DOCSLIB.ORG

Chapter 8 Change of Variables, Parametrizations, Surface Integrals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 8 Change of Variables, Parametrizations, Surface Integrals Chapter 8 Change of Variables, Parametrizations, Surface Integrals x0. The transformation formula In evaluating any integral, if the integral depends on an auxiliary function of the variables involved, it is often a good idea to change variables and try to simplify the integral. The formula which allows one to pass from the original integral to the new one is called the transformation formula (or change of variables formula). It should be noted that certain conditions need to be met before one can achieve this, and we begin by reviewing the one variable situation. Let D be an open interval, say (a; b); in R , and let ' : D! R be a 1-1 , C1 mapping (function) such that '0 6= 0 on D: Put D¤ = '(D): By the hypothesis on '; it's either increasing or decreasing everywhere on D: In the former case D¤ = ('(a);'(b)); and in the latter case, D¤ = ('(b);'(a)): Now suppose we have to evaluate the integral Zb I = f('(u))'0(u) du; a for a nice function f: Now put x = '(u); so that dx = '0(u) du: This change of variable allows us to express the integral as Z'(b) Z I = f(x) dx = sgn('0) f(x) dx; '(a) D¤ where sgn('0) denotes the sign of '0 on D: We then get the transformation formula Z Z f('(u))j'0(u)j du = f(x) dx D D¤ This generalizes to higher dimensions as follows: Theorem Let D be a bounded open set in Rn;' : D! Rn a C1, 1-1 mapping whose Jacobian determinant det(D') is everywhere non-vanishing on D; D¤ = '(D); and f an integrable function on D¤: Then we have the transformation formula Z Z Z Z ¢ ¢ ¢ f('(u))j det D'(u)j du1::: dun = ¢ ¢ ¢ f(x) dx1::: dxn: D D¤ 1 Of course, when n = 1; det D'(u) is simply '0(u); and we recover the old formula. This theorem is quite hard to prove, and we will discuss the 2-dimensional case in detail in x1. In any case, this is one of the most useful things one can learn in Calculus, and one should feel free to (properly) use it as much as possible. It is helpful to note that if ' is linear, i.e., given by a linear transformation with associated matrix M, then D'(u) is just M. In other words, the Jacobian determinant is constant in this case. Note also that ' is C1, even C1, in this case, and moreover ' is 1 ¡ 1 i® M is invertible, i.e., has non-zero determinant. Hence ' is 1 ¡ 1 i® det D'(u) 6= 0. But this is special to the linear situation. Many of the cases where change of variables is used to perform integration are when ' is not linear. x1. The formula in the plane Let D be an open set in R2: We will call a mapping ' : D! R2 as above primitive if it is either of the form (P 1) (u; v) ! (u; g(u; v)) or of the form (P 2) (u; v) ! (h(u; v); v); with @g=@v; @h=@u nowhere vanishing on D: (If @g=@v or @h=@u vanishes at a ¯nite set of points, the argument below can be easily extended.) We will now prove the transformation formula when ' is a composition of two primitive transformations, one of type (P1) and the other of type (P2). For simplicity, let us assume that the functions @g=@v(u; v) and @h=@u(u; v) are always positive. (If either of them is negative, it is elementary to modify the argu- ¤ ment.) Put D1 = f(h(u; v); v)j(u; v) 2 Dg and D = f(x; g(x; v))j(x; v) 2 D1g: By hypothesis, D¤ = '(D): Enclose D1 in a closed rectangle R; and look at the intersection P of D1 with a partition of R; which is bounded by the lines x = xm; m = 1; 2; :::; and v = vr; r = ¤ 1; 2; :::; with the subrectangles Rmr being of sides 4x = l and 4v = k: Let R ; ¤ respectively Rmr, denote the image of R, respectively Rmr; under (u; v) ! (u; g(u; v)): ¤ Then each Rmr is bounded by the parallel lines x = xm and x = xm + l and by the arcs of the two curves