Second Order ODEs. Second Order ODEs. In general second order ODEs contain terms involving y, dy But here only consider equations of the form

Size: px
Start display at page:

Download "Second Order ODEs. Second Order ODEs. In general second order ODEs contain terms involving y, dy But here only consider equations of the form"

Transcription

1 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 A, B and C are constants i.e. they are independent of x and y. These are known as homogeneous equations. Solution: The solution has the form y = Ke λx where λ is a constant. Substitute y = Ke λx in to the ODE to determine the values of λ. The equation for λ is quadratic so in general there are two values for λ which satisfy the equation and so two functions y which are solutions to the ODE. slide 1

2 Demonstration: Divide equation 1 by A (A is not zero) to get d 2 y dx 2 + bdy dx + cy = 0 Now substitute y = Ke λx Kλ 2 e λx + bkλe λx + cke λx = 0 Since e λx can never be 0, can divide equation by Ke λx to get λ 2 + bλ + c = 0 This equation is called the auxiliary equation of the ODE. In general there are two values for λ which we will call λ 1 and λ 2, which can be complex. The general solution is then y = K 1 e λ 1x + K 2 e λ 2x slide 2

3 BUT If the two values of λ are real and equal, need to consider a general solution of the form y = (Kx + M)e λx where K and M are constants. slide 3

4 Example 1: Find the general solution to the equation d 2 y dx 2 + dy dx 2y = 0 Example 2: Find the general solution to the equation d 2 y dx 2 4dy dx + 4y = 0 Example 3: Find the equation of motion of a ball of mass m moving along the x-axis where the force on the ball is proportional to the distance along the axis. slide 4

5 Calculus in A Multi-dimensional World Know about calculus of one variable: If y = f(x), then the slope of the function f(x) at some point x 1 is given by the derivative df evaluated at x dx 1. The area A under the curve between x 1 and x 2 is the integral A = x2 x 1 f(x) dx How do we extend these ideas to functions of several variables? e.g. temperature in a room (or star): T(x, y, z), pressure in a gas: P(V, T), displacement of the Earth s surface in an earthquake: Z(x,y,t) slide 5

6 Partial Differentiation Consider a function f(x,y). The partial derivative f x f x = lim f(x + δx,y) f(x,y) δx 0 is defined by δx ( ) f Often written as or f x. Measures how f changes when x changes. x Evaluated by differentiating f(x,y) w.r.t x, treating y as a constant. Similarly there is a partial derivative with respect to y f y = lim δy 0 f(x,y + δy) f(x,y) δy slide 6

7 f is the rate of f(x,y) in the x direction at the point (x,y). x f is the rate of f(x,y) in the y direction at the point (x,y). y Higher derivatives are also defined and can be computed: 2 f x = f 2 f xx, 2 y, 2 f 2 x y = ( ) f 2 f = f xy, x y y x = y ( ) f x = f yx Example 4: Calculate the first and second partial derivatives of f(x,y) = x 2 xy + 4y 2 Example 5: Calculate the first partial derivatives of f(x,y) = sin(x 2 y) slide 7

8 Total Differentials What is the total change in a function f(x,y) if both x and y change by small amounts δx and δy? δf = f(x + δx,y + δy) f(x,y) δf = f(x + δx,y + δy) f(x,y) f(x,y + δy) + f(x,y + δy) f(x + δx,y + δy) f(x,y + δy) f(x,y + δy) f(x,y) = + δx δy = f f δx + δy plus higher order powers of δx and δy x y So in the limit δx 0 and δy 0, df = f x df is called the total differential of f. dx + f y dy δy slide 8

9 Application: 1) Suppose h(x,y) is our height above sea level at a position (x,y) and we walk along a path x(t), y(t) where t is the time. How fast do we gain height? We want dh dt. Use dh = h h dx + x y dy So dh dt = h dx x dt + h dy which is the generalization of the chain rule. y dt slide 9

10 2) Now suppose we want the rate of change of height with respect to x along the path. We want dh dx. Since we know the path, we know y(x) and dy = dy dx so dx dh = h h dx + x y dy = h h dx + x y which gives dh dx = h x + h y dy dx dy dx dx The total derivative of h w.r.t to x along the path y(x). slide 10

