CBE 6333, R. Levicky 1. Orthogonal Curvilinear Coordinates
|
|
- Esther Adams
- 6 years ago
- Views:
Transcription
1 CBE 6333, R. Levicky 1 Orthogonal Curvilinear Coordinates Introduction. Rectangular Cartesian coordinates are convenient when solving problems in which the geometry of a problem is well described by the coordinates,, and. For example, in Fig. 1, the geometry of the flow is easily described by saying that it occurs in the direction, with the velocity v = v 1 δ 1 being zero at the boundaries = 0 and = d (i.e. no-slip boundary condition). But what about the flow in Fig. 2? Here, the flow could be described by stating that it occurs in the θ direction, with v θ equal to zero at r = r 1 and to ω r 2 at r = r 2. Such a description uses cylindrical, not Cartesian, coordinates. It is not necessary to use cylindrical coordinates, but their use does simplify the description (and mathematical solution) of a problem like the one in Fig. 2. If instead the flow in Figure 2 is described in Cartesian coordinates, the description becomes more convoluted and may go as follows: the components v 1 and v 2 of the velocity v 1 δ 1 + v 2 δ 2 add so as to make a fluid element travel in a circle, with both components equal to zero at ( ) 1/2 = r 1 and obeying the condition v v 2 2 = ω 2 r 2 2 at ( ) 1/2 = r 2. This description is correct, but solving a problem possessing a cylindrical geometry using Cartesian coordinates will be much more cumbersome. v1( x2) d Figure 1. Flow between two flat plates. r2 r 1 Figure 2. Circular flow in an annulus. The outer wall of the annulus is rotating with an angular velocity ω. ω v θ ( r) For geometries as in Fig. 2, proper use of "orthogonal curvilinear coordinates" can simplify the solution to the problem. What are orthogonal curvilinear coordinates? The most familiar examples (there are many others) are cylindrical and spherical coordinates as illustrated in Figures 3 and 4. The cylindrical and spherical coordinate systems are termed "curvilinear" because some of the coordinates change along curves. For instance, in cylindrical coordinates, θ changes along a curve that can be thought of as forming a circle about the origin. The Cartesian coordinate system is not curvilinear since all of the coordinates change along straight lines. The curve along which one coordinate changes while the other coordinates remain fixed is a coordinate curve for that coordinate. Coordinate curves for θ in the cylindrical coordinate system describe circles around the origin, while those for r are lines that radiate outward from the origin. Two coordinate curves for θ and one for r are shown in Figure 3. Coordinate curves for θ in the spherical geometry (Figure 4) describe semicircles (0 < θ < 180 o ), while those for φ describe circles about the origin. In addition to being curvilinear, the cylindrical and spherical coordinate systems are also "orthogonal" because coordinate curves corresponding to different coordinates are perpendicular to one another. For example, in cylindrical coordinates, the coordinate
2 CBE 6333, R. Levicky 2 curves for r are perpendicular to those for θ and. The coordinate curves for θ are perpendicular to those for r and, etc. Clearly, the Cartesian coordinate system is also orthogonal. r r Figure 3. Cylindrical coordinates. The Figure 4. Spherical coordinates. The coordinates are r, θ and. coordinates are r, θ and φ. Unit basis vectors are defined at each point in the coordinate system as sketched in Figures 5 and 6. In general, the direction of a basis vector changes with position. In cylindrical coordinates, both δ r and δ θ change direction with position as illustrated in Fig 5, while the direction of δ x3 is independent of position. In the spherical coordinate system, all three basis vectors δ r, δ θ and δ φ change direction with position. The magnitudes of the basis vectors do not change, since by definition the length of a basis vector is always unity. Figure 5. Basis vectors for the cylindrical coordinate system. Figure 6. Basis vectors for the spherical coordinate system. n important fact to recall: the length s of an arc on a circle is given by the product of the circle's radius r and the angle ζ (ζ is in radians) through which the arc sweeps; s = rζ, see Figure 7. This relation, especially in its differential form ds = r dζ, will be useful in the following discussion.
