ENGI Gradient, Divergence, Curl Page 5.01

Size: px
Start display at page:

Download "ENGI Gradient, Divergence, Curl Page 5.01"

Transcription

1 ENGI Gradient, Divergence, Curl Page e Gradient Operator A brief review is provided ere for te gradient operator in bot Cartesian and ortogonal non-cartesian coordinate systems. Sections in tis Capter: 5.0 Gradient, Divergence, Curl and Laplacian (Cartesian) 5.0 Differentiation in Ortogonal Curvilinear Coordinate Systems 5.0 Summary able for te Gradient Operator 5.04 Derivatives of Basis Vectors 5.0 Gradient, Divergence, Curl and Laplacian (Cartesian) Let z be a function of two independent variables (x, y), so tat z = f (x, y). e function z = f (x, y) defines a surface in. At any point (x, y) in te x-y plane, te direction in wic one must travel in order to experience te greatest possible rate of increase in z at tat point is te direction of te gradient vector, f ˆ f f i ˆj x y e magnitude of te gradient vector is tat greatest possible rate of increase in z at tat point. e gradient vector is not constant everywere, unless te surface is a plane. (e symbol is usually pronounced del ). e concept of te gradient vector can be extended to functions of any number of variables. If u = f (x, y, z, t), ten f f f f f x y z t. If v is a function of position r and time t, wile position is in turn a function of time, ten by te cain rule of differentiation, d v v dx v dy v dz v dr v v dt x dt y dt z dt t dt t d v v v v dt t wic is of use in te study of fluid dynamics.

2 ENGI Gradient, Divergence, Curl Page 5.0 e gradient operator can also be applied to vectors via te scalar (dot) and vector (cross) products: e divergence of a vector field F(x, y, z) is div F F F F F F F F x y z x y z A region free of sources and sinks will ave zero divergence: te total flux into any region is balanced by te total flux out from tat region. e curl of a vector field F(x, y, z) is ˆi curl F F ˆj F kˆ F F F x y z F F y z x F F F z x y In an irrotational field, curl Wenever or Proof: F 0. F for some twice differentiable potential function, curl F 0 curl grad F F F F x y z 0 ˆ i 0 x x y z z y curl ˆ j 0 y y z x x z ˆ k 0 z z x y y x

3 ENGI Gradient, Divergence, Curl Page 5.0 Among many identities involving te gradient operator is div curl F F 0 for all twice-differentiable vector functions F Proof: div curl F F F F F F F x y z y z x z x y F F F F F F 0 x y x z y z y x z x z y e divergence of te gradient of a scalar function is te Laplacian: div grad f f f f f f x y z for all twice-differentiable scalar functions f. In ortogonal non-cartesian coordinate systems, te expressions for te gradient operator are not as simple.

4 ENGI Curvilinear Gradient Page Differentiation in Ortogonal Curvilinear Coordinate Systems For any ortogonal curvilinear coordinate system (u, u, u ) in te unit tangent vectors along te curvilinear axes are ˆ r e, u were te scale factors i r u i., ˆi i i i e displacement vector r can ten be written as r ueˆ ˆ ˆ ue ue, were te unit vectors ê i form an ortonormal basis for. 0 i j eˆi e ˆ j i j i j e differential displacement vector dr is (by te Cain Rule) r r r dr du du du du eˆ du eˆ du e ˆ u u u and te differential arc lengt ds is given by ds dr dr du du du e element of volume dv is x, y, z dv du du du du du du u, u, u Jacobian x y u u x y u u x y u u z u z u z u Example 5.0.: Find te scale factor θ for te sperical polar coordinate system x, y, z r sin cos, r sin sin, r cos : r x y z r cos cos r cos sin r sin r r r r cos cos cos sin sin r cos cos sin sin cos sin du du du r r r

5 ENGI Gradient Summary Page Summary able for te Gradient Operator eˆ eˆ ˆ e Gradient operator u u u eˆ V eˆ ˆ V e V Gradient V u u u Divergence F F F F u u u Curl F eˆ F u eˆ F u eˆ F u Laplacian V = u V u u V u u V u Scale factors: Cartesian: x = y = z =. Cylindrical polar: = z =, =. Sperical polar: r =, = r, = r sin. Example 5.0.: e Laplacian of V in sperical polars is sin V V r sin V V r sin r r sin or V V V cot V V V r r r r r sin

