Directional Derivative and the Gradient Operator
|
|
- Donna Gallagher
- 5 years ago
- Views:
Transcription
1 Chapter 4 Directional Derivative and the Gradient Operator The equation z = f(x, y) defines a surface in 3 dimensions. We can write this as z f(x, y) = 0, or g(x, y, z) = 0, where g(x, y, z) = z f(x, y). The general equation of a surface is g(x, y, z) = c, (4.1) where c is a parameter. Each value of c labels one member of the family of surfaces. g has a magnitude but no direction. Thus, g is a scalar function of x, y and z. Example 4.1 Consider g(x, y, z) = x 2 + y 2 + z 2 = a 2, where c = a 2. This describes the family of concentric spheres centred at the origin and with radius a. An example is shown in Figure 4.1. Consider two surfaces (S 1 and S 2 ) on which g is equal to c 1 and c 2 respectively. This is illustrated in Figure 4.2. Suppose the point lies on surface S 1 and Q on S 2. At, g = c 1 and at Q g = c 2. Thus, the value of g changes from c 1 to c 2 as we move along the path Q. For general and Q, we can calculate the rate of change of g along the line Q. This means we calculate the directional derivative. Suppose the path is the straight line joining and Q. The unit vector û is parallel to Q and has cartesian components û = (l, m, n) li + mj + nk, 57
2 58CHATER 4. DIRECTIONAL DERIVATIVE AND THE GRADIENT OERATOR Figure 4.1: The sphere x 2 + y 2 + z 2 = 1. and i, j, k are the unit vectors along the x, y and z axes. As û is a unit vector û = 1. This means that û 2 = û û = l 2 + m 2 + n 2 = 1. This result follows directly from the scalar (or dot) product of vectors. then A = (A x, A y, A z ) = A x i + A y j + A z k, B = (B x, B y, B z ) = B x i + B y j + B z k, A B = A x B x + A y B y + A z B z. Define the coordinates of as (x 0, y 0, z 0 ) and Q as (x, y, z). If Q is a distance s from in the direction of û, the coordinates of Q are or OQ = O + sû, x = x 0 + ls, y = y 0 + ms, z = z 0 ns. (4.2) As s is varied ( < s < + ) then any point of the line may be reached. This is called the parametric equation for the line. The coordinates of Q may be written as (x(s), y(s), z(s)). Note that the vector Q is Q = sû.
3 59 Figure 4.2: Two surfaces labelled by the constants c 1 and c 2. The path between the points and Q is indicated. Since û is a unit vector, s represents the distance from to Q. The variation of g along the line is g(x, y, z) = g(x(s), y(s), z(s)), on using (4.2). Using the Chain Rule ( ) ( dg g = ds x dx ds + g y dy ds + g z dz ) ds Here the subscript is used to indicate that the derivatives are evaluated at the point. Using (4.2), this may be rearranged to give ( ) ( dg g = ds x l + g y m + g ) z n. Note that the right hand side is equivalent to the scalar product of the two vectors ( g x i + g y j + g ) z k (li + mj + nk).
4 60CHATER 4. DIRECTIONAL DERIVATIVE AND THE GRADIENT OERATOR Thus, ( ) ( dg g = ds x i + g y j + g ) z k û, (4.3) which is called the directional derivative of g along the direction û at the point. The vector g x i + g y j + g k g, (4.4) z is so important in mathematics that it is given the special name of the gradient of the scalar function g(x, y, z). It is denoted by g, and is also called either grad g or the gradient of g. Note that is a vector operator. It converts a scalar function into a vector function. We can think of as the vector operator = i x + j y + k z The symbol is called grad, del or nabla. Thus, the directional derivative of g(x, y, z) along û at (x 0, y 0, z 0 ) is dg ds = ( g û) x 0,y 0,z 0. Note that both terms on the right hand side are vectors and we take the scalar product of two vectors to produce the directional derivative. Example 4.2 Find the directional derivative of g = xy 2 z 3, in the direction u = 2i + 6j + 3k at the point = (1, 1, 1). First we need g = i g x + j g y + k g z = i(y2 z 3 ) + j(2xz 3 ) + k(3xy 2 z 2 ). Note that g is a vector. At (1, 1, 1), we have ( g) = i + 2j + 3k. Next we need to calculate the unit vector so that û u (2i + 6j + 3k) = = 2 u i j k.