3 CBE 6333, R. Levicky 3 Figure 7. Transformation of Coordinates. The coordinates of a point in space, expressed in terms of two different coordinate systems, can be related by transformation of coordinates equations. Let's say the first coordinate system employs q 1, q 2, and q 3 as coordinates, and the second employs,, and. Then the transformation of coordinates equations have the general form and = (q 1, q 2, q 3 ) = (q 1, q 2, q 3 ) = (q 1, q 2, q 3 ) (1) q 1 = q 1 (,, ) q 2 = q 2 (,, ) q 3 = q 3 (,, ) (2) For example, if,, and are the Cartesian coordinates and q 1, q 2, and q 3 are the cylindrical coordinates r, θ, and z, then and = r cosθ = r sinθ = (3) r = ( ) 1/2 θ = tan -1 ( / ) = (4) Equations 3 and 4 are the transformation of coordinates equations between the Cartesian and the cylindrical coordinate systems. For spherical coordinates, the transformation equations are and = r sinθ cosφ = r sinθ sinφ = r cosθ (5) r = ( ) 1/2 θ = cos -1 ( / ( ) 1/2 ) φ = tan -1 ( / ) (6) Basis Vectors. The basis vectors possess unit magnitude and point along the direction of coordinate curves as illustrated in Figures 5 and 6. Because the Cartesian basis vectors are constant in direction as well as magnitude, expressing curvilinear basis vectors in terms of the Cartesian ones can simplify mathematical derivations in some instances. Therefore, it is desired to derive expressions for the
4 CBE 6333, R. Levicky 4 curvilinear basis vectors in terms of the Cartesian basis. To begin, we first define the position p of a point in space in terms of a Cartesian coordinate system: p = (q 1, q 2, q 3 ) δ 1 + (q 1, q 2, q 3 ) δ 2 + (q 1, q 2, q 3 ) δ 3 (6b) Expressions for the Cartesian coordinates x i in expression 6b in terms of the cylindrical coordinates are given by equations 3; for the spherical coordinate system, by equations 5. unit basis vector δ i in the direction of the coordinate q i can be expressed as p δ i = q i p q i (7) 1/2 p p p where = is the magnitude of the vector p. Note that the vector p is q i q i q i q i q i tangent to (i.e. points along) the coordinate curve for q i (Figure 8). The division by the magnitude p scales the vector p p to unit magnitude; thus, is often referred to as the "scale factor" h i. q i q i q i The result of equation (7) is a unit basis vector whose direction points along the coordinate curve for q i and whose length is normalized to unity by dividing by p ; in other words, this is the unit basis q i vector corresponding to the coordinate q i. Figure 8 Equation (7) works trivially for the Cartesian coordinate system. In the Cartesian system, p = δ 1 + δ 2 + δ 3 so that, for example, δ 1 = ( p/ ) / ( p/ ) = δ 1 / (δ 1 δ 1 ) 1/2 = δ 1 / 1 = δ 1. Here, the scale factor h 1 = (δ 1 δ 1 ) 1/2 = 1. What about the cylindrical coordinate system? In the cylindrical system, using equations (3) Then, p = r cosθ δ 1 + r sinθ δ 2 + δ 3 (8)
5 CBE 6333, R. Levicky 5 δ r = ( p/ r) / ( p/ r) = (cosθ δ 1 + sinθ δ 2 ) / [(cosθ δ 1 + sinθ δ 2 ) (cosθ δ 1 + sinθ δ 2 )] 1/2 = (cosθ δ 1 + sinθ δ 2 ) / [cos 2 θ + sin 2 θ] 1/2 δ r = cosθ δ 1 + sinθ δ 2 (9) Note that the scale factor h r = [(cosθ δ x + sinθ δ y ) (cosθ δ x + sinθ δ y )] 1/2 = [cos 2 θ + sin 2 θ] 1/2 = 1. Similarly, δ θ = ( p/ θ) / ( p/ θ) = (- r sinθ δ 1 + r cosθ δ 2 ) / [(- r sinθ δ 1 + r cosθ δ 2 ) (- r sinθ δ 1 + r cosθ δ 2 )] 1/2 = (- r sinθ δ 1 + r cosθ δ 2 ) / [r 2 sin 2 θ + r 2 cos 2 θ] 1/2 = (- r sinθ δ 1 + r cosθ δ 2 ) / [r 2 (sin 2 θ + cos 2 θ)] 1/2 = (- r sinθ δ 1 + r cosθ δ 2 ) / r δ θ = - sinθ δ 1 + cosθ δ 2 (10) Note that the scale factor h θ = ( p/ θ) = r. To summarize, for the cylindrical coordinate system δ r = cosθ δ 1 + sinθ δ 2 δ θ = - sinθ δ 1 + cosθ δ 2 δ x3 = δ 3 (11) h r = 1 h θ = r h x3 = 1 (12) By similar procedures it can be shown that for the spherical coordinate system, δ r = sinθ cos φ δ 1 + sinθ sin φ δ 2 + cosθ δ 3 δ θ = cosθ cos φ δ 1 + cosθ sin φ δ 2 - sinθ δ 3 δ φ = - sin φ δ 1 + cos φ δ 2 (13) h r = 1 h θ = r h φ = r sinθ (14) From expressions 11 and 13 it should be clear that the direction of δ r and δ θ in the cylindrical system, and the direction of all three basis vectors in the spherical coordinate system, changes with position. Expressions 11 and 13 are the desired equations that express the curvilinear basis vectors in terms of their Cartesian counterparts. How do we apply these equations? For instance, an angular velocity v = v θ δ θ expressed in the cylindrical coordinate system can be converted to v = - v θ sinθ δ 1 + v θ cosθ δ 2 by using equation 11 for δ θ. If the expression θ = tan -1 ( / ) is also substituted (see equations 4), the velocity will then be purely expressed in terms of Cartesian coordinate variables and basis vectors.
