M PROOF OF THE DIVERGENCE THEOREM AND STOKES THEOREM
|
|
- Bertina Osborne
- 7 years ago
- Views:
Transcription
1 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 of the Divergence Theorem Let F be a smooth vector field defined on a solid region with boundar surface oriented outward. e wish to show that F d div F d. For the Divergence Theorem, we use the same approach as we used for Green s Theorem; first prove the theorem for rectangular regions, then use the change of variables formula to prove it for regions parameteried b rectangular regions, finall paste such regions together to form general regions. Proof for ectangular Solids with Sides Parallel to the es onsider a smooth vector field F defined on the rectangular solid : a b, c d, e f. (See Figure M.50). e start b computing the flu of F through the two faces of perpendicular to the -ais, 1 2, both oriented outward: F d F d f d e c f d e c F 1 (a,, ) d d + f d e c (F 1 (b,, ) F 1 (a,, )) d d. F 1 (b,, ) d d the Fundamental Theorem of alculus, F 1 (b,, ) F 1 (a,, ) b a F 1 d, so F d + F d 1 2 f d b e c a F 1 d d d F 1 d. a similar argument, we can show 4 F d + F d F 2 d 5 6 F d + F d F d. dding these, we get F d This is the Divergence Theorem for the region. ( F1 + F 2 + F ) d div F d.
2 Theor Supplement Section M 69 1 ) left) (bottom) 4 1 right) Figure M.50: ectangular solid in -space S left) s u S 2 S 6 S 5 (bottom) S 4 t S1 right) (left) 2 5 (bottom) 6 4 Figure M.51: rectangular solid in stu-space the corresponding curved solid in -space Proof for egions Parameteried b ectangular Solids Now suppose we have a smooth change of coordinates (s, t, u), (s, t, u), (s, t, u). onsider a curved solid in -space corresponding to a rectangular solid in stu-space. See Figure M.51. e suppose that the change of coordinates is one-to-one on the interior of, that its Jacobian determinant is positive on. e prove the Divergence Theorem for using the Divergence Theorem for. Let be the boundar of. To prove the Divergence Theorem for, we must show that F d div F d. First we epress the flu through as a flu integral in stu-space over S, the boundar of the rectangular region. In vector notation the change of coordinates is The face 1 of is parameteried b so on this face r r (s, t, u) (s, t, u) i + (s, t, u) j + (s, t, u) k. r r (a, t, u), c t d, e u f, d ± r r dt du. In fact, in order to make d point outward, we must choose the negative sign. (Problem on page 7 shows how this follows from the fact that the Jacobian determinant is positive.) Thus, if S 1 is the face s a of, F d r F S 1 r dt du, 1 The outward pointing area element on S 1 is d S i dt du. Therefore, if we choose a vector field G on stu-space whose component in the s-direction is we have G 1 F r r, F d G ds. 1 S 1
3 70 Theor Supplement Section M Similarl, if we define the t u components of G b then G 2 F r r G F r r, i S i F d G ds, i 2,..., 6. (See Problem 4.) dding the integrals for all the faces, we find that F d G ds. where Since we have alread proved the Divergence Theorem for the rectangular region, we have G ds div G d, div G + G 2 Problems 5 6 on page 7 show that + G 2 + G (,, ) (s, t, u) S S + G. ( F1 + F 2 + F ). So, b the three-variable change of variables formula on page 61, ( div F F1 d + F 2 + F ) d d d ( F1 + F 2 + F ) (,, ) (s, t, u) ds dt du ( G1 + G 2 + G ) ds dt du div G d. In summar, we have shown that F d div F d S G d S div G d. the Divergence Theorem for rectangular solids, the right-h sides of these equations are equal, so the left-h sides are equal also. This proves the Divergence Theorem for the curved region. Pasting egions Together s in the proof of Green s Theorem, we prove the Divergence Theorem for more general regions b pasting smaller regions together along common faces. Suppose the solid region is formed b pasting together solids 1 2 along a common face, as in Figure M.52. The surface which bounds is formed b joining the surfaces 1 2 which bound 1 2, then deleting the common face. The outward flu integral of a vector field F through 1 includes the integral across the common face, the outward flu integral of F through 2
4 Theor Supplement Section M 71 ommon face 2 1 Figure M.52: egion formed b pasting together 1 2 includes the integral over the same face, but oriented in the opposite direction. Thus, when we add the integrals together, the contributions from the common face cancel, we get the flu integral through. Thus we have F d F d + F d. 1 2 ut we also have div F d div F d + div F d. 1 2 So the Divergence Theorem for follows from the Divergence Theorem for 1 2. Hence we have proved the Divergence Theorem for an region formed b pasting together regions that can be smoothl parameteried b rectangular solids. Eample 1 Let be a spherical ball of radius 2, centered at the origin, with a concentric ball of radius 1 removed. Using spherical coordinates, show that the proof of the Divergence Theorem we have given applies to. Solution e cut into two hollowed hemispheres like the one shown in Figure M.5,. In spherical coordinates, is the rectangle 1 ρ 2, 0 φ π, 0 θ π. Each face of this rectangle becomes part of the boundar of. The faces ρ 1 ρ 2 become the inner outer hemispherical surfaces that form part of the boundar of. The faces θ 0 θ π become the two halves of the flat part of the boundar of. The faces φ 0 φ π become line segments along the -ais. e can form b pasting together two solid regions like along the flat surfaces where θ constant. θ π φ π θ 0 φ π ρ 1 ρ 2 φ π 2 ρ Figure M.5: The hollow hemisphere the corresponding rectangular region in ρθφ-space 1 π θ