11 Fields A field is a quantity which varies with position. This quantity can be a single number, a scalar, in which case the field is a scalar field e.g. T(x, y, z) Examples: temperature, altitude of land, density Alternatively the quantity can be a vector, in which case the field is a vector field Example vector fields: velocity of a fluid, electric field At each point in space the quantity has both a magnitude and a direction. The value of a field at a given point does not depend on the coordinate system. slide 11

12 Three different representations of a scalar field: slide 12

13 Example vector field: v(x,y) = ˆr = r = xi + yj xi + yj (x 2 + y 2 ) 1/2 r = r = x 2 + y 2, ˆr = r r Diverging vector field slide 13

14 v(x, y) = yi + xj (x 2 + y 2 ) 3/2 Curling vector field slide 14

15 Field Lines Field lines show the direction of a vector field, but they don t show any information about the magnitude. How to Calculate the field lines Write a 2-d field as v(x,y) = v x (x,y)i + v y (x,y)j. The tanget to the field line makes an angle θ to the x-axis where tanθ = v y(x,y) v x (x,y). Let the field line be y = f(x), but on this field line the slope of the tanget to the field line is dy dx = v y(x,y), which we can integrate to find the equation to the v x (x,y) field line. Example: Calculate the field lines of the vector field v(x, y) = yi + xj (x 2 + y 2 ) 3/2 Field lines: x 2 + y 2 = c slide 15

16 The Gradient (Grad) What s the slope (derivative) of a scalar field H(x,y)? slide 16

17 Consider the temperature field T(x, y, z) at the two points (x, y, z) and (x + dx,y + dy,z + dz). The difference in temperature between these two points is dt where dt = T T dx + x y dy + T z dz But the vector between these two points is dr = dxi + dy j + dz k. So we can write ( dt = i T ) x + j T y + k T (dxi + dy j + dz k ) z dt = T dr where T = i T x + j T y + k T z which is a vector field called the gradient, or grad of the scalar field T. Also slide 17

18 Expect to see the gradient of scalar fields in various physical laws, e.g. heat flow in 1-D h = k T while in 3-D h = k T x Heat flows in the direction of grad T, the direction of decreasing T and nrmal to the isotherms. slide 18

19 Properties of The Gradient 0) T(x,y,z) is a scalar field 1) T(x,y,z) is a vector field 2) dt = T(r + dr) T(r) = T dr is the change in T between points r and r + dr If û is a unit vector parallel to dr so dr = û ds (ds = dr ) then dt = T ûds so 3) dt = T û is the directional derivative - the rate of change of T in the û ds direction of the unit vector û 4) T û is a maximum when T is parallel to û 5) T is the maximum rate of change and it is in the direction T Consider the surface T =constant. Any point P on this surface dt = 0 for û ds any vector u tangent to the surface at P. i.e. T dt = 0 so û ds 6) T is normal to the surface of constant T. slide 19

Section 4.3 Vector Fields

Section 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 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

Sec. 1.1: Basics of Vectors

Sec. 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 information

Ma 1c Practical - Solutions to Homework Set 7

Ma 1c Practical - Solutions to Homework Set 7 Ma 1c Practical - olutions to omework et 7 All exercises are from the Vector Calculus text, Marsden and Tromba (Fifth Edition) Exercise 7.4.. Find the area of the portion of the unit sphere that is cut

More information

Tangent Planes, Linear Approximations and Differentiability

Tangent Planes, Linear Approximations and Differentiability Jim Lambers MAT 80 Spring Semester 009-10 Lecture 5 Notes These notes correspond to Section 114 in Stewart and Section 3 in Marsden and Tromba Tangent Planes, Linear Approximations and Differentiability

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

231 Outline Solutions Tutorial Sheet 4, 5 and November 2007

231 Outline Solutions Tutorial Sheet 4, 5 and November 2007 31 Outline Solutions Tutorial Sheet 4, 5 and 6. 1 Problem Sheet 4 November 7 1. heck that the Jacobian for the transformation from cartesian to spherical polar coordinates is J = r sin θ. onsider the hemisphere

More information

Math 23b Practice Final Summer 2011

Math 23b Practice Final Summer 2011 Math 2b Practice Final Summer 211 1. (1 points) Sketch or describe the region of integration for 1 x y and interchange the order to dy dx dz. f(x, y, z) dz dy dx Solution. 1 1 x z z f(x, y, z) dy dx dz