y = g(x; vr) and y = g(x; vr + k): Then we have xZm+l ¤ area(Rmr) = (g(x; vr + k) ¡ g(x; vr)) dx: xm By the mean value theorem of 1-variable integral calculus, we can write ¤ 0 0 area(Rmr) = l[g(xm; vr + k) ¡ g(xm; vr)]; 2 0 for some point xm in (xm; xm +l): By the mean value theorem of 1-variable di®erential calculus, we get @g area(R¤ ) = lk (x0 ; v0 ); mr @v m r 0 0 for some xm 2 (xm; xm + l) and vr 2 (vr; vr + k): So, for any function f which is integrable on D¤; we obtain ZZ X 0 0 0 @g 0 0 f(x; y) dx dy = lim klf(xm; g(xm; vr)) (xm; vr): P @v m;r D¤ RR @g The expression on the right tends to the integral f(x; g(x; v)) @v (x; v) dx dv: Thus D1 we get the identity RR RR @g f(x; y) dx dy = f(x; g(x; v)) @v (x; v) dx dv ¤ D D1 By applying the same argument, with the roles of x; y (respectively g; h) switched, we obtain RR RR @g @g @h f(x; g(x; v)) @v (x; v) dx dy = f(h(u; v); g(h(u; v); v)) @v (h(u; v); v) @u (u; v) du dv D1 D Since ' = g ± h we get by chain rule that @h @g det D'(u; v) = (u; v) (h(u; v); v); @u @v which is by hypothesis > 0: Thus we get RR RR f(x; y) dx dy = f('(u; v))j det D'(u; v)j du dv D¤ D as asserted in the Theorem. How to do the general case of '? The fact is, we can subdivide D into a ¯nite union of subregions, on each of which ' can be realized as a composition of primitive trans- formations. We refer the interested reader to chapter 3, volume 2, of "Introduction to Calculus and Analysis" by R. Courant and F. John. x2. Examples 3 (1) Let © = f(x; y) 2 R2jx2 +y2 · a2g; with a > 0: We know that A = area(©) = ¼a2: But let us now do this by using polar coordinates. Put D = f(r; θ) 2 R2j0 < r < a; 0 < θ < 2¼g; and de¯ne ' : D! © by '(r; θ) = (r cos θ; r sin θ): Then D is a connected, bounded open set, and ' is C1, with image D¤ which is the complement of a negligible set in ©; hence any integration over D¤ will be the same as doing it over ©. Moreover, @' @' = (cos θ; sin θ) and = (¡r sin θ; r cos θ): @r @θ Hence µ ¶ cos θ ¡r sin θ det(D') = det = r; sin θ r cos θ which is positive on D. So ' is 1 ¡ 1. By the transformation formula, we get ZZ ZZ Za Z2¼ A = dx dy = j det D'(r; θ)j dr dθ = r dr dθ = © D 0 0 Za Z2¼ Za ¸ r2 a = r dr dθ = 2¼ r dr = 2¼ = ¼a2: 2 0 0 0 0 We can justify the iterated integration above by noting that the coordinate func- tion (r; θ) ! r on the open rectangular region D is continuous, thus allowing us to use Fubini. RR (2) Compute the integral I = R xdxdy where R is the region f(r; Á) j 1 · r · 2; 0 · Á · ¼=4g. R R i2 p p 2 ¼=4 ¼=4 r3 2 7 2 We have I = 1 0 r cos(Á)rdrdÁ = sin(Á)]0 ¢ 3 = 2 (8=3 ¡ 1=3) = 6 : 1 (3) Let © be the region insideRR the parallelogram bounded by y = 2x; y = 2x¡2; y = x and y = x + 1: Evaluate I = xy dx dy. © The parallelogram is spanned by the vectors (1; 2) and (2; 2), so it seems reasonable to make the linear change of variable µ ¶ µ ¶ µ ¶ x 1 2 u = y 2 2 v 4 since then we'll have to integrate over the box [0; 1] £ [0; 1] in the u ¡ v-region. The Jacobian matrix of this transformation is of course constant and has determinant ¡2. So we get Z 1 Z 1 Z 1 Z 1 I = (u + 2v)(2u + 2v)j ¡ 2jdudv = 2 (2u2 + 6uv + 4v2)dudv 0 0 0 0 Z 1 Z 1 3 2 2 ¤1 2 = 2 2u =3 + 3u v + 4v u 0 dv = 2 (2=3 + 3v + 4v )dv 0 0 2 3 ¤1 = 2(2v=3 + 3v =2 + 4v =3 0) = 2(2=3 + 3=2 + 4=3) = 7: (4) Find the volume of the cylindrical region in R3 de¯ned by W = f(x; y; z)jx2 + y2 · a2; 0 · z · hg; where a; h are positive real numbers. We need to compute ZZZ I = vol(W ) = dx dy dz: W It is convenient here to introduce the cylindrical coordinates given by the trans- formation ' : D! R3 given by '(r; θ; z) = (r cos θ; r sin θ; z); where D = f0 < r < a; 0 < θ < 2¼; 0 < z < hg: It is easy to see ' is 1-1, C1 and onto the interior D¤ of W: Again, since the boundary of W is negligible, we may just integrate over D¤.