5 72 Theor Supplement Section M Proof of Stokes Theorem onsider an oriented surface, bounded b the curve. e want to prove Stokes Theorem: curl F d F d r. e suppose that has a smooth parameteriation r r (s, t), so that corresponds to a region in the st-plane, corresponds to the boundar of. See Figure M.54. e prove Stokes Theorem for the surface a continuousl differentiable vector field F b epressing the integrals on both sides of the theorem in terms of s t, using Green s Theorem in the st-plane. First, we convert the line integral F d r into a line integral around : F d r F r ds + F r dt. So if we define a 2-dimensional vector field G (G 1, G 2 ) on the st-plane b then G 1 F r F d r G 2 F r, G d s, using s to denote the position vector of a point in the st-plane. hat about the flu integral curl F d that occurs on the other side of Stokes Theorem? In terms of the parameteriation, curl F d In Problem 7 on page 74 we show that Hence e have alread seen that curl F r r ds dt. curl F r r G 2. curl F d F d r ( G2 G ) 1 ds dt. G d s. Green s Theorem, the right-h sides of the last two equations are equal. Hence the left-h sides are equal as well, which is what we had to prove for Stokes Theorem. t s Figure M.54: region in the st-plane the corresponding surface in -space; the curve corresponds to the boundar of
6 Theor Supplement Section M 7 Problems for Section M 1. Let be a solid circular clinder along the -ais, with a smaller concentric clinder removed. Parameterie b a rectangular solid in rθ-space, where r, θ, are clindrical coordinates. 2. In this section we proved the Divergence Theorem using the coordinate definition of divergence. Now we use the Divergence Theorem to show that the coordinate definition is the same as the geometric definition. Suppose F is smooth in a neighborhood of ( 0, 0, 0), let U be the ball of radius with center ( 0, 0, 0). Let m be the minimum value of div F on U let M be the maimum value. (a) Let S be the sphere bounding U. Show that S F d m M. olume of U (b) Eplain wh we can conclude that lim 0 S F d div F olume of U ( 0, 0, 0). (c) Eplain wh the statement in part (b) remains true if we replace U with a cube of side, centered at ( 0, 0, 0). Problems 6 fill in the details of the proof of the Divergence Theorem.. Figure M.51 on page 69 shows the solid region in space parameteried b a rectangular solid in stuspace using the continuousl differentiable change of coordinates r r (s, t, u), a s b, c t d, e u f. ( ) Suppose that r r r is positive. (a) Let 1 be the face of corresponding to the face s a of. Show that r, if it is not ero, points into. (b) Show that r r is an outward pointing normal on 1. (c) Find an outward pointing normal on 2, the face of where s b. 4. Show that for the other five faces of the solid in the proof of the Divergence Theorem (see page 70): i F d S i G d S, i 2,, 4, 5, Suppose that F is a continuousl differentiable vector field that a, b, c are vectors. In this problem we prove the formula grad( F b c ) a + grad( F c a ) b + grad( F a b ) c ( a b c ) div F. (a) Interpreting the divergence as flu densit, eplain wh the formula makes sense. [Hint: onsider the flu out of a small parallelepiped with edges parallel to a, b, c.] (b) Sa how man terms there are in the epansion of the left-h side of the formula in artesian coordinates, without actuall doing the epansion. (c) rite down all the terms on the left-h side that contain F 1/. Show that these terms add up to a b c F1. (d) rite down all the terms that contain F 1/. Show that these add to ero. (e) Eplain how the epressions involving the other seven partial derivatives will work out, how this verifies that the formula holds. 6. Let F be a smooth vector field in -space, let (s, t, u), (s, t, u), (s, t, u) be a smooth change of variables, which we will write in vector form as r r (s, t, u) (s, t, u) i +(s, t, u) j +(s, t, u) k. Define a vector field G (G 1, G 2, G ) on stu-space b G 1 F r r G F r r. (a) Show that + G2 + G G 2 F r r F r r + F r r + F r r. (b) Let r 0 r (s 0, t 0, u 0), let a r ( r 0), r b ( r 0), Use the chain rule to show that ( ) G1 + G2 + G r r 0 r c ( r 0). grad( F b c ) a + grad( F c a ) b + grad( F a b ) c.
