DOCSLIB.ORG

Vector Calculus Theorems with Proof

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Vector Calculus Theorems with Proof Part IA | Vector Calculus Theorems with proof 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. Symmetric and antisymmetric tensors. Revision of principal axes and diagonalization. Quotient theorem. Examples including inertia and conductivity. [5] 1 Contents IA Vector Calculus (Theorems with proof) Contents 0 Introduction 4 1 Derivatives and coordinates 5 1.1 Derivative of functions . .5 1.2 Inverse functions . .5 1.3 Coordinate systems . .5 2 Curves and Line 6 2.1 Parametrised curves, lengths and arc length . .6 2.2 Line integrals of vector fields . .6 2.3 Gradients and Differentials . .6 2.4 Work and potential energy . .6 3 Integration in R2 and R3 7 3.1 Integrals over subsets of R2 .....................7 3.2 Change of variables for an integral in R2 ..............7 3.3 Generalization to R3 .........................8 3.4 Further generalizations . .8 4 Surfaces and surface integrals 9 4.1 Surfaces and Normal . .9 4.2 Parametrized surfaces and area . .9 4.3 Surface integral of vector fields . .9 4.4 Change of variables in R2 and R3 revisited . .9 5 Geometry of curves and surfaces 10 6 Div, Grad, Curl and r 11 6.1 Div, Grad, Curl and r ........................ 11 6.2 Second-order derivatives . 11 7 Integral theorems 12 7.1 Statement and examples . 12 7.1.1 Green's theorem (in the plane) . 12 7.1.2 Stokes' theorem . 12 7.1.3 Divergence/Gauss theorem . 12 7.2 Relating and proving integral theorems . 12 8 Some applications of integral theorems 17 8.1 Integral expressions for div and curl . 17 8.2 Conservative fields and scalar products . 17 8.3 Conservation laws . 18 9 Orthogonal curvilinear coordinates 19 9.1 Line, area and volume elements . 19 9.2 Grad, Div and Curl . 19 2 Contents IA Vector Calculus (Theorems with proof) 10 Gauss' Law and Poisson's equation 21 10.1 Laws of gravitation . 21 10.2 Laws of electrostatics . 21 10.3 Poisson's Equation and Laplace's equation . 21 11 Laplace's and Poisson's equations 22 11.1 Uniqueness theorems . 22 11.2 Laplace's equation and harmonic functions . 23 11.2.1 The mean value property . 23 11.2.2 The maximum (or minimum) principle . 23 11.3 Integral solutions of Poisson's equations . 24 11.3.1 Statement and informal derivation . 24 11.3.2 Point sources and δ-functions* . 24 12 Maxwell's equations 25 12.1 Laws of electromagnetism . 25 12.2 Static charges and steady currents . 25 12.3 Electromagnetic waves . 25 13 Tensors and tensor fields 26 13.1 Definition . 26 13.2 Tensor algebra . 26 13.3 Symmetric and antisymmetric tensors . 26 13.4 Tensors, multi-linear maps and the quotient rule . 26 13.5 Tensor calculus . 26 14 Tensors of rank 2 28 14.1 Decomposition of a second-rank tensor . 28 14.2 The inertia tensor . 28 14.3 Diagonalization of a symmetric second rank tensor . 28 15 Invariant and isotropic tensors 29 15.1 Definitions and classification results . 29 15.2 Application to invariant integrals . 30 3 0 Introduction IA Vector Calculus (Theorems with proof) 0 Introduction 4 1 Derivatives and coordinates IA Vector Calculus (Theorems with proof) 1 Derivatives and coordinates 1.1 Derivative of functions Proposition. 0 0 F (x) = Fi (x)ei: Proposition. d df dg (fg) = g + f dt dt dt d dg dh (g · h) = · h + g · dt dt dt d dg dh (g × h) = × h + g × dt dt dt Note that the order of multiplication must be retained in the case of the cross product. Theorem. The gradient is @f rf = ei @xi Theorem (Chain rule). Given a function f(r(u)), df dr @f dx = rf · = i : du du @xi du Theorem. The derivative of F is given by @yj Mji = : @xi Theorem (Chain rule). Suppose g : Rp ! Rn and f : Rn ! Rm. Suppose that p n m the coordinates of the vectors in R ; R and R are ua; xi and yr respectively. By the chain rule, @y @y @x r = r i ; @ua @xi @ua with summation implied. Writing in matrix form, M(f ◦ g)ra = M(f)riM(g)ia: Alternatively, in operator form, @ @x @ = i : @ua @ua @xi 1.2 Inverse functions 1.3 Coordinate