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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. -
CHAPTER 1 VECTOR ANALYSIS ̂ ̂ Distribution Rule Vector Components
8/29/2016 CHAPTER 1 1. VECTOR ALGEBRA VECTOR ANALYSIS Addition of two vectors 8/24/16 Communicative Chap. 1 Vector Analysis 1 Vector Chap. Associative Definition Multiplication by a scalar Distribution Dot product of two vectors Lee Chow · , & Department of Physics · ·· University of Central Florida ·· 2 Orlando, FL 32816 • Cross-product of two vectors Cross-product follows distributive rule but not the commutative rule. 8/24/16 8/24/16 where , Chap. 1 Vector Analysis 1 Vector Chap. Analysis 1 Vector Chap. In a strict sense, if and are vectors, the cross product of two vectors is a pseudo-vector. Distribution rule A vector is defined as a mathematical quantity But which transform like a position vector: so ̂ ̂ 3 4 Vector components Even though vector operations is independent of the choice of coordinate system, it is often easier 8/24/16 8/24/16 · to set up Cartesian coordinates and work with Analysis 1 Vector Chap. Analysis 1 Vector Chap. the components of a vector. where , , and are unit vectors which are perpendicular to each other. , , and are components of in the x-, y- and z- direction. 5 6 1 8/29/2016 Vector triple products · · · · · 8/24/16 8/24/16 · ( · Chap. 1 Vector Analysis 1 Vector Chap. Analysis 1 Vector Chap. · · · All these vector products can be verified using the vector component method. It is usually a tedious process, but not a difficult process. = - 7 8 Position, displacement, and separation vectors Infinitesimal displacement vector is given by The location of a point (x, y, z) in Cartesian coordinate ℓ as shown below can be defined as a vector from (0,0,0) 8/24/16 In general, when a charge is not at the origin, say at 8/24/16 to (x, y, z) is given by (x’, y’, z’), to find the field produced by this Chap. -
Vector Calculus Operators
Vector Calculus Operators July 6, 2015 1 Div, Grad and Curl The divergence of a vector quantity in Cartesian coordinates is defined by: −! @ @ @ r · ≡ x + y + z (1) @x @y @z −! where = ( x; y; z). The divergence of a vector quantity is a scalar quantity. If the divergence is greater than zero, it implies a net flux out of a volume element. If the divergence is less than zero, it implies a net flux into a volume element. If the divergence is exactly equal to zero, then there is no net flux into or out of the volume element, i.e., any field lines that enter the volume element also exit the volume element. The divergence of the electric field and the divergence of the magnetic field are two of Maxwell's equations. The gradient of a scalar quantity in Cartesian coordinates is defined by: @ @ @ r (x; y; z) ≡ ^{ + |^ + k^ (2) @x @y @z where ^{ is the unit vector in the x-direction,| ^ is the unit vector in the y-direction, and k^ is the unit vector in the z-direction. Evaluating the gradient indicates the direction in which the function is changing the most rapidly (i.e., slope). If the gradient of a function is zero, the function is not changing. The curl of a vector function in Cartesian coordinates is given by: 1 −! @ @ @ @ @ @ r × ≡ ( z − y )^{ + ( x − z )^| + ( y − x )k^ (3) @y @z @z @x @x @y where the resulting function is another vector function that indicates the circulation of the original function, with unit vectors pointing orthogonal to the plane of the circulation components. -
Divergence, Gradient and Curl Based on Lecture Notes by James
Divergence, gradient and curl Based on lecture notes by James McKernan One can formally define the gradient of a function 3 rf : R −! R; by the formal rule @f @f @f grad f = rf = ^{ +| ^ + k^ @x @y @z d Just like dx is an operator that can be applied to a function, the del operator is a vector operator given by @ @ @ @ @ @ r = ^{ +| ^ + k^ = ; ; @x @y @z @x @y @z Using the operator del we can define two other operations, this time on vector fields: Blackboard 1. Let A ⊂ R3 be an open subset and let F~ : A −! R3 be a vector field. The divergence of F~ is the scalar function, div F~ : A −! R; which is defined by the rule div F~ (x; y; z) = r · F~ (x; y; z) @f @f @f = ^{ +| ^ + k^ · (F (x; y; z);F (x; y; z);F (x; y; z)) @x @y @z 1 2 3 @F @F @F = 1 + 2 + 3 : @x @y @z The curl of F~ is the vector field 3 curl F~ : A −! R ; which is defined by the rule curl F~ (x; x; z) = r × F~ (x; y; z) ^{ |^ k^ = @ @ @ @x @y @z F1 F2 F3 @F @F @F @F @F @F = 3 − 2 ^{ − 3 − 1 |^+ 2 − 1 k:^ @y @z @x @z @x @y Note that the del operator makes sense for any n, not just n = 3. So we can define the gradient and the divergence in all dimensions. However curl only makes sense when n = 3. Blackboard 2. The vector field F~ : A −! R3 is called rotation free if the curl is zero, curl F~ = ~0, and it is called incompressible if the divergence is zero, div F~ = 0. -