7 74 INDEX (c) Use Problem 5 to show that + G2 + G ( ) (,, ) F1 (s, t, u) + F2 + F. 7. This problem completes the proof of Stokes Theorem. Let F be a smooth vector field in -space, let S be a surface parameteried b r r (s, t). Let r 0 r (s 0, t 0) be a fied point on S. e define a vector field in st-space as on page 72: G 1 F r G 2 F r. (a) Let a r ( r 0), r b ( r 0). Show that G2 ( r 0) ( r 0) grad( F a ) b grad( F b ) a. (b) Use Problem 0 on page 961 of the tetbook to show curl F r r G2 G1.
88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationTriple Integrals in Cylindrical or Spherical Coordinates
Triple Integrals in Clindrical or Spherical Coordinates. Find the volume of the solid ball 2 + 2 + 2. Solution. Let be the ball. We know b #a of the worksheet Triple Integrals that the volume of is given
More informationSolutions - Homework sections 17.7-17.9
olutions - Homework sections 7.7-7.9 7.7 6. valuate xy d, where is the triangle with vertices (,, ), (,, ), and (,, ). The three points - and therefore the triangle between them - are on the plane x +
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationSolutions to Practice Problems for Test 4
olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,
More informationFundamental 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
More informationAB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss
AB2.5: urfaces and urface Integrals. Divergence heorem of Gauss epresentations of surfaces or epresentation of a surface as projections on the xy- and xz-planes, etc. are For example, z = f(x, y), x =
More informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 4 AREAS AND VOLUMES This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.
More informationThis makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5
1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationThe small increase in x is. and the corresponding increase in y is. Therefore
Differentials For a while now, we have been using the notation dy to mean the derivative of y with respect to. Here is any variable, and y is a variable whose value depends on. One of the reasons that
More informationMath 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)
More informationMULTIPLE INTEGRALS. h 2 (y) are continuous functions on [c, d] and let f(x, y) be a function defined on R. Then
MULTIPLE INTEGALS 1. ouble Integrals Let be a simple region defined by a x b and g 1 (x) y g 2 (x), where g 1 (x) and g 2 (x) are continuous functions on [a, b] and let f(x, y) be a function defined on.
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph
More informationCOMPLEX STRESS TUTORIAL 3 COMPLEX STRESS AND STRAIN
COMPLX STRSS TUTORIAL COMPLX STRSS AND STRAIN This tutorial is not part of the decel unit mechanical Principles but covers elements of the following sllabi. o Parts of the ngineering Council eam subject
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationVector Calculus Solutions to Sample Final Examination #1
Vector alculus s to Sample Final Examination #1 1. Let f(x, y) e xy sin(x + y). (a) In what direction, starting at (,π/), is f changing the fastest? (b) In what directions starting at (,π/) is f changing
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationVector 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)
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular
More informationReview 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.
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationEE2 Mathematics : Vector Calculus
EE2 Mathematics : Vector alculus J. D. Gibbon Professor J. D Gibbon, Dept of Mathematics) j.d.gibbon@ic.ac.uk http://www2.imperial.ac.uk/ jdg These notes are not identical word-for-word with m lectures
More informationv w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More informationPartial Derivatives. @x f (x; y) = @ x f (x; y) @x x2 y + @ @x y2 and then we evaluate the derivative as if y is a constant.