systems 5 2 Curves and Line IA Vector Calculus (Theorems with proof) 2 Curves and Line 2.1 Parametrised curves, lengths and arc length Proposition. Let s denote the arclength of a curve r(u). Then ds dr 0 = ± = ±|r (u)j du du with the sign depending on whether it is in the direction of increasing or decreasing arclength. Proposition. ds = ±|r0(u)jdu 2.2 Line integrals of vector fields 2.3 Gradients and Differentials Theorem. If F = rf(r), then Z F · dr = f(b) − f(a); C where b and a are the end points of the curve. In particular, the line integral does not depend on the curve, but the end points only. This is the vector counterpart of the fundamental theorem of H calculus. A special case is when C is a closed curve, then C F · dr = 0. Proof. Let r(u) be any parametrization of the curve, and suppose a = r(α), b = r(β). Then Z Z Z dr F · dr = rf · dr = rf · du: C C du So by the chain rule, this is equal to Z β d β (f(r(u))) du = [f(r(u))]α = f(b) − f(a): α du Proposition. If F = rf for some f, then @F @F i = j : @xj @xi 2 This is because both are equal to @ f=@xi@xj. Proposition. d(λf + µg) = λdf + µdg d(fg) = (df)g + f(dg) 2.4 Work and potential energy 6 3 Integration in R2 and R3 IA Vector Calculus (Theorems with proof) 3 Integration in R2 and R3 3.1 Integrals over subsets of R2 Proposition. Z Z Z ! f(x; y) dA = f(x; y) dx dy: D Y xy with xy ranging over fx :(x; y) 2 Dg. Theorem (Fubini's theorem). If f is a continuous function and D is a compact (i.e. closed and bounded) subset of R2, then ZZ ZZ f dx dy = f dy dx: While we have rather strict conditions for this theorem, it actually holds in many more cases, but those situations have to be checked manually. Proposition. dA = dx dy in Cartesian coordinates. Proposition. Take separable f(x; y) = h(y)g(x) and D be a rectangle f(x; y): a ≤ x ≤ b; c ≤ y ≤ dg. Then Z Z b ! Z d ! f(x; y) dx dy = g(x) dx h(y) dy D a c 3.2 Change of variables for an integral in R2 Proposition. Suppose we have a change of variables (x; y) $ (u; v) that is smooth and invertible, with regions D; D0 in one-to-one correspondence. Then Z Z f(x; y) dx dy = f(x(u; v); y(u; v))jJj du dv; D D0 where @x @x @(x; y) @u @v J = = @(u; v) @y @y @u @v is the Jacobian. In other words, dx dy = jJj du dv: Proof. Since we are writing (x(u; v); y(u; v)), we are actually transforming from (u; v) to (x; y) and not the other way round. Suppose we start with an area δA0 = δuδv in the (u; v) plane. Then by Taylors' theorem, we have @x @x δx = x(u + δu; v + δv) − x(u; v) ≈ δu + δv: @u @v We have a similar expression for δy and we obtain δx @x @x δu ≈ @u @v δy @y @y δv @u @v 7 3 Integration in R2 and R3 IA Vector Calculus (Theorems with proof) Recall from Vectors and Matrices that the determinant of the matrix is how much it scales up an area. So the area formed by δx and δy is jJj times the area formed by δu and δv. Hence dx dy = jJj du dv: 3.3 Generalization to R3 Proposition. dV = dx dy dz. Proposition. Z Z f dx dy dz = fjJj du dv dw; V V with @x @x @x @u @v @w @(x; y; z) @y @y @y J = = @(u; v; w) @u @v @w @z @z @z @u @v @w Proposition.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • A Quotient Rule Integration by Parts Formula Jennifer Switkes ([email protected]), California State Polytechnic Univer- Sity, Pomona, CA 91768
    A Quotient Rule Integration by Parts Formula Jennifer Switkes ([email protected]), California State Polytechnic Univer- sity, Pomona, CA 91768 In a recent calculus course, I introduced the technique of Integration by Parts as an integration rule corresponding to the Product Rule for differentiation. I showed my students the standard derivation of the Integration by Parts formula as presented in [1]: By the Product Rule, if f (x) and g(x) are differentiable functions, then d f (x)g(x) = f (x)g(x) + g(x) f (x). dx Integrating on both sides of this equation, f (x)g(x) + g(x) f (x) dx = f (x)g(x), which may be rearranged to obtain f (x)g(x) dx = f (x)g(x) − g(x) f (x) dx. Letting U = f (x) and V = g(x) and observing that dU = f (x) dx and dV = g(x) dx, we obtain the familiar Integration by Parts formula UdV= UV − VdU. (1) My student Victor asked if we could do a similar thing with the Quotient Rule. While the other students thought this was a crazy idea, I was intrigued. Below, I derive a Quotient Rule Integration by Parts formula, apply the resulting integration formula to an example, and discuss reasons why this formula does not appear in calculus texts. By the Quotient Rule, if f (x) and g(x) are differentiable functions, then ( ) ( ) ( ) − ( ) ( ) d f x = g x f x f x g x . dx g(x) [g(x)]2 Integrating both sides of this equation, we get f (x) g(x) f (x) − f (x)g(x) = dx.
    [Show full text]
  • Laplace Transform
    Chapter 7 Laplace Transform The Laplace transform can be used to solve differential equations. Be- sides being a different and efficient alternative to variation of parame- ters and undetermined coefficients, the Laplace method is particularly advantageous for input terms that are piecewise-defined, periodic or im- pulsive. The direct Laplace transform or the Laplace integral of a function f(t) defined for 0 ≤ t< 1 is the ordinary calculus integration problem 1 f(t)e−stdt; Z0 succinctly denoted L(f(t)) in science and engineering literature. The L{notation recognizes that integration always proceeds over t = 0 to t = 1 and that the integral involves an integrator e−stdt instead of the usual dt. These minor differences distinguish Laplace integrals from the ordinary integrals found on the inside covers of calculus texts. 7.1 Introduction to the Laplace Method The foundation of Laplace theory is Lerch's cancellation law 1 −st 1 −st 0 y(t)e dt = 0 f(t)e dt implies y(t)= f(t); (1) R R or L(y(t)= L(f(t)) implies y(t)= f(t): In differential equation applications, y(t) is the sought-after unknown while f(t) is an explicit expression taken from integral tables. Below, we illustrate Laplace's method by solving the initial value prob- lem y0 = −1; y(0) = 0: The method obtains a relation L(y(t)) = L(−t), whence Lerch's cancel- lation law implies the solution is y(t)= −t. The Laplace method is advertised as a table lookup method, in which the solution y(t) to a differential equation is found by looking up the answer in a special integral table.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • CHAPTER 3: Derivatives
    CHAPTER 3: Derivatives 3.1: Derivatives, Tangent Lines, and Rates of Change 3.2: Derivative Functions and Differentiability 3.3: Techniques of Differentiation 3.4: Derivatives of Trigonometric Functions 3.5: Differentials and Linearization of Functions 3.6: Chain Rule 3.7: Implicit Differentiation 3.8: Related Rates • Derivatives represent slopes of tangent lines and rates of change (such as velocity). • In this chapter, we will define derivatives and derivative functions using limits. • We will develop short cut techniques for finding derivatives. • Tangent lines correspond to local linear approximations of functions. • Implicit differentiation is a technique used in applied related rates problems. (Section 3.1: Derivatives, Tangent Lines, and Rates of Change) 3.1.1 SECTION 3.1: DERIVATIVES, TANGENT LINES, AND RATES OF CHANGE LEARNING OBJECTIVES • Relate difference quotients to slopes of secant lines and average rates of change. • Know, understand, and apply the Limit Definition of the Derivative at a Point. • Relate derivatives to slopes of tangent lines and instantaneous rates of change. • Relate opposite reciprocals of derivatives to slopes of normal lines. PART A: SECANT LINES • For now, assume that f is a polynomial function of x. (We will relax this assumption in Part B.) Assume that a is a constant. • Temporarily fix an arbitrary real value of x. (By “arbitrary,” we mean that any real value will do). Later, instead of thinking of x as a fixed (or single) value, we will think of it as a “moving” or “varying” variable that can take on different values. The secant line to the graph of f on the interval []a, x , where a < x , is the line that passes through the points a, fa and x, fx.
