Multivariable Calculus PEIMS Code: N1110018 Abbreviation: MULTCAL Grade Level(S): 11-12 Number of Credits: 1.0
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MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009)
MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009) Required Topics • Multivariable functions • 3-D space o Distance o Equations of Spheres o Graphing Planes Spheres Miscellaneous function graphs Sections and level curves (contour diagrams) Level surfaces • Planes o Equations of planes, in different forms o Equation of plane from three points o Linear approximation and tangent planes • Partial Derivatives o Compute using the definition of partial derivative o Compute using differentiation rules o Interpret o Approximate, given contour diagram, table of values, or sketch of sections o Signs from real-world description o Compute a gradient vector • Vectors o Arithmetic on vectors, pictorially and by components o Dot product compute using components v v v v v ⋅ w = v w cosθ v v v v u ⋅ v = 0⇔ u ⊥ v (for nonzero vectors) o Cross product Direction and length compute using components and determinant • Directional derivatives o estimate from contour diagram o compute using the lim definition t→0 v v o v compute using fu = ∇f ⋅ u ( u a unit vector) • Gradient v o v geometry (3 facts, and understand how they follow from fu = ∇f ⋅ u ) Gradient points in direction of fastest increase Length of gradient is the directional derivative in that direction Gradient is perpendicular to level set o given contour diagram, draw a gradient vector • Chain rules • Higher-order partials o compute o mixed partials are equal (under certain conditions) • Optimization o Locate and -
Vector Calculus and Multiple Integrals Rob Fender, HT 2018
Vector Calculus and Multiple Integrals Rob Fender, HT 2018 COURSE SYNOPSIS, RECOMMENDED BOOKS Course syllabus (on which exams are based): Double integrals and their evaluation by repeated integration in Cartesian, plane polar and other specified coordinate systems. Jacobians. Line, surface and volume integrals, evaluation by change of variables (Cartesian, plane polar, spherical polar coordinates and cylindrical coordinates only unless the transformation to be used is specified). Integrals around closed curves and exact differentials. Scalar and vector fields. The operations of grad, div and curl and understanding and use of identities involving these. The statements of the theorems of Gauss and Stokes with simple applications. Conservative fields. Recommended Books: Mathematical Methods for Physics and Engineering (Riley, Hobson and Bence) This book is lazily referred to as “Riley” throughout these notes (sorry, Drs H and B) You will all have this book, and it covers all of the maths of this course. However it is rather terse at times and you will benefit from looking at one or both of these: Introduction to Electrodynamics (Griffiths) You will buy this next year if you haven’t already, and the chapter on vector calculus is very clear Div grad curl and all that (Schey) A nice discussion of the subject, although topics are ordered differently to most courses NB: the latest version of this book uses the opposite convention to polar coordinates to this course (and indeed most of physics), but older versions can often be found in libraries 1 Week One A review of vectors, rotation of coordinate systems, vector vs scalar fields, integrals in more than one variable, first steps in vector differentiation, the Frenet-Serret coordinate system Lecture 1 Vectors A vector has direction and magnitude and is written in these notes in bold e.g. -
The Divergence As the Rate of Change in Area Or Volume