6 CBE 6333, R. Levicky 6 Length, rea and Volume Elements. For a set of orthogonal curvilinear coordinates q 1, q 2 and q 3, with corresponding scaling factors h 1, h 2, and h 3, a differential displacement in position dp is given by (note the use of the summation convention) dp = h 1 dq 1 δ 1 + h 2 dq 2 δ 2 + h 3 dq 3 δ 3 = h i dq i δ i (15) For the Cartesian coordinate system, all of the scale factors equal one, and equation (15) becomes dp = d δ 1 + d δ 2 + d δ 3 (16) as seen previously. For the cylindrical coordinate system, using equations (12) for the scale factors, dp = dr δ r + r dθ δ θ + d δ x3 (17) Why are the scale factors h i needed in equation 15 for dp? Earlier, it was mentioned (Figure 7) that the length of an arc on the circumference of a circle is equal to the product of the central angle that spans the arc times the radius of the circle on whose circumference the arc lies. Therefore, the distance that is traversed when θ changes by an infinitesimal amount dθ is equal to rdθ (Figure 9). Since dp expresses a change in distance, rdθ must be used in equation 17 rather than just simply dθ (dθ is a change in angular position, and is not a distance). The distance corresponding to differential changes in the various coordinates is: Figure 9. n object initially at position p is displaced by rdθ δ θ to a final position p + rdθ δ θ. dθ rdθ rdθ Table I. Coordinate System Coordinate Distance corresponding to an infinitesimal change in coordinate Cartesian d d d Cylindrical r dr θ rdθ d
7 CBE 6333, R. Levicky 7 Spherical r dr θ rdθ φ r sinθ dφ The volume dv of an infinitesimal volume element is obtained as usual; i.e. it is given by the product of the lengths of the three sides defining the length, depth and width of the element. Each of the three sides of the volume element is taken to lie along one of the coordinate directions. Note that the sides are assured to be mutually orthogonal since only orthogonal coordinate systems are being considered. In general dv = h 1 h 2 h 3 dq 1 dq 2 dq 3 (18) For the Cartesian, cylindrical and spherical systems, equation 18 evaluates to Table II. Coordinate System dv Cartesian d d d (19a) Cylindrical dr rdθ d = r dr dθ d (19b) Spherical dr rdθ rsinθ dφ = r 2 sinθ dr dθ dφ (19c) By similar reasoning, the area d of an infinitesimal area element is given by the product of the lengths of its sides. For instance, a differential area element in the Cartesian - plane is d = d d. differential area element in the θ- cylindrical surface is r dθ d (Figure 10). differential area element in the θ-φ spherical surface is r 2 sinθ dθ dφ, etc. d d dθ d rdθ d = dx1dx2 d = rdθdx 3 Figure 10. Derivatives of Basis Vectors with Respect to Coordinate Variables. s pointed out earlier (Figures 5 and 6), in general the direction of a basis vector can change from point to point. If a change in coordinate position can cause a change in the direction of a basis vector, that means that the basis vector must have a nonzero derivative with respect to at least some of the
8 CBE 6333, R. Levicky 8 coordinate variables q i. For example, in the cylindrical coordinate system the basis vectors can be written (equations 11) δ r = cosθ δ 1 + sinθ δ 2 δ θ = - sinθ δ 1 + cosθ δ 2 δ 3 = δ 3 (11) The derivative of δ r with respect to θ then becomes δ r / θ = - sinθ δ 1 + cosθ δ 2 = δ θ. Therefore, the derivative δ r / θ is nonzero. By similar calculations, it can be shown (you should verify these equations by directly considering expressions 11 above): δ r / r = 0 δ r / θ = δ θ δ r / = 0 δ θ / r = 0 δ θ / θ = - δ r δ θ / = 0 δ x3 / r = 0 δ x3 / θ = 0 δ x3 / = 0 (20) For the spherical coordinate system, the basis vectors are given by equations 13 δ r = sinθ cos φ δ 1 + sinθ sin φ δ 2 + cosθ δ 3 δ θ = cosθ cos φ δ 1 + cosθ sin φ δ 2 - sinθ δ 3 δ φ = - sin φ δ 1 + cos φ δ 2 (13) The derivatives of the basis