A1:Orthogonal Coordinate Systems

Size: px
Start display at page:

Download "A1:Orthogonal Coordinate Systems"

Transcription

1 A1:Orthogonal Coordinate Systems A1.1 General Change of Variables Suppose that we express x and y as a function of two other variables u and by the equations We say that these equations are defining a transformation (or mapping) from points (u,v) in a uv-cartesian plane to points (x,y) in the xy-plane. The transformation is one-to-one from the set S in the uv-plane onto the set D in the xy-plane provided that: (i) Every point in S gets mapped to a point in D (ii) Every point in D is the image of a point in S, and (iii) Different points in S get mapped to different point in D With these restrictions, the defining equations can be solved for u and v as function of x and y by the inverse transformation as The Implicit Function Theorem guarantees that the system of equations can be solved for certain variables as function of other variables under the circumstance that the Jacobian determinant (J) satisfies A one-to-one transformation can be used to transform a double integral of the form to a double integral over the corresponding set S in the uv-plane. Under the transformation, the integrand f (x,y) becomes g(u,v) = f (x(u,v),y(u,v)). Now we have to discover how express the area element da = dxdy in terms of the area element dudv in the uv-plane. For any fixed value of u, the transformation equations define a parametric curve (with v as variable) in the xy-plane called a u-curve. Similarly, for a fixed v the equations define a parametric curve called a v-curve. Consider the differential area element bounded by the u-curves corresponding to nearby values u and u + du and the v-curves with nearby values v and v + dv. The image in the xy-plane of the area element dudv in the uv-plane is shown in Figure A.1

2 The vector PQ along the v-curve is given by Figure A.1: Differential area element Applying the Chain Rule we have that a change in x and y can be written as However, dv = 0 along the v-curve and PQ reduces to Similarly, the vector PR can be expressed as Taking the limit as du and dv approach zero the area element is approximately a parallelogram with an area given by Note that the above expression involves the absolute value of the Jacobian determinant (J) defined earlier. Consequently, if the functions x and y have continuous first partial derivatives with respect to u and w on the domain S, under the given transformation the double integral over D can be written as a double integral over S, hence

3 The change of variables extends from a double integral to triple (and higher order) integrals. Consider the transformation Near any point where the Jacobian (J) is nonzero, the transformation scales volume elements according to Under the assumption that x, y and z have continuous first partial derivatives with respect to u, v and w and that the transformation is one-to-one, then Note that it is not necessary for S or D to be closed or that the transformation is one-to-one at the boundary of S. A1.2 Orthogonal Curvilinear Coordinates We denote by xyz-space the system of Cartesian coordinates (x,y,z) in R 3. A different system of coordinates [u,v,w] in xyz-space can be defined by a continuous transformation of the form If the transformation is one-to-one from a region D in uvw-space onto a region R in xyz-space, then a point P in R can be represented by a triple [u,v,w], a unique point Q from the corresponding Cartesian uvw-space that the transformation maps to P. In this case we say that the transformation defines a curvilinear coordinate system in R and call [u,v,w] the curvilinear coordinates of P with respect to that system. Note that [u,v,w] are Cartesian coordinates in their own space (uvw-space), but are curvilinear coordinates in xyz-space. Typically it is reasonable to require the transformation to be only locally one-to-one. Thus, there may be more than one point Q in some small sub region of D that gets mapped to a point P by the transformation. Points that are not even locally one-to-one are called singular points. Now let [u,v,w] be a curvilinear coordinate system in xyz-space, and let P 0 be a nonsingular point for the system. The transformation is locally one-to-one near P 0. Let P 0 have curvilinear coordinates [u 0,v 0,w 0 ]. The plane with equation u = u 0 in uvw-space gets mapped by the transformation to a surface in xyz-space passing through P 0. We call this surface a u-surface with the parametric equations

4 Similarly, we have the v-surface v = v 0 and w-surface w = w 0. We say that [u,v,w] is an orthogonal curvilinear coordinate system in xyz-space if, for every nonsingular point P 0 in xyz-space, each of three coordinate surfaces u = u 0, v = v 0, and w = w 0 intersect the other two at P 0 at right angles. Pairs of coordinate surfaces through a point intersect along a coordinate curve through that point. The coordinate surfaces v = v 0 and w = w 0 intersect along the u-curve with parametric equations A unit vector u tangent to the curve u-curve through P 0 is normal to the coordinate surface u = u 0. Similar statements hold for the unit vectors v and w. For an orthogonal curvilinear coordinate system, the three vectors u, v, and w form a local basis of mutually perpendicular unit vectors at any non singular point P 0. A1.2.1 Scale Factors and Differential Elements Assume that [u,v,w] are orthogonal curvilinear coordinates in the xyz-space, the coordinate surfaces are smooth at any nonsingular point, and that the local basis vectors u, v, and w at any such point from a right-handed-triad. The position vector of a point P in xyz-space can be expressed in terms of the curvilinear coordinates as If we hold v = v 0 and w = w 0 fixed and let u vary, then r = r (u,v 0,w 0 ) defines a u-curve in xyzspace. At any point P on this curve, the vector is tangent to the u-curve at P. In general, the three vectors are tangent respectively to the, u-curve, v-curve, and the w-curve through P. They are also normal respectively to the u-surface, v-surface, and w-surface, so they are mutually perpendicular. The lengths of these three vectors are called the scale factors of the coordinate system. The scale factors are nonzero at a nonsingular point P, so the local basis at P can be obtained by dividing the tangent vectors of the coordinate curves by their lengths.