6 ENGI Gradient Summary Page 5.06 Example 5.0. A potential function V r is sperically symmetric, (tat is, its value depends only on te distance r from te origin), due solely to a point source at te origin. ere are no oter sources or sinks anywere in V r.. Deduce te functional form of V r is sperically symmetric V r,, f r In any regions not containing any sources of te vector field, te divergence of te vector field F V (and terefore te Laplacian of te associated potential function V) must be zero. erefore, for all r 0, div F V V 0 But sin V r sin V r sin r r sin d V dv sin r r sin dr dr d dv dv dv r 0 r B B r dr dr dr dr V Br A, were A, B are arbitrary constants of integration. erefore te potential function must be of te form B V r,, A r is is te standard form of te potential function associated wit a force tat obeys te inverse square law F. r

7 ENGI Basis Vectors Page Derivatives of Basis Vectors d Cartesian: ˆ d ˆ d i j kˆ 0 dt dt dt r x ˆi y ˆj z kˆ v x ˆi y ˆj z kˆ Cylindrical Polar Coordinates: x cos, y sin, z z d d ˆ ˆ r ˆ z kˆ dt dt d ˆ d ˆ v ˆ ˆ z kˆ dt dt d kˆ 0 [radial and transverse components of v ] dt Sperical Polar Coordinates. e declination angle θ is te angle between te positive z axis and te radius vector r. 0 < θ < π. e azimut angle is te angle on te x-y plane, measured anticlockwise from te positive x axis, of te sadow of te radius vector. 0 < < π. z = r cos θ. e sadow of te radius vector on te x-y plane as lengt r sin θ. It ten follows tat x = r sin θ cos and y = r sin θ sin. d d ˆ ˆ d r sin ˆ dt dt dt r r r ˆ d ˆ d d rˆ cos ˆ dt dt dt v r rˆ r ˆ r sin ˆ d ˆ d sin rˆ cos ˆ dt dt

8 ENGI Basis Vectors Page 5.08 Example Find te velocity and acceleration in cylindrical polar coordinates for a particle travelling along te elix x = cos t, y = sin t, z = t. Cylindrical polar coordinates: x cos, y sin, z z y x y, tan x 9cos t 9sin t 9 0 sin t tan tan t t cost r ˆ z kˆ z t z dr v ˆ ˆ zkˆ 0 ˆ ˆ kˆ 6 ˆ kˆ dt [e velocity as no radial component te elix remains te same distance from te z axis at all times.] d v a 6 ˆ kˆ 6 ˆ 0 ˆ dt [e acceleration vector points directly at te z axis at all times.] Oter examples are in te problem sets. END OF CHAPER 5

ENGI Gradient, Divergence, Curl Page 5.01

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

More information

ENGI Duffing s Equation Page 4.65

ENGI 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 information

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

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

More information

MATH 332: Vector Analysis Summer 2005 Homework

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

More information

AE301 Aerodynamics I UNIT B: Theory of Aerodynamics

AE301 Aerodynamics I UNIT B: Theory of Aerodynamics AE301 Aerodynamics I UNIT B: Theory of Aerodynamics ROAD MAP... B-1: Mathematics for Aerodynamics B-: Flow Field Representations B-3: Potential Flow Analysis B-4: Applications of Potential Flow Analysis

More information

Gradient Descent etc.

Gradient Descent etc. 1 Gradient Descent etc EE 13: Networked estimation and control Prof Kan) I DERIVATIVE Consider f : R R x fx) Te derivative is defined as d fx) = lim dx fx + ) fx) Te cain rule states tat if d d f gx) )

More information

r t t x t y t z t, y t are zero, then construct a table for all four functions. dy dx 0 and 0 dt dt horizontal tangent vertical tangent

r t t x t y t z t, y t are zero, then construct a table for all four functions. dy dx 0 and 0 dt dt horizontal tangent vertical tangent 3. uggestions for the Formula heets Below are some suggestions for many more formulae than can be placed easily on both sides of the two standard 8½"" sheets of paper for the final examination. It is strongly

More information

The Derivative as a Function

The Derivative as a Function Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a + )

More information

Notes 3 Review of Vector Calculus

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

More information

Section 15.6 Directional Derivatives and the Gradient Vector

Section 15.6 Directional Derivatives and the Gradient Vector Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,

More information

Lecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.

Lecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator. Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative

More information

Summary of various integrals

Summary of various integrals ummary of various integrals Here s an arbitrary compilation of information about integrals Moisés made on a cold ecember night. 1 General things o not mix scalars and vectors! In particular ome integrals

More information

Math Review Night: Work and the Dot Product

Math Review Night: Work and the Dot Product Math Review Night: Work and the Dot Product Dot Product A scalar quantity Magnitude: A B = A B cosθ The dot product can be positive, zero, or negative Two types of projections: the dot product is the parallel

More information

= h. Geometrically this quantity represents the slope of the secant line connecting the points

= h. Geometrically this quantity represents the slope of the secant line connecting the points Section 3.7: Rates of Cange in te Natural and Social Sciences Recall: Average rate of cange: y y y ) ) ), ere Geometrically tis quantity represents te slope of te secant line connecting te points, f (

More information

ENGI 4430 Line Integrals; Green s Theorem Page 8.01

ENGI 4430 Line Integrals; Green s Theorem Page 8.01 ENGI 443 Line Integrals; Green s Theorem Page 8. 8. Line Integrals Two applications of line integrals are treated here: the evaluation of work done on a particle as it travels along a curve in the presence

More information

Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is

Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is Mat 180 www.timetodare.com Section.7 Derivatives and Rates of Cange Part II Section.8 Te Derivative as a Function Derivatives ( ) In te previous section we defined te slope of te tangent to a curve wit

More information

ENGI 4430 Parametric Vector Functions Page dt dt dt

ENGI 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 information

Figure 25:Differentials of surface.

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

More information

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

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

More information

Disclaimer: 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 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 information

ENGI 4430 Surface Integrals Page and 0 2 r

ENGI 4430 Surface Integrals Page and 0 2 r ENGI 4430 Surface Integrals Page 9.01 9. Surface Integrals - Projection Method Surfaces in 3 In 3 a surface can be represented by a vector parametric equation r x u, v ˆi y u, v ˆj z u, v k ˆ where u,

More information

Mathematical Concepts & Notation

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

More information

INTRODUCTION AND MATHEMATICAL CONCEPTS

INTRODUCTION AND MATHEMATICAL CONCEPTS Capter 1 INTRODUCTION ND MTHEMTICL CONCEPTS PREVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips

More information

Continuity and Differentiability of the Trigonometric Functions

Continuity and Differentiability of the Trigonometric Functions [Te basis for te following work will be te definition of te trigonometric functions as ratios of te sides of a triangle inscribed in a circle; in particular, te sine of an angle will be defined to be te

More information

Module 02: Math Review

Module 02: Math Review Module 02: Math Review 1 Module 02: Math Review: Outline Vector Review (Dot, Cross Products) Review of 1D Calculus Scalar Functions in higher dimensions Vector Functions Differentials Purpose: Provide

More information

Dynamics and Relativity

Dynamics and Relativity Dynamics and Relativity Stepen Siklos Lent term 2011 Hand-outs and examples seets, wic I will give out in lectures, are available from my web site www.damtp.cam.ac.uk/user/stcs/dynamics.tml Lecture notes,

More information

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

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

More information

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

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

More information

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

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

More information

Kinematics (2) - Motion in Three Dimensions

Kinematics (2) - Motion in Three Dimensions Kinematics (2) - Motion in Three Dimensions 1. Introduction Kinematics is a branch of mechanics which describes the motion of objects without consideration of the circumstances leading to the motion. 2.

More information

Chapter 3 Vectors. 3.1 Vector Analysis

Chapter 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 information

CBE 6333, R. Levicky 1. Orthogonal Curvilinear Coordinates

CBE 6333, R. Levicky 1. Orthogonal Curvilinear Coordinates 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

More information

Math Review 1: Vectors

Math Review 1: Vectors Math Review 1: Vectors Coordinate System Coordinate system: used to describe the position of a point in space and consists of 1. An origin as the reference point 2. A set of coordinate axes with scales

More information

Name: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ).