The divergence as the rate of change in area or volume Here we give a very brief sketch of the following fact, which should be familiar to you from multivariable calculus: If a region D of the plane moves with velocity given by the vector field V (x; y) = (f(x; y); g(x; y)); then the instantaneous rate of change in the area of D is the double integral of the divergence of V over D: ZZ ZZ rate of change in area = r · V dA = fx + gy dA: D D (An analogous result is true in three dimensions, with volume replacing area.) To see why this should be so, imagine that we have cut D up into a lot of little pieces Pi, each of which is rectangular with width Wi and height Hi, so that the area of Pi is Ai ≡ HiWi. If we follow the vector field for a short time ∆t, each piece Pi should be “roughly rectangular”. Its width would be changed by the the amount of horizontal stretching that is induced by the vector field, which is difference in the horizontal displacements of its two sides. Thisin turn is approximated by the product of three terms: the rate of change in the horizontal @f component of velocity per unit of displacement ( @x ), the horizontal distance across the box (Wi), and the time step ∆t. Thus ! @f ∆W ≈ W ∆t: i @x i See the figure below; the partial derivative measures how quickly f changes as you move horizontally, and Wi is the horizontal distance across along Pi, so the product fxWi measures the difference in the horizontal components of V at the two ends of Wi. -
A Brief Tour of Vector Calculus
A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0. -
Multivariable Calculus Workbook Developed By: Jerry Morris, Sonoma State University
Multivariable Calculus Workbook Developed by: Jerry Morris, Sonoma State University A Companion to Multivariable Calculus McCallum, Hughes-Hallett, et. al. c Wiley, 2007 Note to Students: (Please Read) This workbook contains examples and exercises that will be referred to regularly during class. Please purchase or print out the rest of the workbook before our next class and bring it to class with you every day. 1. To Purchase the Workbook. Go to the Sonoma State University campus bookstore, where the workbook is available for purchase. The copying charge will probably be between $10.00 and $20.00. 2. To Print Out the Workbook. Go to the Canvas page for our course and click on the link \Math 261 Workbook", which will open the file containing the workbook as a .pdf file. BE FOREWARNED THAT THERE ARE LOTS OF PICTURES AND MATH FONTS IN THE WORKBOOK, SO SOME PRINTERS MAY NOT ACCURATELY PRINT PORTIONS OF THE WORKBOOK. If you do choose to try to print it, please leave yourself enough time to purchase the workbook before our next class in case your printing attempt is unsuccessful. 2 Sonoma State University Table of Contents Course Introduction and Expectations .............................................................3 Preliminary Review Problems ........................................................................4 Chapter 12 { Functions of Several Variables Section 12.1-12.3 { Functions, Graphs, and Contours . 5 Section 12.4 { Linear Functions (Planes) . 11 Section 12.5 { Functions of Three Variables . 14 Chapter 13 { Vectors Section 13.1 & 13.2 { Vectors . 18 Section 13.3 & 13.4 { Dot and Cross Products . 21 Chapter 14 { Derivatives of Multivariable Functions Section 14.1 & 14.2 { Partial Derivatives . -
Multivariable and Vector Calculus
Multivariable and Vector Calculus Lecture Notes for MATH 0200 (Spring 2015) Frederick Tsz-Ho Fong Department of Mathematics Brown University Contents 1 Three-Dimensional Space ....................................5 1.1 Rectangular Coordinates in R3 5 1.2 Dot Product7 1.3 Cross Product9 1.4 Lines and Planes 11 1.5 Parametric Curves 13 2 Partial Differentiations ....................................... 19 2.1 Functions of Several Variables 19 2.2 Partial Derivatives 22 2.3 Chain Rule 26 2.4 Directional Derivatives 30 2.5 Tangent Planes 34 2.6 Local Extrema 36 2.7 Lagrange’s Multiplier 41 2.8 Optimizations 46 3 Multiple Integrations ........................................ 49 3.1 Double Integrals in Rectangular Coordinates 49 3.2 Fubini’s Theorem for General Regions 53 3.3 Double Integrals in Polar Coordinates 57 3.4 Triple Integrals in Rectangular Coordinates 62 3.5 Triple Integrals in Cylindrical Coordinates 67 3.6 Triple Integrals in Spherical Coordinates 70 4 Vector Calculus ............................................ 75 4.1 Vector Fields on R2 and R3 75 4.2 Line Integrals of Vector Fields 83 4.3 Conservative Vector Fields 88 4.4 Green’s Theorem 98 4.5 Parametric Surfaces 105 4.6 Stokes’ Theorem 120 4.7 Divergence Theorem 127 5 Topics in Physics and Engineering .......................... 133 5.1 Coulomb’s Law 133 5.2 Introduction to Maxwell’s Equations 137 5.3 Heat Diffusion 141 5.4 Dirac Delta Functions 144 1 — Three-Dimensional Space 1.1 Rectangular Coordinates in R3 Throughout the course, we will use an ordered triple (x, y, z) to represent a point in the three dimensional space. -