5 Hence The volume element in a orthogonal curvilinear coordinate system is the volume of an infinitesimal coordinate box bounded by pairs of u-, v-, and w-surfaces corresponding to values u and u + du, v and v + dv, and w and w + dw. The coordinate box is spanned by the vectors Therefore, the volume element is given by Furthermore, the surface area elements on coordinate surfaces are the respective faces of the coordinate box The arc length elements along the u-, v-, and w-curves are the edges of the coordinate box

6 A1.3 Polar Coordinates [r,θ] A point P with Cartesian coordinates (x,y) can be located by its polar coordinates [r,θ] in R² via the transformation where r is the distance from origin O to the point P and θ is the angle OP makes with the positive x-axis. Positive angles are measured counterclockwise in the interval [0,2π], see Figure A.2. Figure A2: Cartesian- to polar-coordinates The transformation is locally one-to-one from D, the half of the rθ-plane where 0 < r <, to the region R consists of all points in the xy-plane except from the origin. The origin can namely be represent by [0,θ] for any angle θ. Since the transformation is not locally one-to-one at r = 0, we regard the origin in the xy-plane as an singular point for the polar coordinate system. For any nonsingular fixed point [r 0,θ 0 ], the coordinates curves intersect each other at right angles. As expected, [r,θ] is an orthogonal curvilinear coordinates system in the xy-pane. The position vector r = r(r,θ) of a point P in the xy-plane can be expressed in terms of polar coordinates as At any point P, the vectors are tangent to the r- and θ-curve respectively at P. Thus, the local basis consists of the vectors

7 The area element can be determined as and the arc length elements A1.4 Cylindrical Coordinates [r,θ,z] The cylindrical coordinate system [r,θ,z] in R 3 is defined by the transformation where ordinary plane polar coordinates are used in the xy-plane while retaining the Cartesian z coordinate for measuring vertical distance. Figure A3a shows a point P located by its cylindrical coordinates [r,θ,z] and by its Cartesian coordinates (x,y,z). a) b) Figure A3: a) Cylindrical coordinates of a point b) Coordinate surfaces This transformation maps the half space D given by r > 0 onto all of xyz-space excluding the z- axis, and is locally one-to-one. All points on the z-axis are singular points of the system since the points [0,θ,z] are identical for any θ. The coordinate surfaces, shown in Figure A3b, are r-surfaces: circular cylinders with axis along the z-axis θ-surfaces: vertical half-plane radiating from the z-axis z-surfaces: horizontal planes

8 which for every nonsingular point intersect each other at right angles and thereby form an orthogonal curvilinear coordinate system. The coordinate curves are: r-curves: horizontal straight half-lines radiating from the z-axis θ-curves: horizontal circles with center on the z-axis z-curves: vertical straight lines The position vector of a point P in the xyz-plane can be expressed in terms of cylindrical coordinates as from which we can form the (right handed) local basis vectors The volume element dv, the area elements on coordinate surfaces, and the arc length elements on coordinate curves are A1.5 Spherical coordinates [ρ,ϕ,θ] In the system of spherical coordinates a point P in xyz-space is represented by the ordered triple [ρ,ϕ,θ] via the transformation It is conventional to consider spherical coordinates restricted in such a way that ρ > 0, 0 ϕ π, and 0 θ 2π (or -π < θ π). Every point not on the z-axis has exactly one spherical representation. Thus, all points on the z-axis are considered as singular points. Figure A4 shows a point P located by its cylindrical coordinates [ρ,ϕ,θ] and by its Cartesian coordinates (x,y,z). Note that the r coordinate in cylindrical coordinates is related to ρ and ϕ by

9 Figure A4: Spherical coordinates of a point For the spherical coordinate system, the coordinate surfaces are: The coordinate curves are: (i) ρ-surfaces: spheres centered at the origin (ii) ϕ-surfaces: vertical cones with vertices at origin (iii) θ-surfaces: vertical half-planes radiating from the z-axis (i) ρ-curves: half-lines radiating from the origin (ii) ϕ-curves: vertical semicircles with centers at the origin (iii) θ-curves: horizontal circles with centers on the z-axis Indeed, for every nonsingular point, the coordinate surfaces intersect each other at right angles. The position vector of a point P in the xyz-plane can be expressed in terms of spherical coordinates as from which we can form the (right handed) local basis vectors The volume element dv, the area elements on coordinate surfaces, and the arc length elements on coordinate curves are

Coordinate Transformations in Advanced Calculus

Coordinate Transformations in Advanced Calculus Coordinate Transformations in Advanced Calculus by Sacha Nandlall T.A. for MATH 264, McGill University Email: sacha.nandlall@mail.mcgill.ca Website: http://www.resanova.com/teaching/calculus/ Fall 2006,

More information

Parametric Surfaces. Substitution

Parametric Surfaces. Substitution Calculus Lia Vas Parametric Surfaces. Substitution Recall that a curve in space is given by parametric equations as a function of single parameter t x = x(t) y = y(t) z = z(t). A curve is a one-dimensional

More information

Worksheet 3.4: Triple Integrals in Cylindrical Coordinates. Warm-Up: Cylindrical Volume Element d V

Worksheet 3.4: Triple Integrals in Cylindrical Coordinates. Warm-Up: Cylindrical Volume Element d V Boise State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates From the Toolbox (what you need from previous classes) Know what the volume element dv represents. Be able to find

More information

Vectors and the Geometry of Space

Vectors and the Geometry of Space Vectors and the Geometry of Space In Figure 11.43, consider the line L through the point P(x 1, y 1, z 1 ) and parallel to the vector. The vector v is a direction vector for the line L, and a, b, and c

More information

Math 2130 Practice Problems Sec Name. Change the Cartesian integral to an equivalent polar integral, and then evaluate.