Chapter 1. Vector Analysis 1.1 Vector Algebra 1.1.1 Vector Operations
Chapter 1. Vector Analysis 1.1 Vector Algebra 1.1.1 Vector Operations Addition is commutative: A + B = B + A Addition is associative: (A + B) + C = A + (B + C) To subtract is to add its opposite: A - B = A + (-B) Dot product (= scalar product) is commutative: A . B = B . A Dot product (= scalar product) is distributive: A . (B + C) = A . B + A . C Cross product (= vector product) is not commutative: B x A = A x B Dot product (= vector product) is distributive: A x (B + C) = A x B + A x C 1.1.2 Vector Algebra: Component form Unit vectors Component form 1.1.2 Vector Algebra: Component form 1.1.3 Triple Products 1.1.3 Triple Products BAC-CAB rule 1.1.4 Position, Displacement, and Separation Vectors Position vector: Infinitesimal displacement vector: Separation vector from source point to field point: 1.1.5 How Vectors transform 1.2 Differential Calculus 1.2.1 “Ordinary” Derivatives 1.2.2 Gradient Gradient of T What’s the physical meaning of the Gradient: Gradient is a vector that points in the direction of maximum increase of a function. Its magnitude gives the slope (rate of increase) along this maximal direction. Gradient represents both the magnitude and the direction of the maximum rate of increase of a scalar function. 1.2.3 The Del Operator: : a vector operator, not a vector. (gradient) Gradient represents both the magnitude and the direction of the maximum rate of increase of a scalar function. (divergence) (curl) 1.2.4 The Divergence div AA A Ay A A x z : scalar, a measure of how much the vector A xyz spread out (diverges) from the point in question : positive (negative if the arrows pointed in) divergence : zero divergence : positive divergence 1.2.5 The Curl curl AAA rot : a vector, a measure of how much the vector A curl (rotate) around the point in question. -
The Laplacian
Basic Mathematics The Laplacian R Horan & M Lavelle The aim of this package is to provide a short self assessment programme for students who want to apply the Laplacian operator. Copyright c 2005 [email protected] , [email protected] Last Revision Date: April 4, 2005 Version 1.0 Table of Contents 1. Introduction (Grad, Div, Curl) 2. The Laplacian 3. The Laplacian of a Product of Fields 4. The Laplacian and Vector Fields 5. Final Quiz Solutions to Exercises Solutions to Quizzes The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials. Section 1: Introduction (Grad, Div, Curl) 3 1. Introduction (Grad, Div, Curl) The vector differential operator ∇ , called “del” or “nabla”, is defined in three dimensions to be: ∂ ∂ ∂ ∇ = i + j + k ∂x ∂y ∂z The result of applying this vector operator to a scalar field is called the gradient of the scalar field: ∂f ∂f ∂f gradf(x, y, z) = ∇f(x, y, z) = i + j + k . ∂x ∂y ∂z (See the package on Gradients and Directional Derivatives.) The scalar product of this vector operator with a vector field F (x, y, z) is called the divergence of the vector field: ∂F ∂F ∂F divF (x, y, z) = ∇ · F = 1 + 2 + 3 . ∂x ∂y ∂z Section 1: Introduction (Grad, Div, Curl) 4 The vector product of the vector ∇ with a vector field F (x, y, z) is the curl of the vector field. It is written as curl F (x, y, z) = ∇ × F ∂F ∂F ∂F ∂F ∂F ∂F ∇ × F (x, y, z) = ( 3 − 2 )i − ( 3 − 1 )j + ( 2 − 1 )k ∂y ∂z ∂x ∂z ∂x ∂y i j k ∂ ∂ ∂ = , ∂x ∂y ∂z F1 F2 F3 where the last line is a formal representation of the line above. -
Vector Derivatives