Load more

Recommended publications
  • Cloaking Via Change of Variables for Second Order Quasi-Linear Elliptic Differential Equations Maher Belgacem, Abderrahman Boukricha
    Cloaking via change of variables for second order quasi-linear elliptic differential equations Maher Belgacem, Abderrahman Boukricha To cite this version: Maher Belgacem, Abderrahman Boukricha. Cloaking via change of variables for second order quasi- linear elliptic differential equations. 2017. hal-01444772 HAL Id: hal-01444772 https://hal.archives-ouvertes.fr/hal-01444772 Preprint submitted on 24 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Cloaking via change of variables for second order quasi-linear elliptic differential equations Maher Belgacem, Abderrahman Boukricha University of Tunis El Manar, Faculty of Sciences of Tunis 2092 Tunis, Tunisia. Abstract The present paper introduces and treats the cloaking via change of variables in the framework of quasi-linear elliptic partial differential operators belonging to the class of Leray-Lions (cf. [14]). We show how a regular near-cloak can be obtained using an admissible (nonsingular) change of variables and we prove that the singular change- of variable-based scheme achieve perfect cloaking in any dimension d ≥ 2. We thus generalize previous results (cf. [7], [11]) obtained in the context of electric impedance tomography formulated by a linear differential operator in divergence form.
    [Show full text]
  • Line, Surface and Volume Integrals
    Line, Surface and Volume Integrals 1 Line integrals Z Z Z Á dr; a ¢ dr; a £ dr (1) C C C (Á is a scalar ¯eld and a is a vector ¯eld) We divide the path C joining the points A and B into N small line elements ¢rp, p = 1;:::;N. If (xp; yp; zp) is any point on the line element ¢rp, then the second type of line integral in Eq. (1) is de¯ned as Z XN a ¢ dr = lim a(xp; yp; zp) ¢ rp N!1 C p=1 where it is assumed that all j¢rpj ! 0 as N ! 1. 2 Evaluating line integrals The ¯rst type of line integral in Eq. (1) can be written as Z Z Z Á dr = i Á(x; y; z) dx + j Á(x; y; z) dy C CZ C +k Á(x; y; z) dz C The three integrals on the RHS are ordinary scalar integrals. The second and third line integrals in Eq. (1) can also be reduced to a set of scalar integrals by writing the vector ¯eld a in terms of its Cartesian components as a = axi + ayj + azk. Thus, Z Z a ¢ dr = (axi + ayj + azk) ¢ (dxi + dyj + dzk) C ZC = (axdx + aydy + azdz) ZC Z Z = axdx + aydy + azdz C C C 3 Some useful properties about line integrals: 1. Reversing the path of integration changes the sign of the integral. That is, Z B Z A a ¢ dr = ¡ a ¢ dr A B 2. If the path of integration is subdivided into smaller segments, then the sum of the separate line integrals along each segment is equal to the line integral along the whole path.
