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A Brief Tour of Vector Calculus

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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. Prelude This is an ongoing notes project to capture the essence of the subject of vector calculus by providing a variety of examples and visualizations, but also to present the main ideas of vector calculus in conceptual a framework that is adequate for the needs of mathematics, physics, and engineering majors. The essential prerequisites are • comfort with college level algebra, analytic geometry and trigonometry, • calculus knowledge including exposure to multivariable functions, partial derivatives and multiple integrals, • the material of my notes on Vector Algebra, and the Equations of Lines and Planes in 3-Space or equivalent, and • the material related to polar, cylindrical and spherical frames in my notes on Curvature, Natural Frames, and Acceleration for Plane and Space Curves, particularly for the optional sections. Definitions and results are sometimes stated in terms of functions of n variables for an arbitrary number n ≥ 2, but examples focus on n = 2 and n = 3. Since it is hard to really see what is happening for n ≥ 4, pictures often show up for examples using n = 2 or 3, and the hope is that you internalize some intuition from these pictures and examples. If you have trouble understanding a statement that uses arbitrary n, just read it with n = 2 or n = 3 and try to understand the underlying geometry in these cases. A warning: my notational conventions sometimes differ from other common sources, so use cau- tion when comparing to other resources on vector calculus. In particular, my spherical coordinate system is not the one in most common use, but is an intuitive convention nonetheless, as explained in Curvature, Natural Frames, and Acceleration for Plane and Space Curves, where they are con- structed so as to adapt geographers' conventions for latitude and longitude. These notes exist primarily to prop up the problems; the exposition and ideas herein are foremost to introduce enough language for the reader to then approach and tackle the various problems provided. Some of the problems are quite standard and are meant to drive home a particular concept, while others encourage you to fill in details omitted in examples, and a few problems are meant to give a flavor of a more advanced but mathematical subject, such as the study of differential equations, or topology. There's also a handful of computational problems meant to satisfy those who merely enjoy the meditative art of symbol pushing, but the bulk of the questions ought to provoke some serious thought about how the objects of vector calculus interact with each other and with mathematical models of the real world. In addition to clarifying notations and terminology, footnotes are often used to exposit on more advanced directions, and hint at how the subject matter broadens and connects to contemporary mathematical thinking and research. Finally, please forgive any typos and the clunkiness of formatting; these and the compan- ion/prerequisite notes are work in progress that were primarily hastily assembled in the midst of my terminal years teaching at University of Massachusetts Amherst as a PhD student. I welcome suggestions as I work to improve these. ii 11/14/19 Multivariate Calculus: Vector Calculus Havens 1. Directional Derivatives, the Gradient and the Del Operator § 1.1. Conceptual Review: Directional Derivatives and the Gradient Recall that partial derivatives are defined by computing a difference quotient in which only one variable is perturbed. This has a geometric interpretation as slicing the graph of the function along a plane parallel to the directions of the coordinate of concern xi and the coordinate of the dependent variable whose value is f(x1; : : : ; xi; : : : ; xn) (so this plane is thus normal to the coordinate directions of the variables held constant), and then measuring the rate of change of the function along the curve of intersection, as a function of the variable xi. In two variables, the slicing for the two partial derivatives corresponds to a picture like that of figure1. Figure 1. Curves C1 and C2 on the graph of a function along planes of constant y and x respectively. This geometric picture suggests that we need not be confined to only know the rate of change of the function along coordinate directions, for we could slice the graph along any plane containing the direction of the dependent variable. This gives rise to the notion of a directional derivative, defined so as to allow us to measure the rate of change of a function f in any of the possible directions we might choose as we leave a point of the function's domain. n Fix a connected domain D ⊂ R , and let f : D! R be a multivariate function of n variables, and r = hx1; : : : xni an arbitrary position vector for a point P of D. We'll often conflate the idea of D as a set of points with the conception of it as a set of positions, and thus will unapologetically write things such as r 2 D to mean that the point (x1; : : : ; xn) with position r is an element of D. Definitions will often be stated in the general context of arbitrary n, but examples and pictures will be specialized to low dimensions. Observe that a \direction" at a point r 2 D may be specified by giving a unit vector, i.e. a vector u^ of length 1. In two dimensions, the set of unit vectors, and thus, of directions, is a circle, while in 1 11/14/19 Multivariate Calculus: Vector Calculus Havens n three dimensions it is the surface of a sphere. The set of unit vectors in R geometrically describes n the origin centered (n − 1)-dimensional sphere in R : n−1 n S = fr 2 R : krk = 1g : n−1 Definition. Given a unit vector u^ 2 S and a function f : D! R of n variables, the directional derivative of f in the direction of u^ at a point r0 2 D is f(r0 + hu^) − f(r0) Du^ f(r0) = lim : h!0 h Again, in two dimensions we can actually see and interpret this limit in terms of the familiar notion of a slope of a tangent line. This, together with our discussion of the gradient in two dimensions, will set the stage for understanding tangent space objects later. Figure 2. The directional derivative computes a slope to a curve of intersection of a vertical plane slicing the graph surface in the direction specified by a unit vector. 3 Recall that the graph Gf of a two-variable function f(x; y) is the locus of points (x; y; z) 2 R satisfying z = f(x; y). For f continuously differentiable, this is a smooth1 surface over D. Fix a point 1 r0 2 D at which we are interested in the directional derivative in the direction of a given u^ 2 S . ^ ^ Note that u^ and k determine a plane Πu^;r0 containing the point r0 + f(r0)k = hx0; y0; f(x0; y0)i, and this plane slices the surface Gf along some curve. In the plane Πu^;r0 , the variation of the curve as one displaces from r0 in the direction of ±u^ is purely in the z direction, and so it is natural to try to study the rate of change of z as one moves along the u^ direction by a small displacement hu^. One sees easily that the directional derivative formula above is precisely the appropriate limit of a difference quotient to capture this rate of change. Observe also that the usual partial derivatives are 3 just directional derivatives along the coordinate directions, e.g. for R with standard rectangular 2 11/14/19 Multivariate Calculus: Vector Calculus Havens coordinates (x; y; z), one has: @ @ @ D = ;D = ;D = : ^ı @x ^ @y k^ @z Proposition 1.1. The directional derivative of f in the direction of u^ at a point r0 2 D may be calculated as n @f D f(r ) = X u (r ) ; u^ 0 i @x 0 i=1 i n where ui are the components of u^ in rectangular coordinates on R and xi are the n variables of f giving the rectangular coordinates of a general vector argument r.
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