    [Show full text]
  • MATH 162: Calculus II Framework for Thurs., Mar
    MATH 162: Calculus II Framework for Thurs., Mar. 29–Fri. Mar. 30 The Gradient Vector Today’s Goal: To learn about the gradient vector ∇~ f and its uses, where f is a function of two or three variables. The Gradient Vector Suppose f is a differentiable function of two variables x and y with domain R, an open region of the xy-plane. Suppose also that r(t) = x(t)i + y(t)j, t ∈ I, (where I is some interval) is a differentiable vector function (parametrized curve) with (x(t), y(t)) being a point in R for each t ∈ I. Then by the chain rule, d ∂f dx ∂f dy f(x(t), y(t)) = + dt ∂x dt ∂y dt 0 0 = [fxi + fyj] · [x (t)i + y (t)j] dr = [fxi + fyj] · . (1) dt Definition: For a differentiable function f(x1, . , xn) of n variables, we define the gradient vector of f to be ∂f ∂f ∂f ∇~ f := , ,..., . ∂x1 ∂x2 ∂xn Remarks: • Using this definition, the total derivative df/dt calculated in (1) above may be written as df = ∇~ f · r0. dt In particular, if r(t) = x(t)i + y(t)j + z(t)k, t ∈ (a, b) is a differentiable vector function, and if f is a function of 3 variables which is differentiable at the point (x0, y0, z0), where x0 = x(t0), y0 = y(t0), and z0 = z(t0) for some t0 ∈ (a, b), then df ~ 0 = ∇f(x0, y0, z0) · r (t0). dt t=t0 • If f is a function of 2 variables, then ∇~ f has 2 components.
    [Show full text]
  • Multivariable Analysis
    1 - Multivariable analysis Roland van der Veen Groningen, 20-1-2020 2 Contents 1 Introduction 5 1.1 Basic notions and notation . 5 2 How to solve equations 7 2.1 Linear algebra . 8 2.2 Derivative . 10 2.3 Elementary Riemann integration . 12 2.4 Mean value theorem and Banach contraction . 15 2.5 Inverse and Implicit function theorems . 17 2.6 Picard's theorem on existence of solutions to ODE . 20 3 Multivariable fundamental theorem of calculus 23 3.1 Exterior algebra . 23 3.2 Differential forms . 26 3.3 Integration . 27 3.4 More on cubes and their boundary . 28 3.5 Exterior derivative . 30 3.6 The fundamental theorem of calculus (Stokes Theorem) . 31 3.7 Fundamental theorem of calculus: Poincar´elemma . 32 3 4 CONTENTS Chapter 1 Introduction The goal of these notes is to explore the notions of differentiation and integration in arbitrarily many variables. The material is focused on answering two basic questions: 1. How to solve an equation? How many solutions can one expect? 2. Is there a higher dimensional analogue for the fundamental theorem of calculus? Can one find a primitive? The equations we will address are systems of non-linear equations in finitely many variables and also ordinary differential equations. The approach will be mostly theoretical, schetching a framework in which one can predict how many solutions there will be without necessarily solving the equation. The key assumption is that everything we do can locally be approximated by linear functions. In other words, everything will be differentiable. One of the main results is that the linearization of the equation predicts the number of solutions and approximates them well locally.