Name: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ). Mat - Final Exam August 3 rd, Name: Answer Key No calculators. Sow your work!. points) All answers sould eiter be,, a finite) real number, or DNE does not exist ). a) Use te grap of te function to evaluate

More information

5. Suggestions for the Formula Sheets

5. Suggestions for the Formula Sheets 5. uggestions for the Formula heets Below are some suggestions for many more formulae than can be placed easily on both sides of the two standard 8½" " sheets of paper for the final examination. It is

More information

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

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

More information

Chapter 1. Vector Algebra and Vector Space

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

More information

Differentiation in higher dimensions

Differentiation in higher dimensions Capter 2 Differentiation in iger dimensions 2.1 Te Total Derivative Recall tat if f : R R is a 1-variable function, and a R, we say tat f is differentiable at x = a if and only if te ratio f(a+) f(a) tends

More information

2.20 Fall 2018 Math Review

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

More information

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

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

More information

ENGI 4430 Line Integrals; Green s Theorem Page 8.01

ENGI 4430 Line Integrals; Green s Theorem Page 8.01 ENGI 4430 Line Integrals; Green s Theorem Page 8.01 8. Line Integrals Two applications of line integrals are treated here: the evaluation of work done on a particle as it travels along a curve in the presence

More information

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

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

More information

HOMEWORK HELP 2 FOR MATH 151

HOMEWORK HELP 2 FOR MATH 151 HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,

More information

Combining functions: algebraic methods

Combining functions: algebraic methods Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)

More information

Notes: DERIVATIVES. Velocity and Other Rates of Change

Notes: DERIVATIVES. Velocity and Other Rates of Change Notes: DERIVATIVES Velocity and Oter Rates of Cange I. Average Rate of Cange A.) Def.- Te average rate of cange of f(x) on te interval [a, b] is f( b) f( a) b a secant ( ) ( ) m troug a, f ( a ) and b,

More information

Vector Calculus. A primer

Vector Calculus. A primer Vector Calculus A primer Functions of Several Variables A single function of several variables: f: R $ R, f x (, x ),, x $ = y. Partial derivative vector, or gradient, is a vector: f = y,, y x ( x $ Multi-Valued

More information

Calculus I, Fall Solutions to Review Problems II

Calculus I, Fall Solutions to Review Problems II Calculus I, Fall 202 - Solutions to Review Problems II. Find te following limits. tan a. lim 0. sin 2 b. lim 0 sin 3. tan( + π/4) c. lim 0. cos 2 d. lim 0. a. From tan = sin, we ave cos tan = sin cos =

More information

Divergence Theorem December 2013

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

More information

Higher Derivatives. Differentiable Functions

Higher Derivatives. Differentiable Functions Calculus 1 Lia Vas Higer Derivatives. Differentiable Functions Te second derivative. Te derivative itself can be considered as a function. Te instantaneous rate of cange of tis function is te second derivative.

More information

Created by T. Madas VECTOR OPERATORS. Created by T. Madas

Created 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 information

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a?

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a? Solutions to Test 1 Fall 016 1pt 1. Te grap of a function f(x) is sown at rigt below. Part I. State te value of eac limit. If a limit is infinite, state weter it is or. If a limit does not exist (but is

More information

11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR

11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR SECTION 11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 633 wit speed v o along te same line from te opposite direction toward te source, ten te frequenc of te sound eard b te observer is were c is

More information

Finding and Using Derivative The shortcuts

Finding and Using Derivative The shortcuts Calculus 1 Lia Vas Finding and Using Derivative Te sortcuts We ave seen tat te formula f f(x+) f(x) (x) = lim 0 is manageable for relatively simple functions like a linear or quadratic. For more complex

More information

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

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

More information

Calculus III. Math 233 Spring Final exam May 3rd. Suggested solutions

Calculus 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 information

Chapter 9 Uniform Circular Motion

Chapter 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 information

Tangent Lines-1. Tangent Lines

Tangent Lines-1. Tangent Lines Tangent Lines- Tangent Lines In geometry, te tangent line to a circle wit centre O at a point A on te circle is defined to be te perpendicular line at A to te line OA. Te tangent lines ave te special property

More information

Math 11 Fall 2018 Practice Final Exam

Math 11 Fall 2018 Practice Final Exam Math 11 Fall 218 Practice Final Exam Disclaimer: This practice exam should give you an idea of the sort of questions we may ask on the actual exam. Since the practice exam (like the real exam) is not long

More information

CURRENT MATERIAL: Vector Calculus.