    [Show full text]
  • Introduction to the Modern Calculus of Variations
    MA4G6 Lecture Notes Introduction to the Modern Calculus of Variations Filip Rindler Spring Term 2015 Filip Rindler Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom [email protected] http://www.warwick.ac.uk/filiprindler Copyright ©2015 Filip Rindler. Version 1.1. Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. This is not an encyclopedic work; many important results are omitted and sometimes I only present a special case of a more general theorem. I have, however, tried to strike a balance between a pure introduction and a text that can be used for later revision of forgotten material. The presentation is based around a few principles: • The presentation is quite “modern” in that I use several techniques which are perhaps not usually found in an introductory text or that have only recently been developed. • For most results, I try to use “reasonable” assumptions, not necessarily minimal ones. • When presented with a choice of how to prove a result, I have usually preferred the (in my opinion) most conceptually clear approach over more “elementary” ones. For example, I use Young measures in many instances, even though this comes at the expense of a higher initial burden of abstract theory. • Wherever possible, I first present an abstract result for general functionals defined on Banach spaces to illustrate the general structure of a certain result.
    [Show full text]
  • An Introduction to Space–Time Exterior Calculus
    mathematics Article An Introduction to Space–Time Exterior Calculus Ivano Colombaro * , Josep Font-Segura and Alfonso Martinez Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain; [email protected] (J.F.-S.); [email protected] (A.M.) * Correspondence: [email protected]; Tel.: +34-93-542-1496 Received: 21 May 2019; Accepted: 18 June 2019; Published: 21 June 2019 Abstract: The basic concepts of exterior calculus for space–time multivectors are presented: Interior and exterior products, interior and exterior derivatives, oriented integrals over hypersurfaces, circulation and flux of multivector fields. Two Stokes theorems relating the exterior and interior derivatives with circulation and flux, respectively, are derived. As an application, it is shown how the exterior-calculus space–time formulation of the electromagnetic Maxwell equations and Lorentz force recovers the standard vector-calculus formulations, in both differential and integral forms. Keywords: exterior calculus; exterior algebra; electromagnetism; Maxwell equations; differential forms; tensor calculus 1. Introduction Vector calculus has, since its introduction by J. W. Gibbs [1] and Heaviside, been the tool of choice to represent many physical phenomena. In mechanics, hydrodynamics and electromagnetism, quantities such as forces, velocities and currents are modeled as vector fields in space, while flux, circulation, divergence or curl describe operations on the vector fields themselves. With relativity theory, it was observed that space and time are not independent but just coordinates in space–time [2] (pp. 111–120). Tensors like the Faraday tensor in electromagnetism were quickly adopted as a natural representation of fields in space–time [3] (pp. 135–144). In parallel, mathematicians such as Cartan generalized the fundamental theorems of vector calculus, i.e., Gauss, Green, and Stokes, by means of differential forms [4].
    [Show full text]
  • 7. Transformations of Variables
    Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 7. Transformations of Variables Basic Theory The Problem As usual, we start with a random experiment with probability measure ℙ on an underlying sample space. Suppose that we have a random variable X for the experiment, taking values in S, and a function r:S→T . Then Y=r(X) is a new random variabl e taking values in T . If the distribution of X is known, how do we fin d the distribution of Y ? This is a very basic and important question, and in a superficial sense, the solution is easy. 1. Show that ℙ(Y∈B) = ℙ ( X∈r −1( B)) for B⊆T. However, frequently the distribution of X is known either through its distribution function F or its density function f , and we would similarly like to find the distribution function or density function of Y . This is a difficult problem in general, because as we will see, even simple transformations of variables with simple distributions can lead to variables with complex distributions. We will solve the problem in various special cases. Transformed Variables with Discrete Distributions 2. Suppose that X has a discrete distribution with probability density function f (and hence S is countable). Show that Y has a discrete distribution with probability density function g given by g(y)=∑ f(x), y∈T x∈r −1( y) 3. Suppose that X has a continuous distribution on a subset S⊆ℝ n with probability density function f , and that T is countable.
