by Arthur Mattuck. 45. Closed urfaces A closed

More information

Introduction and Vectors Lecture 1

Introduction and Vectors Lecture 1 1 Introduction Introduction and Vectors Lecture 1 This is a course on classical Electromagnetism. It is the foundation for more advanced courses in modern physics. All physics of the modern era, from quantum

More information

ES.182A Topic 46 Notes Jeremy Orloff. 46 Extensions and applications of the divergence theorem

ES.182A Topic 46 Notes Jeremy Orloff. 46 Extensions and applications of the divergence theorem E.182A Topic 46 Notes Jeremy Orloff 46 Extensions and applications of the divergence theorem 46.1 el Notation There is a nice notation that captures the essence of grad, div and curl. The del operator

More information

PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.

PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives. PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x

More information

Introduction to Vector Calculus (29) SOLVED EXAMPLES. (d) B. C A. (f) a unit vector perpendicular to both B. = ˆ 2k = = 8 = = 8

Introduction to Vector Calculus (29) SOLVED EXAMPLES. (d) B. C A. (f) a unit vector perpendicular to both B. = ˆ 2k = = 8 = = 8 Introduction to Vector Calculus (9) SOLVED EXAMPLES Q. If vector A i ˆ ˆj k, ˆ B i ˆ ˆj, C i ˆ 3j ˆ kˆ (a) A B (e) A B C (g) Solution: (b) A B (c) A. B C (d) B. C A then find (f) a unit vector perpendicular

More information

Math 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin

Math 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin Math 45 Homework et olutions Points. ( pts) The integral is, x + z y d = x + + z da 8 6 6 where is = x + z 8 x + z = 4 o, is the disk of radius centered on the origin. onverting to polar coordinates then

More information

Divergence Theorem and Its Application in Characterizing

Divergence Theorem and Its Application in Characterizing Divergence Theorem and Its Application in Characterizing Fluid Flow Let v be the velocity of flow of a fluid element and ρ(x, y, z, t) be the mass density of fluid at a point (x, y, z) at time t. Thus,

More information

( ) ( ) ( ) ( ) Calculus III - Problem Drill 24: Stokes and Divergence Theorem

( ) ( ) ( ) ( ) Calculus III - Problem Drill 24: Stokes and Divergence Theorem alculus III - Problem Drill 4: tokes and Divergence Theorem Question No. 1 of 1 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as needed () Pick the 1. Use

More information

1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is

1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order

More information

(You may need to make a sin / cos-type trigonometric substitution.) Solution.

(You may need to make a sin / cos-type trigonometric substitution.) Solution. MTHE 7 Problem Set Solutions. As a reminder, a torus with radii a and b is the surface of revolution of the circle (x b) + z = a in the xz-plane about the z-axis (a and b are positive real numbers, with

More information

Lecture 11 Electrostatic Potential

Lecture 11 Electrostatic Potential Lecture 11 Electrostatic Potential Sections: 4.1, 4.2, 4.3, 4.4 Homework: See homework file differential work Work: Definition W = F L dw = F dl= FdLcosα total work done from to W = F d L, J dl α L F α

More information

Line and Surface Integrals. Stokes and Divergence Theorems

Line and Surface Integrals. Stokes and Divergence Theorems Math Methods 1 Lia Vas Line and urface Integrals. tokes and Divergence Theorems Review of urves. Intuitively, we think of a curve as a path traced by a moving particle in space. Thus, a curve is a function

More information

Notes 3 Review of Vector Calculus

Notes 3 Review of Vector Calculus ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 A ˆ Notes 3 Review of Vector Calculus y ya ˆ y x xa V = x y ˆ x Adapted from notes by Prof. Stuart A. Long 1 Overview Here we present

More information

G G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv

G G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv 1. Matching. Fill in the appropriate letter. 1. ds for a surface z = g(x, y) A. r u r v du dv 2. ds for a surface r(u, v) B. r u r v du dv 3. ds for any surface C. G x G z, G y G z, 1 4. Unit normal N

More information

********************************************************** 1. Evaluate the double or iterated integrals:

********************************************************** 1. Evaluate the double or iterated integrals: Practice problems 1. (a). Let f = 3x 2 + 4y 2 + z 2 and g = 2x + 3y + z = 1. Use Lagrange multiplier to find the extrema of f on g = 1. Is this a max or a min? No max, but there is min. Hence, among the

More information

Math 233. Practice Problems Chapter 15. i j k

Math 233. Practice Problems Chapter 15. i j k Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed

More information

One side of each sheet is blank and may be used as scratch paper.