More information

Taylor Series and stationary points

Taylor 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 information

MATH 200 WEEK 5 - WEDNESDAY DIRECTIONAL DERIVATIVE

MATH 200 WEEK 5 - WEDNESDAY DIRECTIONAL DERIVATIVE WEEK 5 - WEDNESDAY DIRECTIONAL DERIVATIVE GOALS Be able to compute a gradient vector, and use it to compute a directional derivative of a given function in a given direction. Be able to use the fact that

More information

Sec. 14.3: Partial Derivatives. All of the following are ways of representing the derivative. y dx

Sec. 14.3: Partial Derivatives. All of the following are ways of representing the derivative. y dx Math 2204 Multivariable Calc Chapter 14: Partial Derivatives I. Review from math 1225 A. First Derivative Sec. 14.3: Partial Derivatives 1. Def n : The derivative of the function f with respect to the

More information

Chapter 4: Partial differentiation

Chapter 4: Partial differentiation Chapter 4: Partial differentiation It is generally the case that derivatives are introduced in terms of functions of a single variable. For example, y = f (x), then dy dx = df dx = f. However, most of

More information

Definition 3 (Continuity). A function f is continuous at c if lim x c f(x) = f(c).

Definition 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 information

16.2. Line Integrals

16.2. Line Integrals 16. Line Integrals Review of line integrals: Work integral Rules: Fdr F d r = Mdx Ndy Pdz FT r'( t) ds r t since d '(s) and hence d ds '( ) r T r r ds T = Fr '( t) dt since r r'( ) dr d dt t dt dt does

More information

Math 233. Practice Problems Chapter 15. i j k

Math 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 information

Math 207 Honors Calculus III Final Exam Solutions

Math 207 Honors Calculus III Final Exam Solutions Math 207 Honors Calculus III Final Exam Solutions PART I. Problem 1. A particle moves in the 3-dimensional space so that its velocity v(t) and acceleration a(t) satisfy v(0) = 3j and v(t) a(t) = t 3 for

More information

ENGI Partial Differentiation Page y f x

ENGI Partial Differentiation Page y f x ENGI 344 4 Partial Differentiation Page 4-0 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

Module 2: Reflecting on One s Problems

Module 2: Reflecting on One s Problems MATH55 Module : Reflecting on One s Problems Main Math concepts: Translations, Reflections, Graphs of Equations, Symmetry Auxiliary ideas: Working with quadratics, Mobius maps, Calculus, Inverses I. Transformations

More information

Mathematical Economics: Lecture 9

Mathematical Economics: Lecture 9 Mathematical Economics: Lecture 9 Yu Ren WISE, Xiamen University October 17, 2011 Outline 1 Chapter 14: Calculus of Several Variables New Section Chapter 14: Calculus of Several Variables Partial Derivatives

More information

(1 + 2y)y = x. ( x. The right-hand side is a standard integral, so in the end we have the implicit solution. y(x) + y 2 (x) = x2 2 +C.

(1 + 2y)y = x. ( x. The right-hand side is a standard integral, so in the end we have the implicit solution. y(x) + y 2 (x) = x2 2 +C. Midterm 1 33B-1 015 October 1 Find the exact solution of the initial value problem. Indicate the interval of existence. y = x, y( 1) = 0. 1 + y Solution. We observe that the equation is separable, and

More information

EE2007: Engineering Mathematics II Vector Calculus

EE2007: 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 information

Mathematics for Physical Sciences III

Mathematics for Physical Sciences III Mathematics for Physical Sciences III Change of lecturer: First 4 weeks: myself again! Remaining 8 weeks: Dr Stephen O Sullivan Continuous Assessment Test Date to be announced (probably Week 7 or 8) -

More information

EE2007: Engineering Mathematics II Vector Calculus

EE2007: 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 information

MATH 425, FINAL EXAM SOLUTIONS

MATH 425, FINAL EXAM SOLUTIONS MATH 425, FINAL EXAM SOLUTIONS Each exercise is worth 50 points. Exercise. a The operator L is defined on smooth functions of (x, y by: Is the operator L linear? Prove your answer. L (u := arctan(xy u

More information

McGill University April 20, Advanced Calculus for Engineers

McGill University April 20, Advanced Calculus for Engineers McGill University April 0, 016 Faculty of Science Final examination Advanced Calculus for Engineers Math 64 April 0, 016 Time: PM-5PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer Student