    [Show full text]
  • Surface Integrals
    VECTOR CALCULUS 16.7 Surface Integrals In this section, we will learn about: Integration of different types of surfaces. PARAMETRIC SURFACES Suppose a surface S has a vector equation r(u, v) = x(u, v) i + y(u, v) j + z(u, v) k (u, v) D PARAMETRIC SURFACES •We first assume that the parameter domain D is a rectangle and we divide it into subrectangles Rij with dimensions ∆u and ∆v. •Then, the surface S is divided into corresponding patches Sij. •We evaluate f at a point Pij* in each patch, multiply by the area ∆Sij of the patch, and form the Riemann sum mn * f() Pij S ij ij11 SURFACE INTEGRAL Equation 1 Then, we take the limit as the number of patches increases and define the surface integral of f over the surface S as: mn * f( x , y , z ) dS lim f ( Pij ) S ij mn, S ij11 . Analogues to: The definition of a line integral (Definition 2 in Section 16.2);The definition of a double integral (Definition 5 in Section 15.1) . To evaluate the surface integral in Equation 1, we approximate the patch area ∆Sij by the area of an approximating parallelogram in the tangent plane. SURFACE INTEGRALS In our discussion of surface area in Section 16.6, we made the approximation ∆Sij ≈ |ru x rv| ∆u ∆v where: x y z x y z ruv i j k r i j k u u u v v v are the tangent vectors at a corner of Sij. SURFACE INTEGRALS Formula 2 If the components are continuous and ru and rv are nonzero and nonparallel in the interior of D, it can be shown from Definition 1—even when D is not a rectangle—that: fxyzdS(,,) f ((,))|r uv r r | dA uv SD SURFACE INTEGRALS This should be compared with the formula for a line integral: b fxyzds(,,) f (())|'()|rr t tdt Ca Observe also that: 1dS |rr | dA A ( S ) uv SD SURFACE INTEGRALS Example 1 Compute the surface integral x2 dS , where S is the unit sphere S x2 + y2 + z2 = 1.
    [Show full text]
  • Arxiv:2109.03250V1 [Astro-Ph.EP] 7 Sep 2021 by TESS
    Draft version September 9, 2021 Typeset using LATEX modern style in AASTeX62 Efficient and precise transit light curves for rapidly-rotating, oblate stars Shashank Dholakia,1 Rodrigo Luger,2 and Shishir Dholakia1 1Department of Astronomy, University of California, Berkeley, CA 2Center for Computational Astrophysics, Flatiron Institute, New York, NY Submitted to ApJ ABSTRACT We derive solutions to transit light curves of exoplanets orbiting rapidly-rotating stars. These stars exhibit significant oblateness and gravity darkening, a phenomenon where the poles of the star have a higher temperature and luminosity than the equator. Light curves for exoplanets transiting these stars can exhibit deviations from those of slowly-rotating stars, even displaying significantly asymmetric transits depending on the system’s spin-orbit angle. As such, these phenomena can be used as a protractor to measure the spin-orbit alignment of the system. In this paper, we introduce a novel semi-analytic method for generating model light curves for gravity-darkened and oblate stars with transiting exoplanets. We implement the model within the code package starry and demonstrate several orders of magnitude improvement in speed and precision over existing methods. We test the model on a TESS light curve of WASP-33, whose host star displays rapid rotation (v sin i∗ = 86:4 km/s). We subtract the host’s δ-Scuti pulsations from the light curve, finding an asymmetric transit characteristic of gravity darkening. We find the projected spin orbit angle is consistent with Doppler +19:0 ◦ tomography and constrain the true spin-orbit angle of the system as ' = 108:3−15:4 .