Math 2130 Practice Problems Sec Name. Change the Cartesian integral to an equivalent polar integral, and then evaluate. Math 10 Practice Problems Sec 1.-1. Name Change the Cartesian integral to an equivalent polar integral, and then evaluate. 1) 5 5 - x dy dx -5 0 A) 5 B) C) 15 D) 5 ) 0 0-8 - 6 - x (8 + ln 9) A) 1 1 + x

More information

Grad operator, triple and line integrals. Notice: this material must not be used as a substitute for attending the lectures

Grad operator, triple and line integrals. Notice: this material must not be used as a substitute for attending the lectures Grad operator, triple and line integrals Notice: this material must not be used as a substitute for attending the lectures 1 .1 The grad operator Let f(x 1, x,..., x n ) be a function of the n variables

More information

Topic 5-6: Parameterizing Surfaces and the Surface Elements ds and ds. Big Ideas. What We Are Doing Today... Notes. Notes. Notes

Topic 5-6: Parameterizing Surfaces and the Surface Elements ds and ds. Big Ideas. What We Are Doing Today... Notes. Notes. Notes Topic 5-6: Parameterizing Surfaces and the Surface Elements ds and ds. Textbook: Section 16.6 Big Ideas A surface in R 3 is a 2-dimensional object in 3-space. Surfaces can be described using two variables.

More information

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions Calculus III Math Spring 7 In-term exam April th. Suggested solutions This exam contains sixteen problems numbered through 6. Problems 5 are multiple choice problems, which each count 5% of your total

More information

Outcomes List for Math Multivariable Calculus (9 th edition of text) Spring

Outcomes List for Math Multivariable Calculus (9 th edition of text) Spring Outcomes List for Math 200-200935 Multivariable Calculus (9 th edition of text) Spring 2009-2010 The purpose of the Outcomes List is to give you a concrete summary of the material you should know, and

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculus III-Final review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the corresponding position vector. 1) Define the points P = (-,

More information

Integration using Transformations in Polar, Cylindrical, and Spherical Coordinates

Integration using Transformations in Polar, Cylindrical, and Spherical Coordinates ections 15.4 Integration using Transformations in Polar, Cylindrical, and pherical Coordinates Cylindrical Coordinates pherical Coordinates MATH 127 (ection 15.5) Applications of Multiple Integrals The

More information

Section Parametrized Surfaces and Surface Integrals. (I) Parametrizing Surfaces (II) Surface Area (III) Scalar Surface Integrals

Section Parametrized Surfaces and Surface Integrals. (I) Parametrizing Surfaces (II) Surface Area (III) Scalar Surface Integrals Section 16.4 Parametrized Surfaces and Surface Integrals (I) Parametrizing Surfaces (II) Surface Area (III) Scalar Surface Integrals MATH 127 (Section 16.4) Parametrized Surfaces and Surface Integrals

More information

1. Suppose that the equation F (x, y, z) = 0 implicitly defines each of the three variables x, y, and z as functions of the other two:

1. Suppose that the equation F (x, y, z) = 0 implicitly defines each of the three variables x, y, and z as functions of the other two: Final Solutions. Suppose that the equation F (x, y, z) implicitly defines each of the three variables x, y, and z as functions of the other two: z f(x, y), y g(x, z), x h(y, z). If F is differentiable

More information

Worksheet 3.5: Triple Integrals in Spherical Coordinates. Warm-Up: Spherical Coordinates (ρ, φ, θ)

Worksheet 3.5: Triple Integrals in Spherical Coordinates. Warm-Up: Spherical Coordinates (ρ, φ, θ) Boise State Math 275 (Ultman) Worksheet 3.5: Triple Integrals in Spherical Coordinates From the Toolbox (what you need from previous classes) Know what the volume element dv represents. Be able to find

More information

Math Exam III Review

Math Exam III Review Math 213 - Exam III Review Peter A. Perry University of Kentucky April 10, 2019 Homework Exam III is tonight at 5 PM Exam III will cover 15.1 15.3, 15.6 15.9, 16.1 16.2, and identifying conservative vector

More information

Double Integrals, Iterated Integrals, Cross-sections

Double Integrals, Iterated Integrals, Cross-sections Chapter 14 Multiple Integrals 1 ouble Integrals, Iterated Integrals, Cross-sections 2 ouble Integrals over more general regions, efinition, Evaluation of ouble Integrals, Properties of ouble Integrals

More information

PURE MATHEMATICS 212 Multivariable Calculus CONTENTS. Page. 1. Assignment Summary... i 2. Summary Assignments...2

PURE MATHEMATICS 212 Multivariable Calculus CONTENTS. Page. 1. Assignment Summary... i 2. Summary Assignments...2 PURE MATHEMATICS 212 Multivariable Calculus CONTENTS Page 1. Assignment Summary... i 2. Summary...1 3. Assignments...2 i PMTH212, Multivariable Calculus Assignment Summary 2010 Assignment Date to be Posted

More information

MATH 2023 Multivariable Calculus

MATH 2023 Multivariable Calculus MATH 2023 Multivariable Calculus Problem Sets Note: Problems with asterisks represent supplementary informations. You may want to read their solutions if you like, but you don t need to work on them. Set

More information

Chapter 15 Notes, Stewart 7e

Chapter 15 Notes, Stewart 7e Contents 15.2 Iterated Integrals..................................... 2 15.3 Double Integrals over General Regions......................... 5 15.4 Double Integrals in Polar Coordinates..........................