vectors for the spherical coordinate system become δ r / r = 0 δ r / θ = δ θ δ r / φ = sinθ δ φ δ θ / r = 0 δ θ / θ = - δ r δ θ / φ = cosθ δ φ δ φ / r = 0 δ φ / θ = 0 δ φ / φ = - sinθ δ r - cosθ δ θ (21) Each of these expressions can be derived from equations 13. Gradient, Divergence and Curl. The gradient operator takes the derivative of a quantity with respect to a change in position. In orthogonal curvilinear coordinates q 1, q 2 and q 3, the operator is (note that the scaling factors h i in the denominator ensure that a derivative is being taken with respect to distance) = δ 1 + δ 2 + δ 3 h1 q1 h2 q2 h3 q Cartesian = δ1 + δ2 + δ x2 1 x3 (22) (23a)
9 CBE 6333, R. Levicky Cylindrical = δ r + δ + δ 1 r θ r θ x3 1 x3 (23b) Spherical = δ r + δθ + δφ 1 r r θ r sin θ φ (23c) The divergence of a vector, written, can be calculated using the definition of the gradient. For the cylindrical coordinate system, = r (r, θ, )δ r + θ ( r, θ, )δ θ + x3 (r, θ, )δ x3 1 and = δ r + δ δ r θ + r θ x3 x [ r (r, θ, )δ r + θ ( r, θ, )δ θ + x3 (r, θ, )δ x3 ] 3 = δ r ( / r) [ r (r, θ, )δ r + θ ( r, θ, )δ θ + x3 (r, θ, )δ x3 ] + δ θ 1/r ( / θ)[ r (r, θ, )δ r + θ ( r, θ, )δ θ + x3 (r, θ, )δ x3 ] + δ x3 ( / ) [ r (r, θ, )δ r + θ ( r, θ, )δ θ + x3 (r, θ, )δ x3 ] Using the product rule for differentiation = δ r (δ r r / r + r δ r / r + δ θ θ / r + θ δ θ / r + δ x3 x3 / r + x3 δ x3 / r) + δ θ 1/r (δ r r / θ + r δ r / θ + δ θ θ / θ + θ δ θ / θ + δ x3 x3 / θ + x3 δ x3 / θ) + δ x3 (δ r r / + r δ r / + δ θ θ / + θ δ θ / + δ x3 x3 / + x3 δ x3 / ) Substituting equations 20 for the derivatives of the basis vectors yields = δ r (δ r r / r + δ θ θ / r + δ x3 x3 / r) + δ θ 1/r (δ r r / θ + r δ θ + δ θ θ / θ - θ δ r + δ x3 x3 / θ ) + δ x3 (δ r r / + δ θ θ / + δ x3 x3 / ) Performing the dot products (remembering that δ i δ j = δ ij since the basis vectors are mutually orthogonal) = r / r + r /r + 1/r θ / θ + x3 / (24) Expression 24 is the divergence of an arbitrary vector in cylindrical coordinates. n analogous approach could be used to derive in spherical or Cartesian coordinates. The results are = 1 / + 2 / + 3 / (Cartesian) (25a)
10 CBE 6333, R. Levicky 10 = r / r + r /r + 1/r θ / θ + x3 / (cylindrical) (25b) = r / r + 2 r /r + 1/r θ / θ + cosθ θ /(r sinθ) + 1/(r sinθ) φ / φ (spherical) (25c) It can be shown that, for an orthogonal curvilinear system, the divergence can be written as = 1 h1h 2h3 ( h2h3 1 ) + ( h1h 3 2 ) + ( h1h 2 3 ) (26) q1 q2 q3 where the h i are the scale factors for the chosen coordinate system (i.e. equations 12 and 14) and q i are the coordinate variables. Equation 26 will work for any orthogonal curvilinear coordinate system, and will reproduce equations 25a through 25c if the appropriate h i and q i are substituted into it. Expressions for the curl of a vector could be derived using a similar approach to that used in deriving equation 24 for. Here, we simply write the final general formula for, = 1 h1h 2h3 h δ h h 2 2 h δ h h 3 3 h δ h h 1 1 ( ) ( ) + q q 2 ( ) ( ) + q q 3 ( ) ( q q 1 ) (27) Equation 27 can be used to write out in the coordinate system of interest, provided that the coordinate system is orthogonal curvilinear. Concluding Remarks. Some of the above expressions, even if written in simplified form, are rather cumbersome. Fortunately, most of the equations needed for fluid dynamics have already been written down for the coordinate systems of greatest interest, i.e. the Cartesian, cylindrical, and spherical systems. Therefore, it will not be necessary to apply the above equations to express the Navier-Stokes equations in spherical coordinates, for example. The desired expressions can be found in virtually any text on transport phenomena.
Gradient, Divergence and Curl in Curvilinear Coordinates
Gradient, Divergence and Curl in Curvilinear Coordinates Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems.