More information

P1 Calculus II. Partial Differentiation & Multiple Integration. Prof David Murray. dwm/courses/1pd

P1 Calculus II. Partial Differentiation & Multiple Integration. Prof David Murray.   dwm/courses/1pd P1 2017 1 / 39 P1 Calculus II Partial Differentiation & Multiple Integration Prof David Murray david.murray@eng.ox.ac.uk www.robots.ox.ac.uk/ dwm/courses/1pd 4 lectures, MT 2017 P1 2017 2 / 39 Motivation

More information

S12.1 SOLUTIONS TO PROBLEMS 12 (ODD NUMBERS)

S12.1 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 information

Chapter 4. Several-variable calculus. 4.1 Derivatives of Functions of Several Variables Functions of Several Variables

Chapter 4. Several-variable calculus. 4.1 Derivatives of Functions of Several Variables Functions of Several Variables Chapter 4 Several-variable calculus 4.1 Derivatives of Functions of Several Variables 4.1.1 Functions of Several Variables ² A function f of n variables (x 1,x 2,...,x n ) in R n is an entity that operates

More information

Find the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x

Find the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc

More information

MA 201: Partial Differential Equations Lecture - 2

MA 201: Partial Differential Equations Lecture - 2 MA 201: Partial Differential Equations Lecture - 2 Linear First-Order PDEs For a PDE f(x,y,z,p,q) = 0, a solution of the type F(x,y,z,a,b) = 0 (1) which contains two arbitrary constants a and b is said

More information

MATH20411 PDEs and Vector Calculus B

MATH20411 PDEs and Vector Calculus B MATH2411 PDEs and Vector Calculus B Dr Stefan Güttel Acknowledgement The lecture notes and other course materials are based on notes provided by Dr Catherine Powell. SECTION 1: Introctory Material MATH2411

More information

Definition: If y = f(x), then. f(x + x) f(x) y = f (x) = lim. Rules and formulas: 1. If f(x) = C (a constant function), then f (x) = 0.

Definition: If y = f(x), then. f(x + x) f(x) y = f (x) = lim. Rules and formulas: 1. If f(x) = C (a constant function), then f (x) = 0. Definition: If y = f(x), then Rules and formulas: y = f (x) = lim x 0 f(x + x) f(x). x 1. If f(x) = C (a constant function), then f (x) = 0. 2. If f(x) = x k (a power function), then f (x) = kx k 1. 3.

More information

Maxima and Minima. (a, b) of R if

Maxima and Minima. (a, b) of R if Maxima and Minima Definition Let R be any region on the xy-plane, a function f (x, y) attains its absolute or global, maximum value M on R at the point (a, b) of R if (i) f (x, y) M for all points (x,

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

18.02 Multivariable Calculus Fall 2007

18.02 Multivariable Calculus Fall 2007 MIT OpenourseWare http://ocw.mit.edu 8.02 Multivariable alculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.02 Lecture 8. hange of variables.

More information

Gradient operator. In our calculation of dφ along the vector ds, we see that it can be described as the scalar product

Gradient operator. In our calculation of dφ along the vector ds, we see that it can be described as the scalar product Gradient operator In our calculation of dφ along the vector ds, we see that it can be described as the scalar product ( φ dφ = x î + φ y ĵ + φ ) z ˆk ( ) u x dsî + u y dsĵ + u z dsˆk We take dφ = φ ds

More information

Ideas from Vector Calculus Kurt Bryan

Ideas from Vector Calculus Kurt Bryan Ideas from Vector Calculus Kurt Bryan Most of the facts I state below are for functions of two or three variables, but with noted exceptions all are true for functions of n variables..1 Tangent Line Approximation

More information

12. Stresses and Strains

12. Stresses and Strains 12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)

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

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

21-256: Partial differentiation

21-256: 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 information

Mathematics for Chemistry: Exam Set 1

Mathematics for Chemistry: Exam Set 1 Mathematics for Chemistry: Exam Set 1 June 18, 017 1 mark Questions 1. The minimum value of the rank of any 5 3 matrix is 0 1 3. The trace of an identity n n matrix is equal to 1-1 0 n 3. A square matrix