One side of each sheet is blank and may be used as scratch paper. Math 244 Spring 2017 (Practice) Final 5/11/2017 Time Limit: 2 hours Name: No calculators or notes are allowed. One side of each sheet is blank and may be used as scratch paper. heck your answers whenever

More information

Divergence Theorem December 2013

Divergence Theorem December 2013 Divergence Theorem 17.3 11 December 2013 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:

More information

Lecture Notes: Divergence Theorem and Stokes Theorem

Lecture Notes: Divergence Theorem and Stokes Theorem Lecture Notes: ivergence heorem and tokes heorem Yufei ao epartment of omputer cience and Engineering hinese University of Hong Kong taoyf@cse.cuhk.edu.hk In this lecture, we will discuss two useful theorems

More information

AE/ME 339. K. M. Isaac. 9/22/2005 Topic 6 FluidFlowEquations_Introduction. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.

AE/ME 339. K. M. Isaac. 9/22/2005 Topic 6 FluidFlowEquations_Introduction. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept. AE/ME 339 Computational Fluid Dynamics (CFD) 1...in the phrase computational fluid dynamics the word computational is simply an adjective to fluid dynamics.... -John D. Anderson 2 1 Equations of Fluid

More information

HOMEWORK 8 SOLUTIONS

HOMEWORK 8 SOLUTIONS HOMEWOK 8 OLUTION. Let and φ = xdy dz + ydz dx + zdx dy. let be the disk at height given by: : x + y, z =, let X be the region in 3 bounded by the cone and the disk. We orient X via dx dy dz, then by definition

More information

The Divergence Theorem

The Divergence Theorem The Divergence Theorem 5-3-8 The Divergence Theorem relates flux of a vector field through the boundary of a region to a triple integral over the region. In particular, let F be a vector field, and let

More information

52. The Del Operator: Divergence and Curl

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,

More information

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015 Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction

More information

Vector Analysis. Electromagnetic Theory PHYS 401. Fall 2017

Vector Analysis. Electromagnetic Theory PHYS 401. Fall 2017 Vector Analysis Electromagnetic Theory PHYS 401 Fall 2017 1 Vector Analysis Vector analysis is a mathematical formalism with which EM concepts are most conveniently expressed and best comprehended. Many

More information

Divergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem

Divergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem Divergence Theorem 17.3 11 December 213 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:

More information

F3k, namely, F F F (10.7.1) x y z

F3k, namely, F F F (10.7.1) x y z 10.7 Divergence theorem of Gauss riple integrals can be transformed into surface integrals over the boundary surface of a region in space and conversely. he transformation is done by the divergence theorem,

More information

Math 11 Fall 2016 Final Practice Problem Solutions

Math 11 Fall 2016 Final Practice Problem Solutions Math 11 Fall 216 Final Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,

More information

Numerical Heat and Mass Transfer

Numerical Heat and Mass Transfer Master degree in Mechanical Engineering Numerical Heat and Mass Transfer 02-Transient Conduction Fausto Arpino f.arpino@unicas.it Outline Introduction Conduction ü Heat conduction equation ü Boundary conditions

More information

MULTIVARIABLE INTEGRATION

MULTIVARIABLE INTEGRATION MULTIVARIABLE INTEGRATION (PLANE & CYLINDRICAL POLAR COORDINATES) PLANE POLAR COORDINATES Question 1 The finite region on the x-y plane satisfies 1 x + y 4, y 0. Find, in terms of π, the value of I. I

More information

Sections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.

Sections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. MTH 34 Review for Exam 4 ections 16.1-16.8. 5 minutes. 5 to 1 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. Review for Exam 4 (16.1) Line

More information

Multiple Integrals and Vector Calculus: Synopsis

Multiple Integrals and Vector Calculus: Synopsis Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration

More information

51. General Surface Integrals

51. General Surface Integrals 51. General urface Integrals The area of a surface in defined parametrically by r(u, v) = x(u, v), y(u, v), z(u, v) over a region of integration in the input-variable plane is given by d = r u r v da.