5 4.1. NORMALS TO SURFACES AND TANGENT LANES 61 Thus, the directional derivative we require is dg ds = ( g) û = 1 (i + 2j + 3k) (2i + 6j + 3k). 7 Evaluating the scalar product gives the final answer as dg ds = 1 7 ( ) = Note that the rates of change of g(x, y, z) along the x, y and z axes are just g/ x, g/ y and g/ z, from before. To confirm that the directional derivative gives this result, we set û = i. Thus, ( g g i = x i + g y j + g ) z k i = g x. Similarly û = j gives g/ y and û = k gives g/ z. 4.1 Normals to surfaces and tangent planes Given a surface f(x, y, z) = c, and a point on it, the tangent plane (T ) to the surface at is the plane which just touches the surface at. The normal vector, (n), to the surface at is defined as the vector which is orthogonal (perpendicular) to every vector t in T through. This is illustrated in Figure 4.3. Note 1: Since f(x, y, z) is constant on the surface, the directional derivative, evaluated at, along any t will be zero. Thus, ( ) df = ( f) ds t = 0, for any t. (4.5) Thus, ( f) is normal to both the surface (at ) and the tangent plane T. ( f) is parallel to the normal at called n. Example 4.3 Let f(x, y, z) = x y 2 +xz. The surface f = 1 contains the point = (1, 2, 2), (check to see that f(1, 2, 2) = 1). Find a vector parallel to n at. and so, evaluating this at (1, 2, 2) gives f = i(1 + z) + j( 2y) + k(x), ( f) = 3i 4j + k, and is parallel to n at. Note 2: Consider the rate of change of f(x, y, z) at along different directions defined by û. At,
6 62CHATER 4. DIRECTIONAL DERIVATIVE AND THE GRADIENT OERATOR Figure 4.3: The surface f(x, y, z) = c is shown. The tangent plane is labelled by T and a typical vector t lying in the tangent plane passing through is shown. ( ) df = ( f) û = f cos γ, ds û (from A B = AB cos θ). When γ = π/2 we find that (df/ds) γ=π/2 = 0. This is to be expected since û coincides with some t in the tangent plane. Thus, û is in the tangent plane and f is constant at, see (4.5). Evidently, (df/ds) has its maximum value when cos γ = 1, namely when γ = 0. Hence, û coincides with the normal direction (n or f). In this case, df ds = f. Note 3: The equation of the plane T, through 0. If (x, y, z) is in T, then the vector (r r 0 ) must be perpendicular to n. Thus, (r r 0 ) n 0 = 0.
7 4.1. NORMALS TO SURFACES AND TANGENT LANES 63 Figure 4.4: The direction of the normal to the surface, n = f makes an angle γ to the direction defined by u. Therefore, if r = xi + yj + zk and r 0 = x 0 i + y 0 j + z 0 k, then using the definition of f and expanding the scalar product gives the equation of the plane as ( ) ( ) ( ) f f f (x x 0 ) + (y y 0 ) + (z z 0 ) = 0. x 0 y 0 z 0 This is of the form ax + by + cz = d, and is the equation of the plane T. Example 4.4 Find the tangent plane to xy 2 + x 2 z = 7, at the point (1, 2, 3). Thus, f = xy 2 + x 2 z and f = 7. The normal vector is n and may be taken as ( f) (1,2,3). Thus, f = i(y 2 + 2xz) + j(2xy) + k(x 2 ),
8 64CHATER 4. DIRECTIONAL DERIVATIVE AND THE GRADIENT OERATOR at the point (1, 2, 3) we have n = f = 10i + 4j + k. With r 0 = (1, 2, 3), so (r r 0 ) n = 0 gives (x 1) 10 + (y 2) 4 + (z 3) 1 = 0, 10x + 4y + z = 21.