Partial Derivatives Partial Derivatives Just as derivatives can be used to eplore the properties of functions of 1 variable, so also derivatives can be used to eplore functions of 2 variables. In this
More informationIntroduction to Plates
Chapter Introduction to Plates Plate is a flat surface having considerabl large dimensions as compared to its thickness. Common eamples of plates in civil engineering are. Slab in a building.. Base slab
More informationGeometry Notes PERIMETER AND AREA
Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationIf Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule.
Oriented Surfaces and Flux Integrals Let be a surface that has a tangent plane at each of its nonboundary points. At such a point on the surface two unit normal vectors exist, and they have opposite directions.
More informationFINAL 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
More informationCHAPTER 24 GAUSS S LAW
CHAPTER 4 GAUSS S LAW 4. The net charge shown in Fig. 4-40 is Q. Identify each of the charges A, B, C shown. A B C FIGURE 4-40 4. From the direction of the lines of force (away from positive and toward
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationGauss Formulation of the gravitational forces
Chapter 1 Gauss Formulation of the gravitational forces 1.1 ome theoretical background We have seen in class the Newton s formulation of the gravitational law. Often it is interesting to describe a conservative
More informationEQUILIBRIUM STRESS SYSTEMS
EQUILIBRIUM STRESS SYSTEMS Definition of stress The general definition of stress is: Stress = Force Area where the area is the cross-sectional area on which the force is acting. Consider the rectangular
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationMENSURATION. Definition
MENSURATION Definition 1. Mensuration : It is a branch of mathematics which deals with the lengths of lines, areas of surfaces and volumes of solids. 2. Plane Mensuration : It deals with the sides, perimeters
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle - in particular the basic equation representing
More informationChapter 22: Electric Flux and Gauss s Law
22.1 ntroduction We have seen in chapter 21 that determining the electric field of a continuous charge distribution can become very complicated for some charge distributions. t would be desirable if we
More informationGeometry 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
More information9.5 CALCULUS AND POLAR COORDINATES
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
More informationArea of Parallelograms (pages 546 549)
A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular
More informationMECHANICS OF SOLIDS - BEAMS TUTORIAL TUTORIAL 4 - COMPLEMENTARY SHEAR STRESS
MECHANICS OF SOLIDS - BEAMS TUTORIAL TUTORIAL 4 - COMPLEMENTARY SHEAR STRESS This the fourth and final tutorial on bending of beams. You should judge our progress b completing the self assessment exercises.
More informationChapter 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,
More informationGeometry Notes VOLUME AND SURFACE AREA
Volume and Surface Area Page 1 of 19 VOLUME AND SURFACE AREA Objectives: After completing this section, you should be able to do the following: Calculate the volume of given geometric figures. Calculate
More informationES240 Solid Mechanics Fall 2007. Stress field and momentum balance. Imagine the three-dimensional body again. At time t, the material particle ( x, y,
S40 Solid Mechanics Fall 007 Stress field and momentum balance. Imagine the three-dimensional bod again. At time t, the material particle,, ) is under a state of stress ij,,, force per unit volume b b,,,.
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationScalar 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,
More informationSolids. Objective A: Volume of a Solids
Solids Math00 Objective A: Volume of a Solids Geometric solids are figures in space. Five common geometric solids are the rectangular solid, the sphere, the cylinder, the cone and the pyramid. A rectangular
More informationSURFACE TENSION. Definition
SURFACE TENSION Definition In the fall a fisherman s boat is often surrounded by fallen leaves that are lying on the water. The boat floats, because it is partially immersed in the water and the resulting
More informationElectromagnetism - Lecture 2. Electric Fields
Electromagnetism - Lecture 2 Electric Fields Review of Vector Calculus Differential form of Gauss s Law Poisson s and Laplace s Equations Solutions of Poisson s Equation Methods of Calculating Electric
More information4 Constrained Optimization: The Method of Lagrange Multipliers. Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551
Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551 LEVEL CURVES 2 7 2 45. f(, ) ln 46. f(, ) 6 2 12 4 16 3 47. f(, ) 2 4 4 2 (11 18) 48. Sometimes ou can classif the critical
More informationMath 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,
More informationVector Fields and Line Integrals
Vector Fields and Line Integrals 1. Match the following vector fields on R 2 with their plots. (a) F (, ), 1. Solution. An vector, 1 points up, and the onl plot that matches this is (III). (b) F (, ) 1,.
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationMathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationMath 1B, lecture 5: area and volume
Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the
More informationSECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651. 1 x 2 y 2 z 2 4. 1 sx 2 y 2 z 2 2. xy-plane. It is sketched in Figure 11.
SECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651 SOLUTION The inequalities 1 2 2 2 4 can be rewritten as 2 FIGURE 11 1 0 1 s 2 2 2 2 so the represent the points,, whose distance from the origin is
More informationA 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
More informationCHAPTER 29 VOLUMES AND SURFACE AREAS OF COMMON SOLIDS
CHAPTER 9 VOLUMES AND SURFACE AREAS OF COMMON EXERCISE 14 Page 9 SOLIDS 1. Change a volume of 1 00 000 cm to cubic metres. 1m = 10 cm or 1cm = 10 6m 6 Hence, 1 00 000 cm = 1 00 000 10 6m = 1. m. Change
More informationShape Dictionary YR to Y6
Shape Dictionary YR to Y6 Guidance Notes The terms in this dictionary are taken from the booklet Mathematical Vocabulary produced by the National Numeracy Strategy. Children need to understand and use
More informationStokes' Theorem Examples
Stokes' Theorem Eamples Stokes' Theorem relates surface integrals and line integrals. STOKES' TEOREM Let F be a vector field. Let be an oriented surface, and let be the boundar curve of, oriented using
More informationCross 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.
More informationLecture 1 Introduction 1. 1.1 Rectangular Coordinate Systems... 1. 1.2 Vectors... 3. Lecture 2 Length, Dot Product, Cross Product 5. 2.1 Length...
CONTENTS i Contents Lecture Introduction. Rectangular Coordinate Sstems..................... Vectors.................................. 3 Lecture Length, Dot Product, Cross Product 5. Length...................................
More informationMATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationPHY 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
More informationThe Math Circle, Spring 2004
The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the
More informationGeometric Optics Converging Lenses and Mirrors Physics Lab IV
Objective Geometric Optics Converging Lenses and Mirrors Physics Lab IV In this set of lab exercises, the basic properties geometric optics concerning converging lenses and mirrors will be explored. The
More informationChapter 17. Review. 1. Vector Fields (Section 17.1)
hapter 17 Review 1. Vector Fields (Section 17.1) There isn t much I can say in this section. Most of the material has to do with sketching vector fields. Please provide some explanation to support your
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More information1 Symmetries of regular polyhedra
1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an
More information15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More information1 Error in Euler s Method
1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of
More informationF = 0. x ψ = y + z (1) y ψ = x + z (2) z ψ = x + y (3)
MATH 255 FINAL NAME: Instructions: You must include all the steps in your derivations/answers. Reduce answers as much as possible, but use exact arithmetic. Write neatly, please, and show all steps. Scientists
More informationArea of Parallelograms, Triangles, and Trapezoids (pages 314 318)
Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base
More informationChapter 4. Electrostatic Fields in Matter
Chapter 4. Electrostatic Fields in Matter 4.1. Polarization A neutral atom, placed in an external electric field, will experience no net force. However, even though the atom as a whole is neutral, the
More information1. Which of the 12 parent functions we know from chapter 1 are power functions? List their equations and names.
Pre Calculus Worksheet. 1. Which of the 1 parent functions we know from chapter 1 are power functions? List their equations and names.. Analyze each power function using the terminology from lesson 1-.
More informationChapter 19. Mensuration of Sphere
8 Chapter 19 19.1 Sphere: A sphere is a solid bounded by a closed surface every point of which is equidistant from a fixed point called the centre. Most familiar examples of a sphere are baseball, tennis
More information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
More information28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z
28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition
More informationPerimeter, Area, and Volume
Perimeter, Area, and Volume Perimeter of Common Geometric Figures The perimeter of a geometric figure is defined as the distance around the outside of the figure. Perimeter is calculated by adding all
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationSection 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50
Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall
More information6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.
hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,
More informationSURFACE AREA AND VOLUME
SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has
More information10.2 Series and Convergence
10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and
More information3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationAP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:
AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationTHE ELECTROMAGNETIC FIELD DUE TO THE ELECTRONS.
THE ELECTROMAGNETIC FIELD DUE TO THE ELECTRONS 367 Proceedings of the London Mathematical Society Vol 1 1904 p 367-37 (Retyped for readability with same page breaks) ON AN EXPRESSION OF THE ELECTROMAGNETIC
More informationDifferentiation 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
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second
More informationCSU Fresno Problem Solving Session. Geometry, 17 March 2012
CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news
More informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More information