More information

Measuring Lengths The First Fundamental Form

Measuring Lengths The First Fundamental Form Differential Geometry Lia Vas Measuring Lengths The First Fundamental Form Patching up the Coordinate Patches. Recall that a proper coordinate patch of a surface is given by parametric equations x = (x(u,

More information

Fall 2016 Semester METR 3113 Atmospheric Dynamics I: Introduction to Atmospheric Kinematics and Dynamics

Fall 2016 Semester METR 3113 Atmospheric Dynamics I: Introduction to Atmospheric Kinematics and Dynamics Fall 2016 Semester METR 3113 Atmospheric Dynamics I: Introduction to Atmospheric Kinematics and Dynamics Lecture 5 August 31 2016 Topics: Polar coordinate system Conversion of polar coordinates to 2-D

More information

Multiple Integrals. max x i 0

Multiple Integrals. max x i 0 Multiple Integrals 1 Double Integrals Definite integrals appear when one solves Area problem. Find the area A of the region bounded above by the curve y = f(x), below by the x-axis, and on the sides by

More information

MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points.

MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points. MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points. 1. Evaluate the area A of the triangle with the vertices

More information

MA 243 Calculus III Fall Assignment 1. Reading assignments are found in James Stewart s Calculus (Early Transcendentals)

MA 243 Calculus III Fall Assignment 1. Reading assignments are found in James Stewart s Calculus (Early Transcendentals) MA 43 Calculus III Fall 8 Dr. E. Jacobs Assignments Reading assignments are found in James Stewart s Calculus (Early Transcendentals) Assignment. Spheres and Other Surfaces Read. -. and.6 Section./Problems

More information

Applications of Triple Integrals

Applications of Triple Integrals Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals

More information

18.02 Final Exam. y = 0

18.02 Final Exam. y = 0 No books, notes or calculators. 5 problems, 50 points. 8.0 Final Exam Useful formula: cos (θ) = ( + cos(θ)) Problem. (0 points) a) (5 pts.) Find the equation in the form Ax + By + z = D of the plane P

More information

MATH 261 EXAM III PRACTICE PROBLEMS

MATH 261 EXAM III PRACTICE PROBLEMS MATH 6 EXAM III PRACTICE PROBLEMS These practice problems are pulled from actual midterms in previous semesters. Exam 3 typically has 5 (not 6!) problems on it, with no more than one problem of any given

More information

MAT203 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS

MAT203 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT203 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT203 covers essentially the same material as MAT201, but is more in depth and theoretical. Exam problems are often more sophisticated in scope and difficulty

More information

Chapter 15 Vector Calculus

Chapter 15 Vector Calculus Chapter 15 Vector Calculus 151 Vector Fields 152 Line Integrals 153 Fundamental Theorem and Independence of Path 153 Conservative Fields and Potential Functions 154 Green s Theorem 155 urface Integrals

More information

MATH 2400, Analytic Geometry and Calculus 3

MATH 2400, Analytic Geometry and Calculus 3 MATH 2400, Analytic Geometry and Calculus 3 List of important Definitions and Theorems 1 Foundations Definition 1. By a function f one understands a mathematical object consisting of (i) a set X, called

More information

Background for Surface Integration

Background for Surface Integration Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to

More information

18.02 Multivariable Calculus Fall 2007

18.02 Multivariable Calculus Fall 2007 MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. CV. Changing Variables in

More information

First we consider how to parameterize a surface (similar to a parameterized curve for line integrals). Surfaces will need two parameters.

First we consider how to parameterize a surface (similar to a parameterized curve for line integrals). Surfaces will need two parameters. Math 55 - Vector Calculus II Notes 14.6 urface Integrals Let s develop some surface integrals. First we consider how to parameterize a surface (similar to a parameterized curve for line integrals). urfaces

More information

f (Pijk ) V. may form the Riemann sum: . Definition. The triple integral of f over the rectangular box B is defined to f (x, y, z) dv = lim

f (Pijk ) V. may form the Riemann sum: . Definition. The triple integral of f over the rectangular box B is defined to f (x, y, z) dv = lim Chapter 14 Multiple Integrals..1 Double Integrals, Iterated Integrals, Cross-sections.2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals.3

More information

= f (a, b) + (hf x + kf y ) (a,b) +

= f (a, b) + (hf x + kf y ) (a,b) + Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals

More information

Math 241, Exam 3 Information.

Math 241, Exam 3 Information. Math 241, xam 3 Information. 11/28/12, LC 310, 11:15-12:05. xam 3 will be based on: Sections 15.2-15.4, 15.6-15.8. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

MAC2313 Final A. a. The vector r u r v lies in the tangent plane of S at a given point. b. S f(x, y, z) ds = R f(r(u, v)) r u r v du dv.

MAC2313 Final A. a. The vector r u r v lies in the tangent plane of S at a given point. b. S f(x, y, z) ds = R f(r(u, v)) r u r v du dv. MAC2313 Final A (5 pts) 1. Let f(x, y, z) be a function continuous in R 3 and let S be a surface parameterized by r(u, v) with the domain of the parameterization given by R; how many of the following are

More information

Math 253, Section 102, Fall 2006 Practice Final Solutions

Math 253, Section 102, Fall 2006 Practice Final Solutions Math 253, Section 102, Fall 2006 Practice Final Solutions 1 2 1. Determine whether the two lines L 1 and L 2 described below intersect. If yes, find the point of intersection. If not, say whether they

More information

Math 241, Final Exam. 12/11/12.