More information

Topic 5.9: Divergence and The Divergence Theorem

Topic 5.9: Divergence and The Divergence Theorem Math 275 Notes (Ultman) Topic 5.9: Divergence and The Divergence Theorem Textbook ection: 16.9 From the Toolbox (what you need from previous classes): Computing partial derivatives. Computing the dot product.

More information

Math 23b Practice Final Summer 2011

Math 23b Practice Final Summer 2011 Math 2b Practice Final Summer 211 1. (1 points) Sketch or describe the region of integration for 1 x y and interchange the order to dy dx dz. f(x, y, z) dz dy dx Solution. 1 1 x z z f(x, y, z) dy dx dz

More information

LABORATORY MODULE. EKT 241/4 ELECTROMAGNETIC THEORY Semester 2 (2009/2010) EXPERIMENT # 2

LABORATORY MODULE. EKT 241/4 ELECTROMAGNETIC THEORY Semester 2 (2009/2010) EXPERIMENT # 2 LABORATORY MODULE EKT 241/4 ELECTROMAGNETIC THEORY Semester 2 (2009/2010) EXPERIMENT # 2 Vector Analysis: Gradient And Divergence Of A Scalar And Vector Field NAME MATRIK # signature DATE PROGRAMME GROUP

More information

Practice problems **********************************************************

Practice problems ********************************************************** Practice problems I will not test spherical and cylindrical coordinates explicitly but these two coordinates can be used in the problems when you evaluate triple integrals. 1. Set up the integral without

More information

Conductors and Dielectrics

Conductors and Dielectrics 5.1 Current and Current Density Conductors and Dielectrics Electric charges in motion constitute a current. The unit of current is the ampere (A), defined as a rate of movement of charge passing a given

More information

EELE 3331 Electromagnetic I Chapter 3. Vector Calculus. Islamic University of Gaza Electrical Engineering Department Dr.

EELE 3331 Electromagnetic I Chapter 3. Vector Calculus. Islamic University of Gaza Electrical Engineering Department Dr. EELE 3331 Electromagnetic I Chapter 3 Vector Calculus Islamic University of Gaza Electrical Engineering Department Dr. Talal Skaik 2012 1 Differential Length, Area, and Volume This chapter deals with integration

More information

송석호 ( 물리학과 )

송석호 ( 물리학과 ) http://optics.hanyang.ac.kr/~shsong 송석호 ( 물리학과 ) Field and Wave Electromagnetics, David K. Cheng Reviews on (Week 1). Vector Analysis 3. tatic Electric Fields (Week ) 4. olution of Electrostatic Problems

More information

Vector Calculus, Maths II

Vector Calculus, Maths II Section A Vector Calculus, Maths II REVISION (VECTORS) 1. Position vector of a point P(x, y, z) is given as + y and its magnitude by 2. The scalar components of a vector are its direction ratios, and represent

More information

Lecture 2: Review of Vector Calculus

Lecture 2: Review of Vector Calculus 1 Lecture 2: Review of Vector Calculus Instructor: Dr. Gleb V. Tcheslavski Contact: gleb@ee.lamar.edu Office Hours: Room 2030 Class web site: www.ee.lamar.edu/gleb/em/in dex.htm 2 Vector norm Foran n-dimensional

More information

MATH 332: Vector Analysis Summer 2005 Homework

MATH 332: Vector Analysis Summer 2005 Homework MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,

More information

MATH 52 FINAL EXAM SOLUTIONS

MATH 52 FINAL EXAM SOLUTIONS MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }

More information

6 Div, grad curl and all that

6 Div, grad curl and all that 6 Div, grad curl and all that 6.1 Fundamental theorems for gradient, divergence, and curl Figure 1: Fundamental theorem of calculus relates df/dx over[a, b] and f(a), f(b). You will recall the fundamental

More information

Chapter 3 - Vector Calculus

Chapter 3 - Vector Calculus Chapter 3 - Vector Calculus Gradient in Cartesian coordinate system f ( x, y, z,...) dr ( dx, dy, dz,...) Then, f f f f,,,... x y z f f f df dx dy dz... f dr x y z df 0 (constant f contour) f dr 0 or f