Laplace Transforms: Theory, Problems, and Solutions
Laplace Transforms: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 43 The Laplace Transform: Basic Definitions and Results 3 44 Further Studies of Laplace Transform 15 45 The Laplace Transform and the Method of Partial Fractions 28 46 Laplace Transforms of Periodic Functions 35 47 Convolution Integrals 45 48 The Dirac Delta Function and Impulse Response 53 49 Solving Systems of Differential Equations Using Laplace Trans- form 61 50 Solutions to Problems 68 2 43 The Laplace Transform: Basic Definitions and Results Laplace transform is yet another operational tool for solving constant coeffi- cients linear differential equations. The process of solution consists of three main steps: • The given \hard" problem is transformed into a \simple" equation. • This simple equation is solved by purely algebraic manipulations. • The solution of the simple equation is transformed back to obtain the so- lution of the given problem. In this way the Laplace transformation reduces the problem of solving a dif- ferential equation to an algebraic problem. The third step is made easier by tables, whose role is similar to that of integral tables in integration. The above procedure can be summarized by Figure 43.1 Figure 43.1 In this section we introduce the concept of Laplace transform and discuss some of its properties. The Laplace transform is defined in the following way. Let f(t) be defined for t ≥ 0: Then the Laplace transform of f; which is denoted by L[f(t)] or by F (s), is defined by the following equation Z T Z 1 L[f(t)] = F (s) = lim f(t)e−stdt = f(t)e−stdt T !1 0 0 The integral which defined a Laplace transform is an improper integral. -
Numerical Methods in Multiple Integration
NUMERICAL METHODS IN MULTIPLE INTEGRATION Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY WAYNE EUGENE HOOVER 1977 LIBRARY Michigan State University This is to certify that the thesis entitled NUMERICAL METHODS IN MULTIPLE INTEGRATION presented by Wayne Eugene Hoover has been accepted towards fulfillment of the requirements for PI; . D. degree in Mathematics w n A fl $11 “ W Major professor I (J.S. Frame) Dag Octdber 29, 1976 ., . ,_.~_ —- (‘sq I MAY 2T3 2002 ABSTRACT NUMERICAL METHODS IN MULTIPLE INTEGRATION By Wayne Eugene Hoover To approximate the definite integral 1:" bl [(1):] f(x1"”’xn)dxl”'dxn an a‘ over the n-rectangle, n R = n [41, bi] , 1' =1 conventional multidimensional quadrature formulas employ a weighted sum of function values m Q“) = Z ij(x,'1:""xjn)- i=1 Since very little is known concerning formulas which make use of partial derivative data, the objective of this investigation is to construct formulas involving not only the traditional weighted sum of function values but also partial derivative correction terms with weights of equal magnitude and alternate signs at the corners or at the midpoints of the sides of the domain of integration, R, so that when the rule is compounded or repeated, the weights cancel except on the boundary. For a single integral, the derivative correction terms are evaluated only at the end points of the interval of integration. In higher dimensions, the situation is somewhat more complicated since as the dimension increases the boundary becomes more complex. Indeed, in higher dimensions, most of the volume of the n-rectande lies near the boundary. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
MULTIVARIABLE CALCULUS Sample Midterm Problems October 1, 2009 INSTRUCTOR: Anar Akhmedov