More information

First and Second Order ODEs

First and Second Order ODEs Civil Engineering 2 Mathematics Autumn 211 M. Ottobre First and Second Order ODEs Warning: all the handouts that I will provide during the course are in no way exhaustive, they are just short recaps. Notation

More information

MATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS

MATH 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 information

Lecture 13 - Wednesday April 29th

Lecture 13 - Wednesday April 29th Lecture 13 - Wednesday April 29th jacques@ucsdedu Key words: Systems of equations, Implicit differentiation Know how to do implicit differentiation, how to use implicit and inverse function theorems 131

More information

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

Math 212-Lecture 8. The chain rule with one independent variable

Math 212-Lecture 8. The chain rule with one independent variable Math 212-Lecture 8 137: The multivariable chain rule The chain rule with one independent variable w = f(x, y) If the particle is moving along a curve x = x(t), y = y(t), then the values that the particle

More information

The Derivative. Appendix B. B.1 The Derivative of f. Mappings from IR to IR

The Derivative. Appendix B. B.1 The Derivative of f. Mappings from IR to IR Appendix B The Derivative B.1 The Derivative of f In this chapter, we give a short summary of the derivative. Specifically, we want to compare/contrast how the derivative appears for functions whose domain

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

Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals

Vector 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 information

Integral Vector Calculus

Integral Vector Calculus ontents 29 Integral Vector alculus 29.1 Line Integrals Involving Vectors 2 29.2 Surface and Volume Integrals 34 29.3 Integral Vector Theorems 54 Learning outcomes In this Workbook you will learn how to

More information

Math 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, 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 information

1 Functions of Several Variables Some Examples Level Curves / Contours Functions of More Variables... 6

1 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 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

Vector Calculus. Dr. D. Sukumar. February 1, 2016

Vector Calculus. Dr. D. Sukumar. February 1, 2016 Vector Calculus Dr. D. Sukumar February 1, 2016 Green s Theorem Tangent form or Ciculation-Curl form c Mdx + Ndy = R ( N x M ) da y Green s Theorem Tangent form or Ciculation-Curl form Stoke s Theorem

More information

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1. MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:

More information

9.7 Gradient of a Scalar Field. Directional Derivative. Mean Value Theorem. Special Cases

9.7 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 information

Math 321 Final Exam. 1. Do NOT write your answers on these sheets. Nothing written on the test papers will be graded.

Math 321 Final Exam. 1. Do NOT write your answers on these sheets. Nothing written on the test papers will be graded. Math 31 Final Exam Instructions 1 Do NOT write your answers on these sheets Nothing written on the test papers will be graded Please begin each section of questions on a new sheet of paper 3 Do not write

More information

EE2007: Engineering Mathematics II Vector Calculus

EE2007: 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 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

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

REVIEW OF DIFFERENTIAL CALCULUS

REVIEW OF DIFFERENTIAL CALCULUS REVIEW OF DIFFERENTIAL CALCULUS DONU ARAPURA 1. Limits and continuity To simplify the statements, we will often stick to two variables, but everything holds with any number of variables. Let f(x, y) be

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

Page Problem Score Max Score a 8 12b a b 10 14c 6 6

Page Problem Score Max Score a 8 12b a b 10 14c 6 6 Fall 14 MTH 34 FINAL EXAM December 8, 14 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 5 1 3 5 4 5 5 5 6 5 7 5 8 5 9 5 1 5 11 1 3 1a

More information

Bridging the Gap between Mathematics and the Physical Sciences

Bridging the Gap between Mathematics and the Physical Sciences Bridging the Gap between Mathematics and the Physical Sciences Department of Mathematics Oregon State University http://www.math.oregonstate.edu/~tevian Mathematics vs. Physics Math vs. Physics Functions

More information

THE IMPORTANCE OF GEOMETRIC REASONING

THE IMPORTANCE OF GEOMETRIC REASONING THE IMPORTANCE OF GEOMETRIC REASONING Tevian Dray & Corinne Manogue Oregon State University I: Mathematics Physics II: The Bridge Project III: Geometric Visualization QUESTIONS I: Suppose T (x, y) = k(x

More information

Integral Theorems. September 14, We begin by recalling the Fundamental Theorem of Calculus, that the integral is the inverse of the derivative,