More information

Mathematics (Course B) Lent Term 2005 Examples Sheet 2

Mathematics (Course B) Lent Term 2005 Examples Sheet 2 N12d Natural Sciences, Part IA Dr M. G. Worster Mathematics (Course B) Lent Term 2005 Examples Sheet 2 Please communicate any errors in this sheet to Dr Worster at M.G.Worster@damtp.cam.ac.uk. Note that

More information

Tangent Planes, Linear Approximations and Differentiability

Tangent Planes, Linear Approximations and Differentiability Jim Lambers MAT 80 Spring Semester 009-10 Lecture 5 Notes These notes correspond to Section 114 in Stewart and Section 3 in Marsden and Tromba Tangent Planes, Linear Approximations and Differentiability

More information

ENGI Gradient, Divergence, Curl Page 5.01

ENGI Gradient, Divergence, Curl Page 5.01 ENGI 94 5. - Gradient, Divergence, Curl Page 5. 5. The Gradient Operator A brief review is provided here for the gradient operator in both Cartesian and orthogonal non-cartesian coordinate systems. Sections

More information

Electromagnetism Physics 15b

Electromagnetism Physics 15b Electromagnetism Physics 15b Lecture #5 Curl Conductors Purcell 2.13 3.3 What We Did Last Time Defined divergence: Defined the Laplacian: From Gauss s Law: Laplace s equation: F da divf = lim S V 0 V Guass

More information

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant

More information

Figure 25:Differentials of surface.

Figure 25:Differentials of surface. 2.5. Change of variables and Jacobians In the previous example we saw that, once we have identified the type of coordinates which is best to use for solving a particular problem, the next step is to do

More information

A Brief Revision of Vector Calculus and Maxwell s Equations

A Brief Revision of Vector Calculus and Maxwell s Equations A Brief Revision of Vector Calculus and Maxwell s Equations Debapratim Ghosh Electronic Systems Group Department of Electrical Engineering Indian Institute of Technology Bombay e-mail: dghosh@ee.iitb.ac.in

More information

Problem Solving 1: Line Integrals and Surface Integrals

Problem Solving 1: Line Integrals and Surface Integrals A. Line Integrals MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics Problem Solving 1: Line Integrals and Surface Integrals The line integral of a scalar function f ( xyz),, along a path C is

More information

Lecture 7 Total Surface and Volume Charges

Lecture 7 Total Surface and Volume Charges Lecture 7 Total Surface and Volume Charges Sections: 2.3 Homework: See homework file the surface element is defined by two line elements: ds= dl dl 1 2 the surface element is a vector Surface Elements

More information

(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0)

(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0) eview Exam Math 43 Name Id ead each question carefully. Avoid simple mistakes. Put a box around the final answer to a question (use the back of the page if necessary). For full credit you must show your

More information

No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers.

No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers. Name: Section: Recitation Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear

More information

P1 Calculus II. Partial Differentiation & Multiple Integration. Prof David Murray. dwm/courses/1pd

P1 Calculus II. Partial Differentiation & Multiple Integration. Prof David Murray.  dwm/courses/1pd P1 2017 1 / 58 P1 Calculus II Partial Differentiation & Multiple Integration Prof David Murray david.murray@eng.ox.ac.uk www.robots.ox.ac.uk/ dwm/courses/1pd 4 lectures, MT 2017 P1 2017 2 / 58 4 Multiple

More information

Mathematical Analysis II, 2018/19 First semester

Mathematical Analysis II, 2018/19 First semester Mathematical Analysis II, 208/9 First semester Yoh Tanimoto Dipartimento di Matematica, Università di Roma Tor Vergata Via della Ricerca Scientifica, I-0033 Roma, Italy email: hoyt@mat.uniroma2.it We basically

More information

Practice problems. 1. Evaluate the double or iterated integrals: First: change the order of integration; Second: polar.