Math 241, Final Exam. 12/11/12. Math, Final Exam. //. No notes, calculator, or text. There are points total. Partial credit may be given. ircle or otherwise clearly identify your final answer. Name:. (5 points): Equation of a line. Find

More information

WW Prob Lib1 Math course-section, semester year

WW Prob Lib1 Math course-section, semester year WW Prob Lib Math course-section, semester year WeBWorK assignment due /25/06 at :00 PM..( pt) Consider the parametric equation x = 7(cosθ + θsinθ) y = 7(sinθ θcosθ) What is the length of the curve for

More information

To graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6. ) is graphed below:

To graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6. ) is graphed below: Polar Coordinates Any point in the plane can be described by the Cartesian coordinates (x, y), where x and y are measured along the corresponding axes. However, this is not the only way to represent points

More information

3. The three points (2, 4, 1), (1, 2, 2) and (5, 2, 2) determine a plane. Which of the following points is in that plane?

3. The three points (2, 4, 1), (1, 2, 2) and (5, 2, 2) determine a plane. Which of the following points is in that plane? Math 4 Practice Problems for Midterm. A unit vector that is perpendicular to both V =, 3, and W = 4,, is (a) V W (b) V W (c) 5 6 V W (d) 3 6 V W (e) 7 6 V W. In three dimensions, the graph of the equation

More information

Jim Lambers MAT 169 Fall Semester Lecture 33 Notes

Jim Lambers MAT 169 Fall Semester Lecture 33 Notes Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 33 Notes These notes correspond to Section 9.3 in the text. Polar Coordinates Throughout this course, we have denoted a point in the plane by an ordered

More information

form are graphed in Cartesian coordinates, and are graphed in Cartesian coordinates.

form are graphed in Cartesian coordinates, and are graphed in Cartesian coordinates. Plot 3D Introduction Plot 3D graphs objects in three dimensions. It has five basic modes: 1. Cartesian mode, where surfaces defined by equations of the form are graphed in Cartesian coordinates, 2. cylindrical

More information

Double Integrals over Polar Coordinate

Double Integrals over Polar Coordinate 1. 15.4 DOUBLE INTEGRALS OVER POLAR COORDINATE 1 15.4 Double Integrals over Polar Coordinate 1. Polar Coordinates. The polar coordinates (r, θ) of a point are related to the rectangular coordinates (x,y)

More information

GEOMETRY IN THREE DIMENSIONS

GEOMETRY IN THREE DIMENSIONS 1 CHAPTER 5. GEOMETRY IN THREE DIMENSIONS 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW GEOMETRY IN THREE DIMENSIONS Contents 1 Geometry in R 3 2 1.1 Lines...............................................

More information

MAC2313 Test 3 A E g(x, y, z) dy dx dz

MAC2313 Test 3 A E g(x, y, z) dy dx dz MAC2313 Test 3 A (5 pts) 1. If the function g(x, y, z) is integrated over the cylindrical solid bounded by x 2 + y 2 = 3, z = 1, and z = 7, the correct integral in Cartesian coordinates is given by: A.

More information

Chapter 5 Partial Differentiation

Chapter 5 Partial Differentiation Chapter 5 Partial Differentiation For functions of one variable, y = f (x), the rate of change of the dependent variable can dy be found unambiguously by differentiation: f x. In this chapter we explore

More information

MATH 234. Excercises on Integration in Several Variables. I. Double Integrals

MATH 234. Excercises on Integration in Several Variables. I. Double Integrals MATH 234 Excercises on Integration in everal Variables I. Double Integrals Problem 1. D = {(x, y) : y x 1, 0 y 1}. Compute D ex3 da. Problem 2. Find the volume of the solid bounded above by the plane 3x

More information

MA 174: Multivariable Calculus Final EXAM (practice) NO CALCULATORS, BOOKS, OR PAPERS ARE ALLOWED. Use the back of the test pages for scrap paper.

MA 174: Multivariable Calculus Final EXAM (practice) NO CALCULATORS, BOOKS, OR PAPERS ARE ALLOWED. Use the back of the test pages for scrap paper. MA 174: Multivariable alculus Final EXAM (practice) NAME lass Meeting Time: NO ALULATOR, BOOK, OR PAPER ARE ALLOWED. Use the back of the test pages for scrap paper. Points awarded 1. (5 pts). (5 pts).

More information

Chapter 6. Curves and Surfaces. 6.1 Graphs as Surfaces

Chapter 6. Curves and Surfaces. 6.1 Graphs as Surfaces Chapter 6 Curves and Surfaces In Chapter 2 a plane is defined as the zero set of a linear function in R 3. It is expected a surface is the zero set of a differentiable function in R n. To motivate, graphs

More information

7/14/2011 FIRST HOURLY PRACTICE IV Maths 21a, O.Knill, Summer 2011

7/14/2011 FIRST HOURLY PRACTICE IV Maths 21a, O.Knill, Summer 2011 7/14/2011 FIRST HOURLY PRACTICE IV Maths 21a, O.Knill, Summer 2011 Name: Start by printing your name in the above box. Try to answer each question on the same page as the question is asked. If needed,

More information

Name: Class: Date: 1. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint.