More informationIntroduction 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 informationSummary: Curvilinear Coordinates
Physics 2460 Electricity and Magnetism I, Fall 2007, Lecture 10 1 Summary: Curvilinear Coordinates 1. Summary of Integral Theorems 2. Generalized Coordinates 3. Cartesian Coordinates: Surfaces of Constant
More informationAN INTRODUCTION TO CURVILINEAR ORTHOGONAL COORDINATES
AN INTRODUCTION TO CURVILINEAR ORTHOGONAL COORDINATES Overview Throughout the first few weeks of the semester, we have studied vector calculus using almost exclusively the familiar Cartesian x,y,z coordinate
More informationReview of Vector Analysis in Cartesian Coordinates
Review of Vector Analysis in Cartesian Coordinates 1 Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers. Scalars are usually
More informationIntroduction to Electromagnetism Prof. Manoj K. Harbola Department of Physics Indian Institute of Technology, Kanpur
Introduction to Electromagnetism Prof. Manoj K. Harbola Department of Physics Indian Institute of Technology, Kanpur Lecture - 12 Line surface area and volume elements in Spherical Polar Coordinates In
More informationENGI 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 information2-5 The Calculus of Scalar and Vector Fields (pp.33-55)
9/6/2005 section_2_5_the_calculus_of_vector_fields_empty.doc 1/9 2-5 The Calculus of Scalar and Vector Fields (pp.33-55) Fields are functions of coordinate variables (e.g.,, ρ, θ) Q: How can we integrate
More informationReview problems for the final exam Calculus III Fall 2003
Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)
More informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclaimer: This is meant to help you start studying. It is not necessarily a complete list of everything you need to know. The MTH 234 final exam mainly consists of standard response questions where students
More informationCurvilinear coordinates
C Curvilinear coordinates The distance between two points Euclidean space takes the simplest form (2-4) in Cartesian coordinates. The geometry of concrete physical problems may make non-cartesian coordinates
More informationCircular Motion Kinematics
Circular Motion Kinematics 8.01 W04D1 Today s Reading Assignment: MIT 8.01 Course Notes Chapter 6 Circular Motion Sections 6.1-6.2 Announcements Math Review Week 4 Tuesday 9-11 pm in 26-152. Next Reading
More informationIntegrals in cylindrical, spherical coordinates (Sect. 15.7)
Integrals in clindrical, spherical coordinates (Sect. 15.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.
More informationChapter 1. Vector Analysis
Chapter 1. Vector Analysis Hayt; 8/31/2009; 1-1 1.1 Scalars and Vectors Scalar : Vector: A quantity represented by a single real number Distance, time, temperature, voltage, etc Magnitude and direction
More informationFundamentals of Applied Electromagnetics. Chapter 2 - Vector Analysis
Fundamentals of pplied Electromagnetics Chapter - Vector nalsis Chapter Objectives Operations of vector algebra Dot product of two vectors Differential functions in vector calculus Divergence of a vector
More information10.1 Curves Defined by Parametric Equation
10.1 Curves Defined by Parametric Equation 1. Imagine that a particle moves along the curve C shown below. It is impossible to describe C by an equation of the form y = f (x) because C fails the Vertical
More informationMath 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 informationVectors and Geometry
Vectors and Geometry Vectors In the context of geometry, a vector is a triplet of real numbers. In applications to a generalized parameters space, such as the space of random variables in a reliability
More informationCIRCULAR MOTION AND ROTATION
1. UNIFORM CIRCULAR MOTION So far we have learned a great deal about linear motion. This section addresses rotational motion. The simplest kind of rotational motion is an object moving in a perfect circle
More information7 Curvilinear coordinates
7 Curvilinear coordinates Read: Boas sec. 5.4, 0.8, 0.9. 7. Review of spherical and cylindrical coords. First I ll review spherical and cylindrical coordinate systems so you can have them in mind when
More informationGreen s, Divergence, Stokes: Statements and First Applications
Math 425 Notes 12: Green s, Divergence, tokes: tatements and First Applications The Theorems Theorem 1 (Divergence (planar version)). Let F be a vector field in the plane. Let be a nice region of the plane
More informationThe choice of origin, axes, and length is completely arbitrary.
Polar Coordinates There are many ways to mark points in the plane or in 3-dim space for purposes of navigation. In the familiar rectangular coordinate system, a point is chosen as the origin and a perpendicular
More informationMATH 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 informationLecture 2 : Curvilinear Coordinates
Lecture 2 : Curvilinear Coordinates Fu-Jiun Jiang October, 200 I. INTRODUCTION A. Definition and Notations In 3-dimension Euclidean space, a vector V can be written as V = e x V x + e y V y + e z V z with
More informationMultiple 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(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 informationGeneral review: - a) Dot Product
General review: - a) Dot Product If θ is the angle between the vectors a and b, then a b = a b cos θ NOTE: Two vectors a and b are orthogonal, if and only if a b = 0. Properties of the Dot Product If a,
More informationexample consider flow of water in a pipe. At each point in the pipe, the water molecule has a velocity
Module 1: A Crash Course in Vectors Lecture 1: Scalar and Vector Fields Objectives In this lecture you will learn the following Learn about the concept of field Know the difference between a scalar field
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:
More information2-5 The Calculus of Scalar and Vector Fields (pp.33-55)
9/9/2004 sec 2_5 empty.doc 1/5 2-5 The Calculus of Scalar and Vector Fields (pp.33-55) Q: A: 1. 4. 2. 5. 3. 6. A. The Integration of Scalar and Vector Fields 1. The Line Integral 9/9/2004 sec 2_5 empty.doc
More informationWeek 7: Integration: Special Coordinates
Week 7: Integration: Special Coordinates Introduction Many problems naturally involve symmetry. One should exploit it where possible and this often means using coordinate systems other than Cartesian coordinates.