    [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]
  • The Laplace Operator in Polar Coordinates in Several Dimensions∗
    The Laplace operator in polar coordinates in several dimensions∗ Attila M´at´e Brooklyn College of the City University of New York January 15, 2015 Contents 1 Polar coordinates and the Laplacian 1 1.1 Polar coordinates in n dimensions............................. 1 1.2 Cartesian and polar differential operators . ............. 2 1.3 Calculating the Laplacian in polar coordinates . .............. 2 2 Changes of variables in harmonic functions 4 2.1 Inversionwithrespecttotheunitsphere . ............ 4 1 Polar coordinates and the Laplacian 1.1 Polar coordinates in n dimensions Let n ≥ 2 be an integer, and consider the n-dimensional Euclidean space Rn. The Laplace operator Rn n 2 2 in is Ln =∆= i=1 ∂ /∂xi . We are interested in solutions of the Laplace equation Lnf = 0 n 2 that are spherically symmetric,P i.e., is such that f depends only on r=1 xi . In order to do this, we need to use polar coordinates in n dimensions. These can be definedP as follows: for k with 1 ≤ k ≤ n 2 k 2 define rk = i=1 xi and put for 2 ≤ k ≤ n write P (1) rk−1 = rk sin φk and xk = rk cos φk. rk−1 = rk sin φk and xk = rk cos φk. The polar coordinates of the point (x1,x2,...,xn) will be (rn,φ2,φ3,...,φn). 2 2 2 In case n = 2, we can write y = x1, x = x2. The polar coordinates (r, θ) are defined by r = x + y , (2) x = r cos θ and y = r sin θ, so we can take r2 = r and φ2 = θ.
    [Show full text]
  • Surface Integrals
    V9. Surface Integrals Surface integrals are a natural generalization of line integrals: instead of integrating over a curve, we integrate over a surface in 3-space. Such integrals are important in any of the subjects that deal with continuous media (solids, fluids, gases), as well as subjects that deal with force fields, like electromagnetic or gravitational fields. Though most of our work will be spent seeing how surface integrals can be calculated and what they are used for, we first want to indicate briefly how they are defined. The surface integral of the (continuous) function f(x,y,z) over the surface S is denoted by (1) f(x,y,z) dS . ZZS You can think of dS as the area of an infinitesimal piece of the surface S. To define the integral (1), we subdivide the surface S into small pieces having area ∆Si, pick a point (xi,yi,zi) in the i-th piece, and form the Riemann sum (2) f(xi,yi,zi)∆Si . X As the subdivision of S gets finer and finer, the corresponding sums (2) approach a limit which does not depend on the choice of the points or how the surface was subdivided. The surface integral (1) is defined to be this limit. (The surface has to be smooth and not infinite in extent, and the subdivisions have to be made reasonably, otherwise the limit may not exist, or it may not be unique.) 1. The surface integral for flux. The most important type of surface integral is the one which calculates the flux of a vector field across S.
    [Show full text]
  • Surface Integrals, Stokes's Theorem and Gauss's Theorem
    Surface Integrals, Stokes's Theorem and Gauss's Theorem Specifying a surface: ² Parametric form: X(s; t) = (x(s; t); y(s; t); z(s; t)) in some region D in R2. ² Explicit function: z = g(x; y) where g is a function in some region D in R2. ² Implicit function: g(x; y; z) = 0, for some function g of three variables. Normal vector and unit normal vector: Normal vectors and unit normal vectors to a surface play an important role in surface integrals. Here are some ways to obtain these vectors. ² Geometric reasoning. For surfaces of simple shape (e.g., planes or spherical surfaces), the easiest way to get normal vectors is by geometric arguments. For example, if the surface is spherical and centered at the origin, then x is a normal vector. If the surface is horizontal or vertical, one can take one of the unit basis vectors (or its negative) as normal vectors. Such geometric reasoning requires virtually no calculation, and is usually the quickest way to evaluate a surface integral in situations where it can be applied. ² Surfaces given by an equation g(x; y; z) = 0. In this case, the gradient of g, rg, is normal to the surface. Normalizing this vector, we get a unit normal vector, n = rg=krgk. Similarly, if the surface is given by an equation z = g(x; y), the vector (gx; gy; 1) is normal to the surface. ² Surfaces given by a parametric representation X(s; t). In this case, the vector N = Ts£ Tt is a normal vector, called the standard normal vector of the given parametrization.
    [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]