Name: Class: Date: 1. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. . Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f (x, y) = x y, x + y = 8. Set up the triple integral of an arbitrary continuous function

More information

) in the k-th subbox. The mass of the k-th subbox is M k δ(x k, y k, z k ) V k. Thus,

) in the k-th subbox. The mass of the k-th subbox is M k δ(x k, y k, z k ) V k. Thus, 1 Triple Integrals Mass problem. Find the mass M of a solid whose density (the mass per unit volume) is a continuous nonnegative function δ(x, y, z). 1. Divide the box enclosing into subboxes, and exclude

More information

MATH 2400: CALCULUS 3 MAY 9, 2007 FINAL EXAM

MATH 2400: CALCULUS 3 MAY 9, 2007 FINAL EXAM MATH 4: CALCULUS 3 MAY 9, 7 FINAL EXAM I have neither given nor received aid on this exam. Name: 1 E. Kim................ (9am) E. Angel.............(1am) 3 I. Mishev............ (11am) 4 M. Daniel...........

More information

302 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES. 4. Function of several variables, their domain. 6. Limit of a function of several variables

302 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES. 4. Function of several variables, their domain. 6. Limit of a function of several variables 302 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES 3.8 Chapter Review 3.8.1 Concepts to Know You should have an understanding of, and be able to explain the concepts listed below. 1. Boundary and interior points

More information

Mathematically, the path or the trajectory of a particle moving in space in described by a function of time.

Mathematically, the path or the trajectory of a particle moving in space in described by a function of time. Module 15 : Vector fields, Gradient, Divergence and Curl Lecture 45 : Curves in space [Section 45.1] Objectives In this section you will learn the following : Concept of curve in space. Parametrization

More information

Practice problems from old exams for math 233 William H. Meeks III December 21, 2009

Practice problems from old exams for math 233 William H. Meeks III December 21, 2009 Practice problems from old exams for math 233 William H. Meeks III December 21, 2009 Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These

More information

Multivariate Calculus Review Problems for Examination Two

Multivariate Calculus Review Problems for Examination Two Multivariate Calculus Review Problems for Examination Two Note: Exam Two is on Thursday, February 28, class time. The coverage is multivariate differential calculus and double integration: sections 13.3,

More information

Multivariate Calculus: Review Problems for Examination Two

Multivariate Calculus: Review Problems for Examination Two Multivariate Calculus: Review Problems for Examination Two Note: Exam Two is on Tuesday, August 16. The coverage is multivariate differential calculus and double integration. You should review the double

More information

Let and be a differentiable function. Let Then be the level surface given by

Let and be a differentiable function. Let Then be the level surface given by Module 12 : Total differential, Tangent planes and normals Lecture 35 : Tangent plane and normal [Section 35.1] > Objectives In this section you will learn the following : The notion tangent plane to a

More information

Calculus IV. Exam 2 November 13, 2003

Calculus IV. Exam 2 November 13, 2003 Name: Section: Calculus IV Math 1 Fall Professor Ben Richert Exam November 1, Please do all your work in this booklet and show all the steps. Calculators and note-cards are not allowed. Problem Possible

More information

Math 11 Fall 2016 Section 1 Monday, October 17, 2016

Math 11 Fall 2016 Section 1 Monday, October 17, 2016 Math 11 Fall 16 Section 1 Monday, October 17, 16 First, some important points from the last class: f(x, y, z) dv, the integral (with respect to volume) of f over the three-dimensional region, is a triple

More information

Preliminary Mathematics of Geometric Modeling (3)

Preliminary Mathematics of Geometric Modeling (3) Preliminary Mathematics of Geometric Modeling (3) Hongxin Zhang and Jieqing Feng 2006-11-27 State Key Lab of CAD&CG, Zhejiang University Differential Geometry of Surfaces Tangent plane and surface normal

More information

10/4/2011 FIRST HOURLY PRACTICE VI Math 21a, Fall Name:

10/4/2011 FIRST HOURLY PRACTICE VI Math 21a, Fall Name: 10/4/2011 FIRST HOURLY PRACTICE VI Math 21a, Fall 2011 Name: MWF 9 Chao Li MWF 9 Thanos Papaïoannou MWF 10 Emily Riehl MWF 10 Jameel Al-Aidroos MWF 11 Oliver Knill MWF 11 Tatyana Kobylyatskaya MWF 12 Tatyana

More information

PURE MATHEMATICS 212 Multivariable Calculus CONTENTS. Page. 1. Assignment Summary... i 2. Introduction Timetable Assignments...

PURE MATHEMATICS 212 Multivariable Calculus CONTENTS. Page. 1. Assignment Summary... i 2. Introduction Timetable Assignments... PURE MATHEMATICS 212 Multivariable Calculus CONTENTS Page 1. Assignment Summary... i 2. Introduction...1 3. Timetable... 3 4. Assignments...5 i PMTH212, Multivariable Calculus Assignment Summary 2009

More information

QUIZ 4 (CHAPTER 17) SOLUTIONS MATH 252 FALL 2008 KUNIYUKI SCORED OUT OF 125 POINTS MULTIPLIED BY % POSSIBLE

QUIZ 4 (CHAPTER 17) SOLUTIONS MATH 252 FALL 2008 KUNIYUKI SCORED OUT OF 125 POINTS MULTIPLIED BY % POSSIBLE QUIZ 4 (CHAPTER 17) SOLUTIONS MATH 5 FALL 8 KUNIYUKI SCORED OUT OF 15 POINTS MULTIPLIED BY.84 15% POSSIBLE 1) Reverse the order of integration, and evaluate the resulting double integral: 16 y dx dy. Give