More informationOrthogonal Curvilinear Coordinates
Chapter 5 Orthogonal Curvilinear Coordinates Last update: 22 Nov 21 Syllabus section: 4. Orthogonal curvilinear coordinates; length of line element; grad, div and curl in curvilinear coordinates; spherical
More informationMATH 280 Multivariate Calculus Fall Integrating a vector field over a curve
MATH 280 Multivariate alculus Fall 2012 Definition Integrating a vector field over a curve We are given a vector field F and an oriented curve in the domain of F as shown in the figure on the left below.
More informationMath 20C Homework 2 Partial Solutions
Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we
More informationIII. TRANSFORMATION RELATIONS
III. TRANSFORMATION RELATIONS The transformation relations from cartesian coordinates to a general curvilinear system are developed here using certain concepts from differential geometry and tensor analysis,
More informationChapter 9 Uniform Circular Motion
9.1 Introduction Chapter 9 Uniform Circular Motion Special cases often dominate our study of physics, and circular motion is certainly no exception. We see circular motion in many instances in the world;
More informationSECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.
SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),
More informationENGI Gradient, Divergence, Curl Page 5.01
ENGI 940 5.0 - Gradient, Divergence, Curl Page 5.0 5. e Gradient Operator A brief review is provided ere for te gradient operator in bot Cartesian and ortogonal non-cartesian coordinate systems. Sections
More informationx + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the
1.(8pts) Find F ds where F = x + ye z + ze y, y + xe z + ze x, z and where T is the T surface in the pictures. (The two pictures are two views of the same surface.) The boundary of T is the unit circle
More informationIntegral Theorems. September 14, We begin by recalling the Fundamental Theorem of Calculus, that the integral is the inverse of the derivative,
Integral Theorems eptember 14, 215 1 Integral of the gradient We begin by recalling the Fundamental Theorem of Calculus, that the integral is the inverse of the derivative, F (b F (a f (x provided f (x
More information- 1 - θ 1. n 1. θ 2. mirror. object. image
TEST 5 (PHY 50) 1. a) How will the ray indicated in the figure on the following page be reflected by the mirror? (Be accurate!) b) Explain the symbols in the thin lens equation. c) Recall the laws governing
More informationUNIT 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 informationCoordinates 2D and 3D Gauss & Stokes Theorems
Coordinates 2 and 3 Gauss & Stokes Theorems Yi-Zen Chu 1 2 imensions In 2 dimensions, we may use Cartesian coordinates r = (x, y) and the associated infinitesimal area We may also employ polar coordinates
More informationProblem 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 informationMultiple 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 informationChapter 3 Vectors. 3.1 Vector Analysis
Chapter 3 Vectors 3.1 Vector nalysis... 1 3.1.1 Introduction to Vectors... 1 3.1.2 Properties of Vectors... 1 3.2 Coordinate Systems... 6 3.2.1 Cartesian Coordinate System... 6 3.2.2 Cylindrical Coordinate
More informationRigid Object. Chapter 10. Angular Position. Angular Position. A rigid object is one that is nondeformable
Rigid Object Chapter 10 Rotation of a Rigid Object about a Fixed Axis A rigid object is one that is nondeformable The relative locations of all particles making up the object remain constant All real objects
More informationMATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm.
MATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm. Bring a calculator and something to write with. Also, you will be allowed to bring in one 8.5 11 sheet of paper
More informationChapter 9 Overview: Parametric and Polar Coordinates
Chapter 9 Overview: Parametric and Polar Coordinates As we saw briefly last year, there are axis systems other than the Cartesian System for graphing (vector coordinates, polar coordinates, rectangular
More informationAppendix: Orthogonal Curvilinear Coordinates. We define the infinitesimal spatial displacement vector dx in a given orthogonal coordinate system with
Appendix: Orthogonal Curvilinear Coordinates Notes: Most of the material presented in this chapter is taken from Anupam G (Classical Electromagnetism in a Nutshell 2012 (Princeton: New Jersey)) Chap 2
More informationCircular Motion Kinematics 8.01 W03D1
Circular Motion Kinematics 8.01 W03D1 Announcements Open up the Daily Concept Questions page on the MITx 8.01x Webpage. Problem Set 2 due Tue Week 3 at 9 pm Week 3 Prepset due Friday Week 3 at 8:30 am
More informationSOLUTIONS 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 informationthe Cartesian coordinate system (which we normally use), in which we characterize points by two coordinates (x, y) and
2.5.2 Standard coordinate systems in R 2 and R Similarly as for functions of one variable, integrals of functions of two or three variables may become simpler when changing coordinates in an appropriate
More informationPART ONE DYNAMICS OF A SINGLE PARTICLE
PART ONE DYNAMICS OF A SINGLE PARTICLE 1 Kinematics of a Particle 1.1 Introduction One of the main goals of this book is to enable the reader to take a physical system, model it by using particles or rigid
More informationLecture 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 informationMATH H53 : Final exam
MATH H53 : Final exam 11 May, 18 Name: You have 18 minutes to answer the questions. Use of calculators or any electronic items is not permitted. Answer the questions in the space provided. If you run out
More informationUniversity of Illinois at Chicago Department of Physics
University of Illinois at Chicago Department of Physics Electromagnetism Qualifying Examination January 4, 2017 9.00 am - 12.00 pm Full credit can be achieved from completely correct answers to 4 questions.