Integral 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

IMP 13 First Hour Exam Solutions

IMP 13 First Hour Exam Solutions IMP 13 First Hour Exam Solutions 1. Two charges are arranged along the x-axis as shown, with q 1 = 4 µc at x = -4 m and q 2 = -9 µc at x = +2 m. (Remember 1 µc = 10-6 C). y * q 1 * q 2 x a. What is the

More information

Math review. Math review

Math review. Math review Math review 1 Math review 3 1 series approximations 3 Taylor s Theorem 3 Binomial approximation 3 sin(x), for x in radians and x close to zero 4 cos(x), for x in radians and x close to zero 5 2 some geometry

More information

Math Divergence and Curl

Math Divergence and Curl Math 23 - Divergence and Curl Peter A. Perry University of Kentucky November 3, 28 Homework Work on Stewart problems for 6.5: - (odd), 2, 3-7 (odd), 2, 23, 25 Finish Homework D2 due tonight Begin Homework

More information

Mathematics for Chemistry: Exam Set 1

Mathematics for Chemistry: Exam Set 1 Mathematics for Chemistry: Exam Set 1 March 19, 017 1 mark Questions 1. The maximum value of the rank of any 5 3 matrix is (a (b3 4 5. The determinant of an identity n n matrix is equal to (a 1 (b -1 0

More information

Leaving Cert Differentiation

Leaving Cert Differentiation Leaving Cert Differentiation Types of Differentiation 1. From First Principles 2. Using the Rules From First Principles You will be told when to use this, the question will say differentiate with respect

More information

MAT 211 Final Exam. Fall Jennings.

MAT 211 Final Exam. Fall Jennings. MAT 211 Final Exam. Fall 218. Jennings. Useful formulas polar coordinates spherical coordinates: SHOW YOUR WORK! x = rcos(θ) y = rsin(θ) da = r dr dθ x = ρcos(θ)cos(φ) y = ρsin(θ)cos(φ) z = ρsin(φ) dv

More information

Directional Derivatives in the Plane

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 information

Solutions to old Exam 3 problems

Solutions 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 information

MATH 280 Multivariate Calculus Fall Integrating a vector field over a curve

MATH 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 information

Solution to Homework 1. Vector Analysis (P201)

Solution to Homework 1. Vector Analysis (P201) Solution to Homework 1. Vector Analysis (P1) Q1. A = 3î + ĵ ˆk, B = î + 3ĵ + 4ˆk, C = 4î ĵ 6ˆk. The sides AB, BC and AC are, AB = 4î ĵ 6ˆk AB = 7.48 BC = 5î + 5ĵ + 1ˆk BC = 15 1.5 CA = î 3ĵ 4ˆk CA = 6

More information

Numerical Optimization Algorithms

Numerical Optimization Algorithms Numerical Optimization Algorithms 1. Overview. Calculus of Variations 3. Linearized Supersonic Flow 4. Steepest Descent 5. Smoothed Steepest Descent Overview 1 Two Main Categories of Optimization Algorithms

More information

Bridging the Gap between Mathematics and the Physical Sciences

Bridging the Gap between Mathematics and the Physical Sciences Bridging the Gap between Mathematics and the Physical Sciences Department of Mathematics Oregon State University http://www.math.oregonstate.edu/~tevian Mathematics vs. Physics Math vs. Physics Functions

More information

16.2 Line Integrals. Lukas Geyer. M273, Fall Montana State University. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21

16.2 Line Integrals. Lukas Geyer. M273, Fall Montana State University. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21 16.2 Line Integrals Lukas Geyer Montana State University M273, Fall 211 Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall 211 1 / 21 Scalar Line Integrals Definition f (x) ds = lim { s i } N f (P i ) s

More information

e x3 dx dy. 0 y x 2, 0 x 1.

e x3 dx dy. 0 y x 2, 0 x 1. Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=

More information

Solutions to Math 41 Second Exam November 5, 2013

Solutions to Math 41 Second Exam November 5, 2013 Solutions to Math 4 Second Exam November 5, 03. 5 points) Differentiate, using the method of your choice. a) fx) = cos 03 x arctan x + 4π) 5 points) If u = x arctan x + 4π then fx) = fu) = cos 03 u and

More information

MAT 211 Final Exam. Spring Jennings. Show your work!

MAT 211 Final Exam. Spring Jennings. Show your work! MAT 211 Final Exam. pring 215. Jennings. how your work! Hessian D = f xx f yy (f xy ) 2 (for optimization). Polar coordinates x = r cos(θ), y = r sin(θ), da = r dr dθ. ylindrical coordinates x = r cos(θ),

More information

Page Points Score Total: 210. No more than 200 points may be earned on the exam.