More information

Math 251 Quiz 5 Fall b. Calculate. 2. Sketch the region. Write as one double integral by interchanging the order of integration: 2

Math 251 Quiz 5 Fall b. Calculate. 2. Sketch the region. Write as one double integral by interchanging the order of integration: 2 Math 251 Quiz 5 Fall 2002 1. a. Calculate 5 1 0 1 x dx dy b. Calculate 1 5 1 0 xdxdy 2. Sketch the region. Write as one double integral by interchanging the order of integration: 0 2 dx 2 x dy f(x,y) +

More information

Polar Coordinates. Calculus 2 Lia Vas. If P = (x, y) is a point in the xy-plane and O denotes the origin, let

Polar Coordinates. Calculus 2 Lia Vas. If P = (x, y) is a point in the xy-plane and O denotes the origin, let Calculus Lia Vas Polar Coordinates If P = (x, y) is a point in the xy-plane and O denotes the origin, let r denote the distance from the origin O to the point P = (x, y). Thus, x + y = r ; θ be the angle

More information

38. Triple Integration over Rectangular Regions

38. Triple Integration over Rectangular Regions 8. Triple Integration over Rectangular Regions A rectangular solid region S in R can be defined by three compound inequalities, a 1 x a, b 1 y b, c 1 z c, where a 1, a, b 1, b, c 1 and c are constants.

More information

Polar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 28 / 46

Polar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 28 / 46 Polar Coordinates Polar Coordinates: Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ)

More information

Calculus III Meets the Final

Calculus III Meets the Final Calculus III Meets the Final Peter A. Perry University of Kentucky December 7, 2018 Homework Review for Final Exam on Thursday, December 13, 6:00-8:00 PM Be sure you know which room to go to for the final!

More information

Workbook. MAT 397: Calculus III

Workbook. MAT 397: Calculus III Workbook MAT 397: Calculus III Instructor: Caleb McWhorter Name: Summer 2017 Contents Preface..................................................... 2 1 Spatial Geometry & Vectors 3 1.1 Basic n Euclidean

More information

What you will learn today

What you will learn today What you will learn today Tangent Planes and Linear Approximation and the Gradient Vector Vector Functions 1/21 Recall in one-variable calculus, as we zoom in toward a point on a curve, the graph becomes

More information

CHAPTER 6: APPLICATIONS OF INTEGRALS

CHAPTER 6: APPLICATIONS OF INTEGRALS (Exercises for Section 6.1: Area) E.6.1 CHAPTER 6: APPLICATIONS OF INTEGRALS SECTION 6.1: AREA 1) For parts a) and b) below, in the usual xy-plane i) Sketch the region R bounded by the graphs of the given

More information

Functions of Several Variables

Functions of Several Variables Functions of Several Variables Directional Derivatives and the Gradient Vector Philippe B Laval KSU April 7, 2012 Philippe B Laval (KSU) Functions of Several Variables April 7, 2012 1 / 19 Introduction

More information

MATH203 Calculus. Dr. Bandar Al-Mohsin. School of Mathematics, KSU

MATH203 Calculus. Dr. Bandar Al-Mohsin. School of Mathematics, KSU School of Mathematics, KSU Theorem The rectangular coordinates (x, y, z) and the cylindrical coordinates (r, θ, z) of a point P are related as follows: x = r cos θ, y = r sin θ, tan θ = y x, r 2 = x 2

More information

SPECIAL TECHNIQUES-II

SPECIAL TECHNIQUES-II SPECIAL TECHNIQUES-II Lecture 19: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Method of Images for a spherical conductor Example :A dipole near aconducting sphere The

More information

To graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6

To graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6 Polar Coordinates Any point in the plane can be described by the Cartesian coordinates (x, y), where x and y are measured along the corresponding axes. However, this is not the only way to represent points

More information

Math 113 Calculus III Final Exam Practice Problems Spring 2003

Math 113 Calculus III Final Exam Practice Problems Spring 2003 Math 113 Calculus III Final Exam Practice Problems Spring 23 1. Let g(x, y, z) = 2x 2 + y 2 + 4z 2. (a) Describe the shapes of the level surfaces of g. (b) In three different graphs, sketch the three cross

More information

Rectangular Coordinates in Space

Rectangular Coordinates in Space Rectangular Coordinates in Space Philippe B. Laval KSU Today Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 1 / 11 Introduction We quickly review one and two-dimensional spaces and then

More information

Final Exam - Review. Cumulative Final Review covers sections and Chapter 12

Final Exam - Review. Cumulative Final Review covers sections and Chapter 12 Final Exam - eview Cumulative Final eview covers sections 11.4-11.8 and Chapter 12 The following is a list of important concepts from each section that will be tested on the Final Exam, but were not covered

More information

UNIVERSITI TEKNOLOGI MALAYSIA SSCE 1993 ENGINEERING MATHEMATICS II TUTORIAL 2. 1 x cos dy dx x y dy dx. y cosxdy dx

UNIVERSITI TEKNOLOGI MALAYSIA SSCE 1993 ENGINEERING MATHEMATICS II TUTORIAL 2. 1 x cos dy dx x y dy dx. y cosxdy dx UNIVESITI TEKNOLOI MALAYSIA SSCE 99 ENINEEIN MATHEMATICS II TUTOIAL. Evaluate the following iterated integrals. (e) (g) (i) x x x sinx x e x y dy dx x dy dx y y cosxdy dx xy x + dxdy (f) (h) (y + x)dy

More information

Math 52 Final Exam March 16, 2009

Math 52 Final Exam March 16, 2009 Math 52 Final Exam March 16, 2009 Name : Section Leader: Josh Lan Xiannan (Circle one) Genauer Huang Li Section Time: 10:00 11:00 1:15 2:15 (Circle one) This is a closed-book, closed-notes exam. No calculators

More information

Put your initials on the top of every page, in case the pages become separated.