More informationENGI Duffing s Equation Page 4.65
ENGI 940 4. - Duffing s Equation Page 4.65 4. Duffing s Equation Among the simplest models of damped non-linear forced oscillations of a mechanical or electrical system with a cubic stiffness term is Duffing
More informationMULTIVARIABLE INTEGRATION
MULTIVARIABLE INTEGRATION (SPHERICAL POLAR COORDINATES) Question 1 a) Determine with the aid of a diagram an expression for the volume element in r, θ, ϕ. spherical polar coordinates, ( ) [You may not
More informationMATHEMATICS 200 December 2013 Final Exam Solutions
MATHEMATICS 2 December 21 Final Eam Solutions 1. Short Answer Problems. Show our work. Not all questions are of equal difficult. Simplif our answers as much as possible in this question. (a) The line L
More informationRotational Kinematics and Dynamics. UCVTS AIT Physics
Rotational Kinematics and Dynamics UCVTS AIT Physics Angular Position Axis of rotation is the center of the disc Choose a fixed reference line Point P is at a fixed distance r from the origin Angular Position,
More informationUnit Circle: The unit circle has radius 1 unit and is centred at the origin on the Cartesian plane. POA
The Unit Circle Unit Circle: The unit circle has radius 1 unit and is centred at the origin on the Cartesian plane THE EQUATION OF THE UNIT CIRCLE Consider any point P on the unit circle with coordinates
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,
More informationAPPENDIX 2.1 LINE AND SURFACE INTEGRALS
2 APPENDIX 2. LINE AND URFACE INTEGRAL Consider a path connecting points (a) and (b) as shown in Fig. A.2.. Assume that a vector field A(r) exists in the space in which the path is situated. Then the line
More informationMATHEMATICS 200 April 2010 Final Exam Solutions
MATHEMATICS April Final Eam Solutions. (a) A surface z(, y) is defined by zy y + ln(yz). (i) Compute z, z y (ii) Evaluate z and z y in terms of, y, z. at (, y, z) (,, /). (b) A surface z f(, y) has derivatives
More informationPHYS 281: Midterm Exam
PHYS 28: Midterm Exam October 28, 200, 8:00-9:20 Last name (print): Initials: No calculator or other aids allowed PHYS 28: Midterm Exam Instructor: B. R. Sutherland Date: October 28, 200 Time: 8:00-9:20am
More informationCourse Notes Math 275 Boise State University. Shari Ultman
Course Notes Math 275 Boise State University Shari Ultman Fall 2017 Contents 1 Vectors 1 1.1 Introduction to 3-Space & Vectors.............. 3 1.2 Working With Vectors.................... 7 1.3 Introduction
More informationMath 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 >
Math 63 Final Problem 1: [ points, 5 points to each part] Given the points P : (1, 1, 1), Q : (1,, ), R : (,, c 1), where c is a parameter, find (a) the vector equation of the line through P and Q. (b)
More informationDistance Formula in 3-D Given any two points P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ) the distance between them is ( ) ( ) ( )
Vectors and the Geometry of Space Vector Space The 3-D coordinate system (rectangular coordinates ) is the intersection of three perpendicular (orthogonal) lines called coordinate axis: x, y, and z. Their
More informationCURRENT MATERIAL: Vector Calculus.