Page Points Score Total: 210. No more than 200 points may be earned on the exam. Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Points Score 3 18 4 18 5 18 6 18 7 18 8 18 9 18 10 21 11 21 12 21 13 21 Total: 210 No more than 200

More information

MATH 2203 Exam 3 Version 2 Solutions Instructions mathematical correctness clarity of presentation complete sentences

MATH 2203 Exam 3 Version 2 Solutions Instructions mathematical correctness clarity of presentation complete sentences MATH 2203 Exam 3 (Version 2) Solutions March 6, 2015 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation.

More information

Lesson 31 - Average and Instantaneous Rates of Change

Lesson 31 - Average and Instantaneous Rates of Change Lesson 31 - Average and Instantaneous Rates of Change IBHL Math & Calculus - Santowski 1 Lesson Objectives! 1. Calculate an average rate of change! 2. Estimate instantaneous rates of change using a variety

More information

Getting started: CFD notation

Getting started: CFD notation PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =

More information

i.e. the conservation of mass, the conservation of linear momentum, the conservation of energy.

i.e. the conservation of mass, the conservation of linear momentum, the conservation of energy. 04/04/2017 LECTURE 33 Geometric Interpretation of Stream Function: In the last class, you came to know about the different types of boundary conditions that needs to be applied to solve the governing equations

More information

MLC Practice Final Exam

MLC Practice Final Exam Name: Section: Recitation/Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 13. Show all your work on the standard

More information

25. Chain Rule. Now, f is a function of t only. Expand by multiplication:

25. 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 information

Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics

Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics 3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)

More information

Vector Calculus - GATE Study Material in PDF

Vector Calculus - GATE Study Material in PDF Vector Calculus - GATE Study Material in PDF In previous articles, we have already seen the basics of Calculus Differentiation and Integration and applications. In GATE 2018 Study Notes, we will be introduced

More information

MA102: Multivariable Calculus

MA102: Multivariable Calculus MA102: Multivariable Calculus Rupam Barman and Shreemayee Bora Department of Mathematics IIT Guwahati Differentiability of f : U R n R m Definition: Let U R n be open. Then f : U R n R m is differentiable

More information

Consider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity

Consider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity 1 Second Order Ordinary Differential Equations 1.1 The harmonic oscillator Consider an ideal pendulum as shown below. θ l Fr mg l θ is the angular acceleration θ is the angular velocity A point mass m

More information

The Geometry of Calculus

The Geometry of Calculus Introduction Department of Mathematics Oregon State University http://math.oregonstate.edu/~tevian Introduction Acknowledgments Places: People: Stuart Boersma, Johnner Barrett, Sam Cook, Aaron Wangberg

More information

Exercises for Multivariable Differential Calculus XM521

Exercises 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 information

Strain analysis.

Strain analysis. Strain analysis ecalais@purdue.edu Plates vs. continuum Gordon and Stein, 1991 Most plates are rigid at the until know we have studied a purely discontinuous approach where plates are

More information

Module Two: Differential Calculus(continued) synopsis of results and problems (student copy)

Module Two: Differential Calculus(continued) synopsis of results and problems (student copy) Module Two: Differential Calculus(continued) synopsis of results and problems (student copy) Srikanth K S 1 Syllabus Taylor s and Maclaurin s theorems for function of one variable(statement only)- problems.

More information

M343 Homework 3 Enrique Areyan May 17, 2013

M343 Homework 3 Enrique Areyan May 17, 2013 M343 Homework 3 Enrique Areyan May 17, 013 Section.6 3. Consider the equation: (3x xy + )dx + (6y x + 3)dy = 0. Let M(x, y) = 3x xy + and N(x, y) = 6y x + 3. Since: y = x = N We can conclude that this

More information

MULTIVARIABLE CALCULUS

MULTIVARIABLE CALCULUS MULTIVARIABLE CALCULUS JOHN QUIGG Contents 13.1 Three-Dimensional Coordinate Systems 2 13.2 Vectors 3 13.3 The Dot Product 5 13.4 The Cross Product 6 13.5 Equations of Lines and Planes 7 13.6 Cylinders

More information