Put your initials on the top of every page, in case the pages become separated. Math 1201, Fall 2016 Name (print): Dr. Jo Nelson s Calculus III Practice for 1/2 of Final, Midterm 1 Material Time Limit: 90 minutes DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO. This exam contains

More information

Chapter 11. Parametric Equations And Polar Coordinates

Chapter 11. Parametric Equations And Polar Coordinates Instructor: Prof. Dr. Ayman H. Sakka Chapter 11 Parametric Equations And Polar Coordinates In this chapter we study new ways to define curves in the plane, give geometric definitions of parabolas, ellipses,

More information

Math Boot Camp: Coordinate Systems

Math Boot Camp: Coordinate Systems Math Boot Camp: Coordinate Systems You can skip this boot camp if you can answer the following question: Staying on a sphere of radius R, what is the shortest distance between the point (0, 0, R) on the

More information

Math 265 Exam 3 Solutions

Math 265 Exam 3 Solutions C Roettger, Fall 16 Math 265 Exam 3 Solutions Problem 1 Let D be the region inside the circle r 5 sin θ but outside the cardioid r 2 + sin θ. Find the area of D. Note that r and θ denote polar coordinates.

More information

MATH 116 REVIEW PROBLEMS for the FINAL EXAM

MATH 116 REVIEW PROBLEMS for the FINAL EXAM MATH 116 REVIEW PROBLEMS for the FINAL EXAM The following questions are taken from old final exams of various calculus courses taught in Bilkent University 1. onsider the line integral (2xy 2 z + y)dx

More information

Polar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 27 / 45

Polar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 27 / 45 : Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ) Chapter 10: Parametric Equations

More information

MATH 200 WEEK 9 - WEDNESDAY TRIPLE INTEGRALS

MATH 200 WEEK 9 - WEDNESDAY TRIPLE INTEGRALS MATH WEEK 9 - WEDNESDAY TRIPLE INTEGRALS MATH GOALS Be able to set up and evaluate triple integrals using rectangular, cylindrical, and spherical coordinates MATH TRIPLE INTEGRALS We integrate functions

More information

Exam 2 Preparation Math 2080 (Spring 2011) Exam 2: Thursday, May 12.

Exam 2 Preparation Math 2080 (Spring 2011) Exam 2: Thursday, May 12. Multivariable Calculus Exam 2 Preparation Math 28 (Spring 2) Exam 2: Thursday, May 2. Friday May, is a day off! Instructions: () There are points on the exam and an extra credit problem worth an additional

More information

14.5 Directional Derivatives and the Gradient Vector

14.5 Directional Derivatives and the Gradient Vector 14.5 Directional Derivatives and the Gradient Vector 1. Directional Derivatives. Recall z = f (x, y) and the partial derivatives f x and f y are defined as f (x 0 + h, y 0 ) f (x 0, y 0 ) f x (x 0, y 0

More information

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Homework - Solutions 3 2 Homework 2 - Solutions 3 3 Homework 3 - Solutions 9 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus

More information

1 Double Integrals over Rectangular Regions

1 Double Integrals over Rectangular Regions Contents ouble Integrals over Rectangular Regions ouble Integrals Over General Regions 7. Introduction.................................... 7. Areas of General Regions............................. 9.3 Region

More information

Math 2374 Spring 2007 Midterm 3 Solutions - Page 1 of 6 April 25, 2007

Math 2374 Spring 2007 Midterm 3 Solutions - Page 1 of 6 April 25, 2007 Math 374 Spring 7 Midterm 3 Solutions - Page of 6 April 5, 7. (3 points) Consider the surface parametrized by (x, y, z) Φ(x, y) (x, y,4 (x +y )) between the planes z and z 3. (i) (5 points) Set up the

More information

Chapter 6 Some Applications of the Integral

Chapter 6 Some Applications of the Integral Chapter 6 Some Applications of the Integral More on Area More on Area Integrating the vertical separation gives Riemann Sums of the form More on Area Example Find the area A of the set shaded in Figure

More information

f x = 2e xy +y(2x+y)e xy = (2+2xy+y 2 )e xy.

f x = 2e xy +y(2x+y)e xy = (2+2xy+y 2 )e xy. gri (rg38778) Homework 11 gri (11111) 1 This print-out should have 3 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. Find lim (x,y) (,) 1

More information

12.7 Tangent Planes and Normal Lines

12.7 Tangent Planes and Normal Lines .7 Tangent Planes and Normal Lines Tangent Plane and Normal Line to a Surface Suppose we have a surface S generated by z f(x,y). We can represent it as f(x,y)-z 0 or F(x,y,z) 0 if we wish. Hence we can

More information

Curvilinear Coordinates

Curvilinear Coordinates Curvilinear Coordinates Cylindrical Coordinates A 3-dimensional coordinate transformation is a mapping of the form T (u; v; w) = hx (u; v; w) ; y (u; v; w) ; z (u; v; w)i Correspondingly, a 3-dimensional

More information