Math 275, section 002 (Ultman) Spring 2012 FINAL EXAM REVIEW The final exam will be held on Wednesday 9 May from 8:00 10:00am in our regular classroom. You will be allowed both sides of two 8.5 11 sheets
More informationVector 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 informationPRACTICE 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 informationThe 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 informationAn Overview of Mechanics
An Overview of Mechanics Mechanics: The study of how bodies react to forces acting on them. Statics: The study of bodies in equilibrium. Dynamics: 1. Kinematics concerned with the geometric aspects of
More information3. On the grid below, sketch and label graphs of the following functions: y = sin x, y = cos x, and y = sin(x π/2). π/2 π 3π/2 2π 5π/2
AP Physics C Calculus C.1 Name Trigonometric Functions 1. Consider the right triangle to the right. In terms of a, b, and c, write the expressions for the following: c a sin θ = cos θ = tan θ =. Using
More informationMagnetostatics. Lecture 23: Electromagnetic Theory. Professor D. K. Ghosh, Physics Department, I.I.T., Bombay
Magnetostatics Lecture 23: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Magnetostatics Up until now, we have been discussing electrostatics, which deals with physics
More informationName: ID: Math 233 Exam 1. Page 1
Page 1 Name: ID: This exam has 20 multiple choice questions, worth 5 points each. You are allowed to use a scientific calculator and a 3 5 inch note card. 1. Which of the following pairs of vectors are
More informationGauss s Law & Potential
Gauss s Law & Potential Lecture 7: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Flux of an Electric Field : In this lecture we introduce Gauss s law which happens to
More informationVectors, metric and the connection
Vectors, metric and the connection 1 Contravariant and covariant vectors 1.1 Contravariant vectors Imagine a particle moving along some path in the 2-dimensional flat x y plane. Let its trajectory be given
More informationBiot-Savart. The equation is this:
Biot-Savart When a wire carries a current, this current produces a magnetic field in the vicinity of the wire. One way of determining the strength and direction of this field is with the Law of Biot-Savart.
More informationSolutions 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 informationASTR 320: Solutions to Problem Set 2
ASTR 320: Solutions to Problem Set 2 Problem 1: Streamlines A streamline is defined as a curve that is instantaneously tangent to the velocity vector of a flow. Streamlines show the direction a massless
More informationSummary for Vector Calculus and Complex Calculus (Math 321) By Lei Li
Summary for Vector alculus and omplex alculus (Math 321) By Lei Li 1 Vector alculus 1.1 Parametrization urves, surfaces, or volumes can be parametrized. Below, I ll talk about 3D case. Suppose we use e
More informationLecture 6 - Introduction to Electricity
Lecture 6 - Introduction to Electricity A Puzzle... We are all familiar with visualizing an integral as the area under a curve. For example, a b f[x] dx equals the sum of the areas of the rectangles of
More informationRIGID BODY MOTION (Section 16.1)
RIGID BODY MOTION (Section 16.1) There are cases where an object cannot be treated as a particle. In these cases the size or shape of the body must be considered. Rotation of the body about its center
More informationEELE 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 informationRotation Basics. I. Angular Position A. Background
Rotation Basics I. Angular Position A. Background Consider a student who is riding on a merry-go-round. We can represent the student s location by using either Cartesian coordinates or by using cylindrical
More informationFluid Dynamics Problems M.Sc Mathematics-Second Semester Dr. Dinesh Khattar-K.M.College
Fluid Dynamics Problems M.Sc Mathematics-Second Semester Dr. Dinesh Khattar-K.M.College 1. (Example, p.74, Chorlton) At the point in an incompressible fluid having spherical polar coordinates,,, the velocity
More informationAPPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
More informationChapter 10. Rotation of a Rigid Object about a Fixed Axis
Chapter 10 Rotation of a Rigid Object about a Fixed Axis Angular Position Axis of rotation is the center of the disc Choose a fixed reference line. Point P is at a fixed distance r from the origin. A small
More informationEQUATIONS OF MOTION: NORMAL AND TANGENTIAL COORDINATES (Section 13.5)
EQUATIONS OF MOTION: NORMAL AND TANGENTIAL COORDINATES (Section 13.5) Today s Objectives: Students will be able to apply the equation of motion using normal and tangential coordinates. APPLICATIONS Race
More information3 Chapter. Gauss s Law
3 Chapter Gauss s Law 3.1 Electric Flux... 3-2 3.2 Gauss s Law (see also Gauss s Law Simulation in Section 3.10)... 3-4 Example 3.1: Infinitely Long Rod of Uniform Charge Density... 3-9 Example 3.2: Infinite
More informationMathematical 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 information3 + 4i 2 + 3i. 3 4i Fig 1b
The introduction of complex numbers in the 16th century was a natural step in a sequence of extensions of the positive integers, starting with the introduction of negative numbers (to solve equations of
More informationMath 302 Outcome Statements Winter 2013
Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a
More informationCurvilinear Coordinates
University of Alabama Department of Physics and Astronomy PH 106-4 / LeClair Fall 2008 Curvilinear Coordinates Note that we use the convention that the cartesian unit vectors are ˆx, ŷ, and ẑ, rather than
More informationSections 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 informationThe region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.
Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.
More information10.1 Review of Parametric Equations
10.1 Review of Parametric Equations Recall that often, instead of representing a curve using just x and y (called a Cartesian equation), it is more convenient to define x and y using parametric equations
More informationANTENNAS. Vector and Scalar Potentials. Maxwell's Equations. E = jωb. H = J + jωd. D = ρ (M3) B = 0 (M4) D = εe
ANTENNAS Vector and Scalar Potentials Maxwell's Equations E = jωb H = J + jωd D = ρ B = (M) (M) (M3) (M4) D = εe B= µh For a linear, homogeneous, isotropic medium µ and ε are contant. Since B =, there
More information