    [Show full text]
  • Vector Calculus and Differential Forms with Applications To
    Vector Calculus and Differential Forms with Applications to Electromagnetism Sean Roberson May 7, 2015 PREFACE This paper is written as a final project for a course in vector analysis, taught at Texas A&M University - San Antonio in the spring of 2015 as an independent study course. Students in mathematics, physics, engineering, and the sciences usually go through a sequence of three calculus courses before go- ing on to differential equations, real analysis, and linear algebra. In the third course, traditionally reserved for multivariable calculus, stu- dents usually learn how to differentiate functions of several variable and integrate over general domains in space. Very rarely, as was my case, will professors have time to cover the important integral theo- rems using vector functions: Green’s Theorem, Stokes’ Theorem, etc. In some universities, such as UCSD and Cornell, honors students are able to take an accelerated calculus sequence using the text Vector Cal- culus, Linear Algebra, and Differential Forms by John Hamal Hubbard and Barbara Burke Hubbard. Here, students learn multivariable cal- culus using linear algebra and real analysis, and then they generalize familiar integral theorems using the language of differential forms. This paper was written over the course of one semester, where the majority of the book was covered. Some details, such as orientation of manifolds, topology, and the foundation of the integral were skipped to save length. The paper should still be readable by a student with at least three semesters of calculus, one course in linear algebra, and one course in real analysis - all at the undergraduate level.
    [Show full text]
  • 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.
    [Show full text]
  • Chap. 8. Vector Differential Calculus. Grad. Div. Curl
    Chap. 8. Vector Differential Calculus. Grad. Div. Curl 8.1. Vector Algebra in 2-Space and 3-Space - Scalar: a quantity only with its magnitude; temperature, speed, mass, volume, … - Vector: a quantity with its magnitude and its direction; velocity, acceleration, force, … (arrow & directed line segment) Initial and termination point Norm of a: length of a vector a. IaI =1: unit vector Equality of a Vectors: a=b: same length and direction. Components of a Vector: P(x1,y1,z1) Q(x2,y2,z2) in Cartesian coordinates. = = []− − − = a PQ x 2 x1, y2 y1,z2 z1 [a1,a 2 ,a3 ] = 2 + 2 + 2 Length in Terms of Components: a a1 a 2 a3 Position Vector: from origin (0,0,0) point A (x,y,z): r=[x,y,z] Vector Addition, Scalar Multiplication b (1) Addition: a + b = []a + b ,a + b ,a + b 1 1 2 2 3 3 a a+b a+ b= b+ a (u+ v) + w= u+ (v+ w) a+ 0= 0+ a= a a+ (-a) = 0 = [] (2) Multiplication: c a c a 1 , c a 2 ,ca3 c(a + b) = ca + cb (c + k) a = ca + ka c(ka) = cka 1a = a 0a = 0 (-1)a = -a = []= + + Unit Vectors: i, j, k a a1,a 2 ,a 3 a1i a 2 j a3 k i = [1,0,0], j=[0,1,0], k=[0,0,1] 8.2. Inner Product (Dot Product) Definition: a ⋅ b = a b cosγ if a ≠ 0, b ≠ 0 a ⋅ b = 0 if a = 0 or b = 0; cosγ = 0 3 ⋅ = + + = a b a1b1 a 2b2 a3b3 aibi i=1 a ⋅ b = 0 (a is orthogonal to b; a, b=orthogonal vectors) Theorem 1: The inner product of two nonzero vectors is zero iff these vectors are perpendicular.
    [Show full text]