Directional Derivatives in the Plane
Directional Derivatives in the Plane P. Sam Johnson April 10, 2017 P. Sam Johnson (NIT Karnataka) Directional Derivatives in the Plane April 10, 2017 1 / 30 Directional Derivatives in the Plane Let z =
More informationVector Calculus, Maths II
Section A Vector Calculus, Maths II REVISION (VECTORS) 1. Position vector of a point P(x, y, z) is given as + y and its magnitude by 2. The scalar components of a vector are its direction ratios, and represent
More informationCreated by T. Madas LINE INTEGRALS. Created by T. Madas
LINE INTEGRALS LINE INTEGRALS IN 2 DIMENSIONAL CARTESIAN COORDINATES Question 1 Evaluate the integral ( x + 2y) dx, C where C is the path along the curve with equation y 2 = x + 1, from ( ) 0,1 to ( )
More informationMAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.
MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant
More information2. (i) Find the equation of the circle which passes through ( 7, 1) and has centre ( 4, 3).
Circle 1. (i) Find the equation of the circle with centre ( 7, 3) and of radius 10. (ii) Find the centre of the circle 2x 2 + 2y 2 + 6x + 8y 1 = 0 (iii) What is the radius of the circle 3x 2 + 3y 2 + 5x
More informationVECTORS IN A STRAIGHT LINE
A. The Equation of a Straight Line VECTORS P3 VECTORS IN A STRAIGHT LINE A particular line is uniquely located in space if : I. It has a known direction, d, and passed through a known fixed point, or II.
More informationExercises for Multivariable Differential Calculus XM521
This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done
More informationThis exam will be over material covered in class from Monday 14 February through Tuesday 8 March, corresponding to sections in the text.
Math 275, section 002 (Ultman) Spring 2011 MIDTERM 2 REVIEW The second midterm will be held in class (1:40 2:30pm) on Friday 11 March. You will be allowed one half of one side of an 8.5 11 sheet of paper
More informationSec. 1.1: Basics of Vectors
Sec. 1.1: Basics of Vectors Notation for Euclidean space R n : all points (x 1, x 2,..., x n ) in n-dimensional space. Examples: 1. R 1 : all points on the real number line. 2. R 2 : all points (x 1, x
More informationMathematics (Course B) Lent Term 2005 Examples Sheet 2
N12d Natural Sciences, Part IA Dr M. G. Worster Mathematics (Course B) Lent Term 2005 Examples Sheet 2 Please communicate any errors in this sheet to Dr Worster at M.G.Worster@damtp.cam.ac.uk. Note that
More 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 informationTaylor Series and stationary points
Chapter 5 Taylor Series and stationary points 5.1 Taylor Series The surface z = f(x, y) and its derivatives can give a series approximation for f(x, y) about some point (x 0, y 0 ) as illustrated in Figure
More informationMath 234. What you should know on day one. August 28, You should be able to use general principles like. x = cos t, y = sin t, 0 t π.
Math 234 What you should know on day one August 28, 2001 1 You should be able to use general principles like Length = ds, Area = da, Volume = dv For example the length of the semi circle x = cos t, y =
More informationSection 4.3 Vector Fields
Section 4.3 Vector Fields DEFINITION: A vector field in R n is a map F : A R n R n that assigns to each point x in its domain A a vector F(x). If n = 2, F is called a vector field in the plane, and if
More information9.5. Lines and Planes. Introduction. Prerequisites. Learning Outcomes
Lines and Planes 9.5 Introduction Vectors are very convenient tools for analysing lines and planes in three dimensions. In this Section you will learn about direction ratios and direction cosines and then
More informationSecond Order ODEs. Second Order ODEs. In general second order ODEs contain terms involving y, dy But here only consider equations of the form
Second Order ODEs Second Order ODEs In general second order ODEs contain terms involving y, dy But here only consider equations of the form A d2 y dx 2 + B dy dx + Cy = 0 dx, d2 y dx 2 and F(x). where
More informationProblem Solving 1: Line Integrals and Surface Integrals
A. Line Integrals MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics Problem Solving 1: Line Integrals and Surface Integrals The line integral of a scalar function f ( xyz),, along a path C is
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 2/13/13, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.2. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241sp13/241.html)
More informationa Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
More informationTangent Plane. Linear Approximation. The Gradient
Calculus 3 Lia Vas Tangent Plane. Linear Approximation. The Gradient The tangent plane. Let z = f(x, y) be a function of two variables with continuous partial derivatives. Recall that the vectors 1, 0,
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 information21-256: Partial differentiation