MULTIVARIABLE CALCULUS Sample Midterm Problems October 1, 2009 INSTRUCTOR: Anar Akhmedov 1. Let P (1, 0, 3), Q(0, 2, 4) and R(4, 1, 6) be points. − − − (a) Find the equation of the plane through the points P , Q and R. (b) Find the area of the triangle with vertices P , Q and R. Solution: The vector P~ Q P~ R = < 1, 2, 1 > < 3, 1, 9 > = < 17, 6, 5 > is the normal vector of this plane,× so equation− of− the− plane× is 17(x 1)+6(y −0)+5(z + 3) = 0, which simplifies to 17x 6y 5z = 32. − − − − − Area = 1 P~ Q P~ R = 1 < 1, 2, 1 > < 3, 1, 9 > = √350 2 | × | 2 | − − − × | 2 2. Let f(x, y)=(x y)3 +2xy + x2 y. Find the linear approximation L(x, y) near the point (1, 2). − − 2 2 2 2 Solution: fx = 3x 6xy +3y +2y +2x and fy = 3x +6xy 3y +2x 1, so − − − − fx(1, 2) = 9 and fy(1, 2) = 2. Then the linear approximation of f at (1, 2) is given by − L(x, y)= f(1, 2)+ fx(1, 2)(x 1) + fy(1, 2)(y 2)=2+9(x 1)+( 2)(y 2). − − − − − 3. Find the distance between the parallel planes x +2y z = 1 and 3x +6y 3z = 3. − − − Use the following formula to find the distance between the given parallel planes ax0+by0+cz0+d D = | 2 2 2 | . Use a point from the second plane (for example (1, 0, 0)) as (x ,y , z ) √a +b +c 0 0 0 and the coefficents from the first plane a = 1, b = 2, c = 1, and d = 1. -
Math 56A: Introduction to Stochastic Processes and Models
Math 56a: Introduction to Stochastic Processes and Models Kiyoshi Igusa, Mathematics August 31, 2006 A stochastic process is a random process which evolves with time. The basic model is the Markov chain. This is a set of “states” together with transition probabilities from one state to another. For example, in simple epidemic models there are only two states: S = “susceptible” and I = “infected.” The probability of going from S to I increases with the size of I. In the simplest model The S → I probability is proportional to I, the I → S probability is constant and time is discrete (for example, events happen only once per day). In the corresponding deterministic model we would have a first order recursion. In a continuous time Markov chain, transition events can occur at any time with a certain prob- ability density. The corresponding deterministic model is a first order differential equation. This includes the “general stochastic epidemic.” The number of states in a Markov chain is either finite or countably infinite. When the collection of states becomes a continuum, e.g., the price of a stock option, we no longer have a “Markov chain.” We have a more general stochastic process. Under very general conditions we obtain a Wiener process, also known as Brownian motion. The mathematics of hedging implies that stock options should be priced as if they are exactly given by this process. Ito’s formula explains how to calculate (or try to calculate) stochastic integrals which give the long term expected values for a Wiener process. This course will be a theoretical mathematics course. -
Part IA — Vector Calculus
Part IA | Vector Calculus Based on lectures by B. Allanach Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. 3 Curves in R 3 Parameterised curves and arc length, tangents and normals to curves in R , the radius of curvature. [1] 2 3 Integration in R and R Line integrals. Surface and volume integrals: definitions, examples using Cartesian, cylindrical and spherical coordinates; change of variables. [4] Vector operators Directional derivatives. The gradient of a real-valued function: definition; interpretation as normal to level surfaces; examples including the use of cylindrical, spherical *and general orthogonal curvilinear* coordinates. Divergence, curl and r2 in Cartesian coordinates, examples; formulae for these oper- ators (statement only) in cylindrical, spherical *and general orthogonal curvilinear* coordinates. Solenoidal fields, irrotational fields and conservative fields; scalar potentials. Vector derivative identities. [5] Integration theorems Divergence theorem, Green's theorem, Stokes's theorem, Green's second theorem: statements; informal proofs; examples; application to fluid dynamics, and to electro- magnetism including statement of Maxwell's equations. [5] Laplace's equation 2 3 Laplace's equation in R and R : uniqueness theorem and maximum principle. Solution of Poisson's equation by Gauss's method (for spherical and cylindrical symmetry) and as an integral. [4] 3 Cartesian tensors in R Tensor transformation laws, addition, multiplication, contraction, with emphasis on tensors of second rank. Isotropic second and third rank tensors.