CURRENT MATERIAL: Vector Calculus. Math 275, section 002 (Ultman) Fall 2011 FINAL EXAM REVIEW The final exam will be held on Wednesday 14 December from 10:30am 12:30pm in our regular classroom. You will be allowed both sides of an 8.5 11

More information

DO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START

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

More information

ENGI 4430 Gauss & Stokes Theorems; Potentials Page 10.01

ENGI 4430 Gauss & Stokes Theorems; Potentials Page 10.01 ENGI 443 Gauss & tokes heorems; Potentials Page.. Gauss Divergence heorem Let be a piecewise-smooth closed surface enclosing a volume in vector field. hen the net flux of F out of is F d F d, N 3 and let

More information

AN INTRODUCTION TO CURVILINEAR ORTHOGONAL COORDINATES

AN 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 information

Review of Vector Analysis in Cartesian Coordinates

Review 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 information

Notes on Planetary Motion

Notes on Planetary Motion (1) Te motion is planar Notes on Planetary Motion Use 3-dimensional coordinates wit te sun at te origin. Since F = ma and te gravitational pull is in towards te sun, te acceleration A is parallel to te

More information

UNIVERSITY OF MANITOBA DEPARTMENT OF MATHEMATICS MATH 1510 Applied Calculus I FIRST TERM EXAMINATION - Version A October 12, :30 am

UNIVERSITY OF MANITOBA DEPARTMENT OF MATHEMATICS MATH 1510 Applied Calculus I FIRST TERM EXAMINATION - Version A October 12, :30 am DEPARTMENT OF MATHEMATICS MATH 1510 Applied Calculus I October 12, 2016 8:30 am LAST NAME: FIRST NAME: STUDENT NUMBER: SIGNATURE: (I understand tat ceating is a serious offense DO NOT WRITE IN THIS TABLE

More information

Multiple Integrals and Vector Calculus: Synopsis

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

More information

Solution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4.

Solution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4. December 09, 20 Calculus PracticeTest s Name: (4 points) Find te absolute extrema of f(x) = x 3 0 on te interval [0, 4] Te derivative of f(x) is f (x) = 3x 2, wic is zero only at x = 0 Tus we only need

More information

Continuity and Differentiability Worksheet

Continuity and Differentiability Worksheet Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;

More information

Figure 21:The polar and Cartesian coordinate systems.

Figure 21:The polar and Cartesian coordinate systems. Figure 21:The polar and Cartesian coordinate systems. Coordinate systems in R There are three standard coordinate systems which are used to describe points in -dimensional space. These coordinate systems

More information

Practice Problem Solutions: Exam 1

Practice Problem Solutions: Exam 1 Practice Problem Solutions: Exam 1 1. (a) Algebraic Solution: Te largest term in te numerator is 3x 2, wile te largest term in te denominator is 5x 2 3x 2 + 5. Tus lim x 5x 2 2x 3x 2 x 5x 2 = 3 5 Numerical

More information

CURRENT MATERIAL: Vector Calculus.

CURRENT 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 information

Solutions to the Multivariable Calculus and Linear Algebra problems on the Comprehensive Examination of January 31, 2014

Solutions to the Multivariable Calculus and Linear Algebra problems on the Comprehensive Examination of January 31, 2014 Solutions to te Multivariable Calculus and Linear Algebra problems on te Compreensive Examination of January 3, 24 Tere are 9 problems ( points eac, totaling 9 points) on tis portion of te examination.

More information

CHAPTER 3: DERIVATIVES

CHAPTER 3: DERIVATIVES (Answers to Exercises for Capter 3: Derivatives) A.3.1 CHAPTER 3: DERIVATIVES SECTION 3.1: DERIVATIVES, TANGENT LINES, and RATES OF CHANGE 1) a) f ( 3) f 3.1 3.1 3 f 3.01 f ( 3) 3.01 3 f 3.001 f ( 3) 3.001

More information

Problem Solving 1: Line Integrals and Surface Integrals

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

More information

Lecture 2: Review of Vector Calculus

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

More information

Test 2 Review. 1. Find the determinant of the matrix below using (a) cofactor expansion and (b) row reduction. A = 3 2 =

Test 2 Review. 1. Find the determinant of the matrix below using (a) cofactor expansion and (b) row reduction. A = 3 2 = Test Review Find te determinant of te matrix below using (a cofactor expansion and (b row reduction Answer: (a det + = (b Observe R R R R R R R R R Ten det B = (((det Hence det Use Cramer s rule to solve:

More information

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

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

More information

Study Guide for Exam #2

Study Guide for Exam #2 Physical Mechanics METR103 November, 000 Study Guide for Exam # The information even below is meant to serve as a guide to help you to prepare for the second hour exam. The absence of a topic or point