Practice problems. 1. Evaluate the double or iterated integrals: First: change the order of integration; Second: polar. Practice problems 1. Evaluate the double or iterated integrals: x 3 + 1dA where = {(x, y) : 0 y 1, y x 1}. 1/ 1 y 0 3y sin(x + y )dxdy First: change the order of integration; Second: polar.. Consider the

More information

18.1. Math 1920 November 29, ) Solution: In this function P = x 2 y and Q = 0, therefore Q. Converting to polar coordinates, this gives I =

18.1. Math 1920 November 29, ) Solution: In this function P = x 2 y and Q = 0, therefore Q. Converting to polar coordinates, this gives I = Homework 1 elected olutions Math 19 November 9, 18 18.1 5) olution: In this function P = x y and Q =, therefore Q x P = x. We obtain the following integral: ( Q I = x ydx = x P ) da = x da. onverting to

More information

MATH2000 Flux integrals and Gauss divergence theorem (solutions)

MATH2000 Flux integrals and Gauss divergence theorem (solutions) DEPARTMENT O MATHEMATIC MATH lux integrals and Gauss divergence theorem (solutions ( The hemisphere can be represented as We have by direct calculation in terms of spherical coordinates. = {(r, θ, φ r,

More information

Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008

Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Code: C-031 Date and time: 17 Nov, 2008, 9:30 A.M. - 12:30 P.M. Maximum Marks: 45 Important Instructions: 1. The question

More information

Tutorial Divergence. (ii) Explain why four of these integrals are zero, and calculate the other two.

Tutorial Divergence. (ii) Explain why four of these integrals are zero, and calculate the other two. (1) Below is a graphical representation of a vector field v with a z-component equal to zero. (a) Draw a box somewhere inside this vector field. The box is 3-dimensional. To make things easy, it is a good

More information

Solutions for the Practice Final - Math 23B, 2016

Solutions for the Practice Final - Math 23B, 2016 olutions for the Practice Final - Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy

More information

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt Jim Lambers MAT 28 ummer emester 212-1 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain

More information

Solutions to Sample Questions for Final Exam

Solutions to Sample Questions for Final Exam olutions to ample Questions for Final Exam Find the points on the surface xy z 3 that are closest to the origin. We use the method of Lagrange Multipliers, with f(x, y, z) x + y + z for the square of the

More information

Mathematical Concepts & Notation

Mathematical Concepts & Notation Mathematical Concepts & Notation Appendix A: Notation x, δx: a small change in x t : the partial derivative with respect to t holding the other variables fixed d : the time derivative of a quantity that

More information

MATHS 267 Answers to Stokes Practice Dr. Jones

MATHS 267 Answers to Stokes Practice Dr. Jones MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the

More information

free space (vacuum) permittivity [ F/m]

free space (vacuum) permittivity [ F/m] Electrostatic Fields Electrostatic fields are static (time-invariant) electric fields produced by static (stationary) charge distributions. The mathematical definition of the electrostatic field is derived

More information

Maxwell s Equations in Differential Form, and Uniform Plane Waves in Free Space

Maxwell s Equations in Differential Form, and Uniform Plane Waves in Free Space C H A P T E R 3 Maxwell s Equations in Differential Form, and Uniform Plane Waves in Free Space In Chapter 2, we introduced Maxwell s equations in integral form. We learned that the quantities involved

More information

Electric Flux Density, Gauss s Law and Divergence

Electric Flux Density, Gauss s Law and Divergence Unit 3 Electric Flux Density, Gauss s Law and Divergence 3.1 Electric Flux density In (approximately) 1837, Michael Faraday, being interested in static electric fields and the effects which various insulating

More information

xy 2 e 2z dx dy dz = 8 3 (1 e 4 ) = 2.62 mc. 12 x2 y 3 e 2z 2 m 2 m 2 m Figure P4.1: Cube of Problem 4.1.

xy 2 e 2z dx dy dz = 8 3 (1 e 4 ) = 2.62 mc. 12 x2 y 3 e 2z 2 m 2 m 2 m Figure P4.1: Cube of Problem 4.1. Problem 4.1 A cube m on a side is located in the first octant in a Cartesian coordinate system, with one of its corners at the origin. Find the total charge contained in the cube if the charge density

More information

53. Flux Integrals. Here, R is the region over which the double integral is evaluated.

53. Flux Integrals. Here, R is the region over which the double integral is evaluated. 53. Flux Integrals Let be an orientable surface within 3. An orientable surface, roughly speaking, is one with two distinct sides. At any point on an orientable surface, there exists two normal vectors,

More information

UNIT 1. INTRODUCTION

UNIT 1. INTRODUCTION UNIT 1. INTRODUCTION Objective: The aim of this chapter is to gain knowledge on Basics of electromagnetic fields Scalar and vector quantities, vector calculus Various co-ordinate systems namely Cartesian,

More information

SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)

SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours) SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please

More information

The exam will have 5 questions covering multiple

The exam will have 5 questions covering multiple Math 2210-1 Notes of 11/14/2018 Math 2210 Exam 3 ummary The exam will have 5 questions covering multiple integrals. Multiple Integrals are integrals of functions of n variables over regions in IR n. For

More information

Use partial integration with respect to y to compute the inner integral (treating x as a constant.)