21-256: Partial differentiation Clive Newstead, Thursday 5th June 2014 This is a summary of the important results about partial derivatives and the chain rule that you should know. Partial derivatives
More informationBSc (Hons) in Computer Games Development. vi Calculate the components a, b and c of a non-zero vector that is orthogonal to
1 APPLIED MATHEMATICS INSTRUCTIONS Full marks will be awarded for the correct solutions to ANY FIVE QUESTIONS. This paper will be marked out of a TOTAL MAXIMUM MARK OF 100. Credit will be given for clearly
More informationOverview of vector calculus. Coordinate systems in space. Distance formula. (Sec. 12.1)
Math 20C Multivariable Calculus Lecture 1 1 Coordinates in space Slide 1 Overview of vector calculus. Coordinate systems in space. Distance formula. (Sec. 12.1) Vector calculus studies derivatives and
More informationx 1. x n i + x 2 j (x 1, x 2, x 3 ) = x 1 j + x 3
Version: 4/1/06. Note: These notes are mostly from my 5B course, with the addition of the part on components and projections. Look them over to make sure that we are on the same page as regards inner-products,
More information(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.
MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.
More informationVector Fields and Line Integrals The Fundamental Theorem for Line Integrals
Math 280 Calculus III Chapter 16 Sections: 16.1, 16.2 16.3 16.4 16.5 Topics: Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals Green s Theorem Curl and Divergence Section 16.1
More information11.1 Three-Dimensional Coordinate System
11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into
More informationMATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS
MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,
More informationEE2007: Engineering Mathematics II Vector Calculus
EE2007: Engineering Mathematics II Vector Calculus Ling KV School of EEE, NTU ekvling@ntu.edu.sg Rm: S2-B2b-22 Ver 1.1: Ling KV, October 22, 2006 Ver 1.0: Ling KV, Jul 2005 EE2007/Ling KV/Aug 2006 EE2007:
More informationg(t) = f(x 1 (t),..., x n (t)).
Reading: [Simon] p. 313-333, 833-836. 0.1 The Chain Rule Partial derivatives describe how a function changes in directions parallel to the coordinate axes. Now we shall demonstrate how the partial derivatives
More informationNATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 24, Time Allowed: 150 Minutes Maximum Marks: 30
NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 24, 2011 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions on the following page 1 INSTRUCTIONS
More informationVectors and Fields. Vectors versus scalars
C H A P T E R 1 Vectors and Fields Electromagnetics deals with the study of electric and magnetic fields. It is at once apparent that we need to familiarize ourselves with the concept of a field, and in
More informationCourse Summary Math 211
Course Summary Math 211 table of contents I. Functions of several variables. II. R n. III. Derivatives. IV. Taylor s Theorem. V. Differential Geometry. VI. Applications. 1. Best affine approximations.
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 informationDefinition 3 (Continuity). A function f is continuous at c if lim x c f(x) = f(c).
Functions of Several Variables A function of several variables is just what it sounds like. It may be viewed in at least three different ways. We will use a function of two variables as an example. z =
More informationEngineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics
Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics Module 10 - Lecture 24 Kinematics of a particle moving on a curve Today,
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 informationSolutions to old Exam 3 problems
Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are
More informationEE2007: Engineering Mathematics II Vector Calculus
EE2007: Engineering Mathematics II Vector Calculus Ling KV School of EEE, NTU ekvling@ntu.edu.sg Rm: S2-B2a-22 Ver: August 28, 2010 Ver 1.6: Martin Adams, Sep 2009 Ver 1.5: Martin Adams, August 2008 Ver
More information4.3 Equations in 3-space
4.3 Equations in 3-space istance can be used to define functions from a 3-space R 3 to the line R. Let P be a fixed point in the 3-space R 3 (say, with coordinates P (2, 5, 7)). Consider a function f :
More informationG G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv
1. Matching. Fill in the appropriate letter. 1. ds for a surface z = g(x, y) A. r u r v du dv 2. ds for a surface r(u, v) B. r u r v du dv 3. ds for any surface C. G x G z, G y G z, 1 4. Unit normal N
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)
More 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 informationHigher Mathematics Course Notes
Higher Mathematics Course Notes Equation of a Line (i) Collinearity: (ii) Gradient: If points are collinear then they lie on the same straight line. i.e. to show that A, B and C are collinear, show that
More informationCalculus III. Math 233 Spring Final exam May 3rd. Suggested solutions
alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.