More information

3.1 Extreme Values of a Function

3.1 Extreme Values of a Function .1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find

More information

Chapter 3 - Vector Calculus

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

More information

SOME PROBLEMS YOU SHOULD BE ABLE TO DO

SOME 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 information

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

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

More information

Brief Review of Vector Algebra

Brief Review of Vector Algebra APPENDIX Brief Review of Vector Algebra A.0 Introduction Vector algebra is used extensively in computational mechanics. The student must thus understand the concepts associated with this subject. The current

More information

ENGI Partial Differentiation Page y f x

ENGI 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 information

1.1. Fields Partial derivatives

1.1. Fields Partial derivatives 1.1. Fields A field associates a physical quantity with a position A field can be also time dependent, for example. The simplest case is a scalar field, where given physical quantity can be described by

More information

2. Temperature, Pressure, Wind, and Minor Constituents.

2. Temperature, Pressure, Wind, and Minor Constituents. 2. Temperature, Pressure, Wind, and Minor Constituents. 2. Distributions of temperature, pressure and wind. Close examination of Figs..7-.0 of MS reveals te following features tat cry out for explanation:

More information

Directional Derivative and the Gradient Operator

Directional Derivative and the Gradient Operator 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).

More information

Lecture Notes Introduction to Vector Analysis MATH 332

Lecture Notes Introduction to Vector Analysis MATH 332 Lecture Notes Introduction to Vector Analysis MATH 332 Instructor: Ivan Avramidi Textbook: H. F. Davis and A. D. Snider, (WCB Publishers, 1995) New Mexico Institute of Mining and Technology Socorro, NM

More information

Fundamentals of Applied Electromagnetics. Chapter 2 - Vector Analysis

Fundamentals 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 information

MATH1901 Differential Calculus (Advanced)

MATH1901 Differential Calculus (Advanced) MATH1901 Dierential Calculus (Advanced) Capter 3: Functions Deinitions : A B A and B are sets assigns to eac element in A eactl one element in B A is te domain o te unction B is te codomain o te unction

More information

Lecture II: Vector and Multivariate Calculus

Lecture II: Vector and Multivariate Calculus Lecture II: Vector and Multivariate Calculus Dot Product a, b R ' ', a ( b = +,- a + ( b + R. a ( b = a b cos θ. θ convex angle between the vectors. Squared norm of vector: a 3 = a ( a. Alternative notation:

More information

MATHS 267 Answers to Stokes Practice Dr. Jones

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

More information

1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t).

1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t). . Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd, periodic function tat as been sifted upwards, so we will use

More information

Differentiation Rules and Formulas

Differentiation Rules and Formulas Differentiation Rules an Formulas Professor D. Olles December 1, 01 1 Te Definition of te Derivative Consier a function y = f(x) tat is continuous on te interval a, b]. Ten, te slope of te secant line

More information

Vector Calculus, Maths II

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

More information

Section 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point.

Section 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point. Popper 6: Review of skills: Find tis difference quotient. f ( x ) f ( x) if f ( x) x Answer coices given in audio on te video. Section.1 Te Definition of te Derivative We are interested in finding te slope

More information

Derivatives. if such a limit exists. In this case when such a limit exists, we say that the function f is differentiable.

Derivatives. if such a limit exists. In this case when such a limit exists, we say that the function f is differentiable. Derivatives 3. Derivatives Definition 3. Let f be a function an a < b be numbers. Te average rate of cange of f from a to b is f(b) f(a). b a Remark 3. Te average rate of cange of a function f from a to

More information

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

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

More information

1 Calculus. 1.1 Gradients and the Derivative. Q f(x+h) f(x)

1 Calculus. 1.1 Gradients and the Derivative. Q f(x+h) f(x) Calculus. Gradients and te Derivative Q f(x+) δy P T δx R f(x) 0 x x+ Let P (x, f(x)) and Q(x+, f(x+)) denote two points on te curve of te function y = f(x) and let R denote te point of intersection of

More information

Differential Operators and the Divergence Theorem

Differential Operators and the Divergence Theorem 1 of 6 1/15/2007 6:31 PM Differential Operators and the Divergence Theorem One of the most important and useful mathematical constructs is the "del operator", usually denoted by the symbol Ñ (which is

More information