Use partial integration with respect to y to compute the inner integral (treating x as a constant.) Math 54 ~ Multiple Integration 4. Iterated Integrals and Area in the Plane Iterated Integrals f ( x, y) dydx = f ( x, y) dy dx b g ( x) b g ( x) a g ( x) a g ( x) Use partial integration with respect to

More information

DO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START

DO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START Math 265 Student name: KEY Final Exam Fall 23 Instructor & Section: This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot also be a communication device

More information

2.20 Fall 2018 Math Review

2.20 Fall 2018 Math Review 2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more

More information

Fundamental Electromagnetics (Chapter 4: Vector Calculus)

Fundamental Electromagnetics (Chapter 4: Vector Calculus) Fundamental Electromagnetics (Chapter 4: Vector Calculus) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 0-760-453 Fax:0-760-4435 Key Point Study differential elements in length, area, and volume useful

More information

Problem Solving 1: The Mathematics of 8.02 Part I. Coordinate Systems

Problem Solving 1: The Mathematics of 8.02 Part I. Coordinate Systems Problem Solving 1: The Mathematics of 8.02 Part I. Coordinate Systems In 8.02 we regularly use three different coordinate systems: rectangular (Cartesian), cylindrical and spherical. In order to become

More information

Chapter 1. Vector Algebra and Vector Space

Chapter 1. Vector Algebra and Vector Space 1. Vector Algebra 1.1. Scalars and vectors Chapter 1. Vector Algebra and Vector Space The simplest kind of physical quantity is one that can be completely specified by its magnitude, a single number, together

More information

Problem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j (0 t 2π). Solution: We assume a > b > 0.

Problem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j (0 t 2π). Solution: We assume a > b > 0. MATH 64: FINAL EXAM olutions Problem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j ( t π). olution: We assume a > b >. A = 1 π (xy yx )dt = 3ab π

More information

Math 211, Fall 2014, Carleton College

Math 211, Fall 2014, Carleton College A. Let v (, 2, ) (1,, ) 1, 2, and w (,, 3) (1,, ) 1,, 3. Then n v w 6, 3, 2 is perpendicular to the plane, with length 7. Thus n/ n 6/7, 3/7, 2/7 is a unit vector perpendicular to the plane. [The negation

More information

Without fully opening the exam, check that you have pages 1 through 12.

Without fully opening the exam, check that you have pages 1 through 12. Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 12. Show all your work on the standard

More information

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9 MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)

More information

The Divergence Theorem

The Divergence Theorem Math 1a The Divergence Theorem 1. Parameterize the boundary of each of the following with positive orientation. (a) The solid x + 4y + 9z 36. (b) The solid x + y z 9. (c) The solid consisting of all points

More information

is the ith variable and a i is the unit vector associated with the ith variable. h i

is the ith variable and a i is the unit vector associated with the ith variable. h i . Chapter 10 Vector Calculus Features Used right( ), product( ),./,.*, listúmat( ), mod( ), For...EndFor, norm( ), unitv( ),

More information

Final Exam. Monday March 19, 3:30-5:30pm MAT 21D, Temple, Winter 2018

Final Exam. Monday March 19, 3:30-5:30pm MAT 21D, Temple, Winter 2018 Name: Student ID#: Section: Final Exam Monday March 19, 3:30-5:30pm MAT 21D, Temple, Winter 2018 Show your work on every problem. orrect answers with no supporting work will not receive full credit. Be

More information

EKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009)

EKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009) EKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009) Dr. Mohamad Hekarl Uzir-chhekarl@eng.usm.my School of Chemical Engineering Engineering Campus, Universiti

More information

Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.

Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr. 1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line

More information

Vectors and Fields. Vectors versus scalars

Vectors and Fields. Vectors versus scalars C H A P T E R 1 Vectors and Fields Electromagnetics deals with the study of electric and magnetic fields. It is at once apparent that we need to familiarize ourselves with the concept of a field, and in

More information