More informationMath 147, Homework 1 Solutions Due: April 10, 2012
1. For what values of a is the set: Math 147, Homework 1 Solutions Due: April 10, 2012 M a = { (x, y, z) : x 2 + y 2 z 2 = a } a smooth manifold? Give explicit parametrizations for open sets covering M
More informationWeek 4: Differentiation for Functions of Several Variables
Week 4: Differentiation for Functions of Several Variables Introduction A functions of several variables f : U R n R is a rule that assigns a real number to each point in U, a subset of R n, For the next
More informationContents. 2 Partial Derivatives. 2.1 Limits and Continuity. Calculus III (part 2): Partial Derivatives (by Evan Dummit, 2017, v. 2.
Calculus III (part 2): Partial Derivatives (by Evan Dummit, 2017, v 260) Contents 2 Partial Derivatives 1 21 Limits and Continuity 1 22 Partial Derivatives 5 23 Directional Derivatives and the Gradient
More informationCIRCLES PART - II Theorem: The condition that the straight line lx + my + n = 0 may touch the circle x 2 + y 2 = a 2 is
CIRCLES PART - II Theorem: The equation of the tangent to the circle S = 0 at P(x 1, y 1 ) is S 1 = 0. Theorem: The equation of the normal to the circle S x + y + gx + fy + c = 0 at P(x 1, y 1 ) is (y
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechanics for Scientists and Engineers David Miller Background mathematics 7 Differential equations Differential equations in one variable A differential equation involves derivatives of some function
More informationDifferential equations. Background mathematics review
Differential equations Background mathematics review David Miller Differential equations First-order differential equations Background mathematics review David Miller Differential equations in one variable
More informationMultivariable Calculus
Multivariable alculus Jaron Kent-Dobias May 17, 2011 1 Lines in Space By space, we mean R 3. First, conventions. Always draw right-handed axes. You can define a L line precisely in 3-space with 2 points,
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 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 informationCreated by T. Madas VECTOR OPERATORS. Created by T. Madas
VECTOR OPERATORS GRADIENT gradϕ ϕ Question 1 A surface S is given by the Cartesian equation x 2 2 + y = 25. a) Draw a sketch of S, and describe it geometrically. b) Determine an equation of the tangent
More informationCore A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document
Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document Background knowledge: (a) The arithmetic of integers (including HCFs and LCMs), of fractions, and of real numbers.
More information12.1. Cartesian Space
12.1. Cartesian Space In most of your previous math classes, we worked with functions on the xy-plane only meaning we were working only in 2D. Now we will be working in space, or rather 3D. Now we will
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 informationMethod of Lagrange Multipliers
Method of Lagrange Multipliers A. Salih Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram September 2013 Lagrange multiplier method is a technique
More informationVector Calculus lecture notes
Vector Calculus lecture notes Thomas Baird December 13, 21 Contents 1 Geometry of R 3 2 1.1 Coordinate Systems............................... 2 1.1.1 Distance................................. 3 1.1.2 Surfaces.................................
More informationME201 ADVANCED CALCULUS MIDTERM EXAMINATION. Instructor: R. Culham. Name: Student ID Number: Instructions
ME201 ADVANCED CALCULUS MIDTERM EXAMINATION February 14, 2017 8:30 am - 10:30 am Instructor: R. Culham Name: Student ID Number: Instructions 1. This is a 2 hour, closed-book examination. 2. Permitted aids
More information4 Partial Differentiation
4 Partial Differentiation Many equations in engineering, physics and mathematics tie together more than two variables. For example Ohm s Law (V = IR) and the equation for an ideal gas, PV = nrt, which
More informationSimple Co-ordinate geometry problems
Simple Co-ordinate geometry problems 1. Find the equation of straight line passing through the point P(5,2) with equal intercepts. 1. Method 1 Let the equation of straight line be + =1, a,b 0 (a) If a=b
More informationSOME PROBLEMS YOU SHOULD BE ABLE TO DO
OME PROBLEM YOU HOULD BE ABLE TO DO I ve attempted to make a list of the main calculations you should be ready for on the exam, and included a handful of the more important formulas. There are no examples
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 informationENGI 4430 Parametric Vector Functions Page dt dt dt
ENGI 4430 Parametric Vector Functions Page 2-01 2. Parametric Vector Functions (continued) Any non-zero vector r can be decomposed into its magnitude r and its direction: r rrˆ, where r r 0 Tangent Vector:
More informationDepartment of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008
Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Code: C-031 Date and time: 17 Nov, 2008, 9:30 A.M. - 12:30 P.M. Maximum Marks: 45 Important Instructions: 1. The question
More informationPrint Your Name: Your Section:
Print Your Name: Your Section: Mathematics 1c. Practice Final Solutions This exam has ten questions. J. Marsden You may take four hours; there is no credit for overtime work No aids (including notes, books,
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 information18.02 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. V9. Surface Integrals Surface
More information9.7 Gradient of a Scalar Field. Directional Derivative. Mean Value Theorem. Special Cases
SEC. 9.7 Gradient of a Scalar Field. Directional Derivative 395 Mean Value Theorems THEOREM Mean Value Theorem Let f(x, y, z) be continuous and have continuous first partial derivatives in a domain D in
More information1 Functions of Several Variables Some Examples Level Curves / Contours Functions of More Variables... 6
Contents 1 Functions of Several Variables 1 1.1 Some Examples.................................. 2 1.2 Level Curves / Contours............................. 4 1.3 Functions of More Variables...........................
More informationPractice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.
1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line
More 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 informationEdexcel past paper questions. Core Mathematics 4. Parametric Equations
Edexcel past paper questions Core Mathematics 4 Parametric Equations Edited by: K V Kumaran Email: kvkumaran@gmail.com C4 Maths Parametric equations Page 1 Co-ordinate Geometry A parametric equation of
More informationDO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START
Math 265 Student name: KEY Final Exam Fall 23 Instructor & Section: This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot also be a communication device
More informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics GROUPS Trinity Term 06 MA3: Advanced Calculus SAMPLE EXAM, Solutions DAY PLACE TIME Prof. Larry Rolen Instructions to Candidates: Attempt
More informationCHAPTER 7 DIV, GRAD, AND CURL
CHAPTER 7 DIV, GRAD, AND CURL 1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: (1 ϕ = ( ϕ, ϕ,, ϕ x 1 x 2 x n
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 informationMath 265H: Calculus III Practice Midterm II: Fall 2014
Name: Section #: Math 65H: alculus III Practice Midterm II: Fall 14 Instructions: This exam has 7 problems. The number of points awarded for each question is indicated in the problem. Answer each question
More informationCreated by T. Madas SURFACE INTEGRALS. Created by T. Madas
SURFACE INTEGRALS Question 1 Find the area of the plane with equation x + 3y + 6z = 60, 0 x 4, 0 y 6. 8 Question A surface has Cartesian equation y z x + + = 1. 4 5 Determine the area of the surface which
More informationMath 233. Practice Problems Chapter 15. i j k
Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed
More informationMaths Higher Prelim Content
Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of
More informationKevin James. MTHSC 206 Section 12.5 Equations of Lines and Planes
MTHSC 206 Section 12.5 Equations of Lines and Planes Definition A line in R 3 can be described by a point and a direction vector. Given the point r 0 and the direction vector v. Any point r on the line
More informationEE2007: Engineering Mathematics II Vector Calculus
EE2007: Engineering Mathematics II Vector Calculus Ling KV School of EEE, NTU ekvling@ntu.edu.sg Rm: S2-B2b-22 Ver 1.1: Ling KV, October 22, 2006 Ver 1.0: Ling KV, Jul 2005 EE2007/Ling KV/Aug 2006 My part:
More informationME201 ADVANCED CALCULUS MIDTERM EXAMINATION. Instructor: R. Culham. Name: Student ID Number: Instructions
ME201 ADVANCED CALCULUS MIDTERM EXAMINATION February 13, 2018 8:30 am - 10:30 am Instructor: R. Culham Name: Student ID Number: Instructions 1. This is a 2 hour, closed-book examination. 2. Permitted aids
More informationSYSTEM OF CIRCLES If d is the distance between the centers of two intersecting circles with radii r 1, r 2 and θ is the
SYSTEM OF CIRCLES Theorem: If d is the distance between the centers of two intersecting circles with radii r 1, r 2 and θ is the 2 2 2 d r1 r2 angle between the circles then cos θ =. 2r r 1 2 Proof: Let
More information3: Mathematics Review
3: Mathematics Review B. Rouben McMaster University Course EP 4D03/6D03 Nuclear Reactor Analysis (Reactor Physics) 015 Sept.-Dec. 015 September 1 Review of: Table of Contents Co-ordinate systems (Cartesian,
More informationDerivatives and Integrals
Derivatives and Integrals Definition 1: Derivative Formulas d dx (c) = 0 d dx (f ± g) = f ± g d dx (kx) = k d dx (xn ) = nx n 1 (f g) = f g + fg ( ) f = f g fg g g 2 (f(g(x))) = f (g(x)) g (x) d dx (ax
More informatione x2 dxdy, e x2 da, e x2 x 3 dx = e
STS26-4 Calculus II: The fourth exam Dec 15, 214 Please show all your work! Answers without supporting work will be not given credit. Write answers in spaces provided. You have 1 hour and 2minutes to complete
More information(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3
Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III
More information25. Chain Rule. Now, f is a function of t only. Expand by multiplication:
25. Chain Rule The Chain Rule is present in all differentiation. If z = f(x, y) represents a two-variable function, then it is plausible to consider the cases when x and y may be functions of other variable(s).
More informationIntroduction to Vector Calculus (29) SOLVED EXAMPLES. (d) B. C A. (f) a unit vector perpendicular to both B. = ˆ 2k = = 8 = = 8
Introduction to Vector Calculus (9) SOLVED EXAMPLES Q. If vector A i ˆ ˆj k, ˆ B i ˆ ˆj, C i ˆ 3j ˆ kˆ (a) A B (e) A B C (g) Solution: (b) A B (c) A. B C (d) B. C A then find (f) a unit vector perpendicular
More informationENGI Partial Differentiation Page y f x
ENGI 3424 4 Partial Differentiation Page 4-01 4. Partial Differentiation For functions of one variable, be found unambiguously by differentiation: y f x, the rate of change of the dependent variable can
More information0, otherwise. Find each of the following limits, or explain that the limit does not exist.
Midterm Solutions 1, y x 4 1. Let f(x, y) = 1, y 0 0, otherwise. Find each of the following limits, or explain that the limit does not exist. (a) (b) (c) lim f(x, y) (x,y) (0,1) lim f(x, y) (x,y) (2,3)
More informationLB 220 Homework 4 Solutions
LB 220 Homework 4 Solutions Section 11.4, # 40: This problem was solved in class on Feb. 03. Section 11.4, # 42: This problem was also solved in class on Feb. 03. Section 11.4, # 43: Also solved in class
More informationMathematical Notation Math Calculus & Analytic Geometry III
Name : Mathematical Notation Math 221 - alculus & Analytic Geometry III Use Word or WordPerect to recreate the ollowing documents. Each article is worth 10 points and can e printed and given to the instructor
More informationS12.1 SOLUTIONS TO PROBLEMS 12 (ODD NUMBERS)
OLUTION TO PROBLEM 2 (ODD NUMBER) 2. The electric field is E = φ = 2xi + 2y j and at (2, ) E = 4i + 2j. Thus E = 2 5 and its direction is 2i + j. At ( 3, 2), φ = 6i + 4 j. Thus the direction of most rapid
More informationDirectional Derivatives and Gradient Vectors. Suppose we want to find the rate of change of a function z = f x, y at the point in the
14.6 Directional Derivatives and Gradient Vectors 1. Partial Derivates are nice, but they only tell us the rate of change of a function z = f x, y in the i and j direction. What if we are interested in
More information