Vector derivatives September 7, 2015 In generalizing the idea of a derivative to vectors, we find several new types of object. Here we look at ordinary derivatives, but also the gradient, divergence and curl. 1 Curves df(x) The derivative of a function of a single variable is familiar from calculus, dx . The simplest generalization of this is when we have a curve in two or three dimensions. A curve is a smoothly parameterized sequence of points in space. You are familiar with this idea from mechanics, where a particle’s position may be parameterized by time, x (t) = (x (t) ; y (t) ; z (t)) At any time t, these three functions give us the position of the moving particle. The complete path of the particle is a “curve”. We can produce a second vector, the velocity vector, from this position vector by differentiation dx (t) v (t) = dt dx (t) dy (t) dz (t) = ; ; dt dt dt We differentiate each of the three functions with respect to the parameter. This result generalizes to ar- bitrary curves and parameterizations. If we have a curve parameterized by any parameter λ, x (λ) = (x (λ) ; y (λ) ; z (λ)), then differentiation gives a tangent vector to the curve at each point, dx (λ) t (λ) = dλ Many parameters can give the same curve. For example, x (t) = (x (t) ; y (t) ; z (t)) 1 = v t; y t; gt2 0x 0y 2 and 1 x (σ) = v σ2; y σ2; gσ4 0x 0y 2 describe the same path in space. However, the length of the tangent vector will depend on which parameter is chosen. -
3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77
3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important concept. For partial derivatives, a similar idea allows us to solve for a function whose partial derivative in one of the variables is given, as seen earlier. However, when several partial derivatives of an unknown function are given simultaneously, there may not be any function with those values! In particular, the gradient field for smooth functions must respect Clairaut’s theorem, so not every vector field is the gradient of some scalar function! This section explores these questions and in answering them, introduces two very important additional vector derivatives, the Curl and Divergence. 3.8.1 Solving rf = v for f given v Looking back: Recall how antiderivatives were introduced in one variable calcu- lus: to find F with a given derivative f, you learned: 1. For f given by a simple formula, guess and check led to an antiderivative F . 2. Any other anti derivative G had the property G = F + C. R x 3. When all else fails, any continuous f has anti-derivative F (x) = a f(t) dt for any fixed a. Earlier in this chapter the corresponding result for a single partial derivative was found to be similar, except that instead of adding constants to get new solutions from old ones, functions independent of the single variable could be added. The gradient vector has partial derivatives in components, so we might expect that this result would allow us to solve the system of equations rG = v. -
Divergence and Curl of COVID19 Spreading in Lower Peninsula of Michigan
Divergence and Curl of COVID19 Spreading in Lower Peninsula of Michigan Yanshuo Wang ( [email protected] ) LLLW LLC Research Article Keywords: Divergence, Curl, COVID19, Transmission Density, Circulation Posted Date: April 27th, 2021 DOI: https://doi.org/10.21203/rs.3.rs-458125/v1 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Divergence and Curl of COVID19 Spreading in Lower Peninsula of Michigan Yanshuo Wang*, Reliability and Data Mining Consultant *Correspondence to: [email protected] 5 Abstract: Divergence and Curl concept in vector field is applied to the COVID19 spreading data for Lower Peninsula of Michigan State, U.S.A. The Divergence is an operator of COVID19 transmission vector field, which produces a scalar field giving the quantity of transmission vector field’s source at each location. The COVID19 10 spreading volume density of the outward flux of transmission field is represented by divergence around a given location. The Curl is an operator of COVID19 transmission field, which describes the circulation of a transmission vector field. The Curl at a location in COVID19 transmission field is represented by a vector whose length and direction denote the magnitude and axis of the maximum 15 circulation. The curl of a transmission field is formally defined as the circulation density at each location of COVID19 transmission field. From data analysis of divergence of curl of Lower Michigan Peninsula, the COVID19 transmission volume flux and circulation can be identified for each County. Summary: 20 This paper is to use vector Divergence and Curl concept to apply to COVID19 confirmed cases in Lower Peninsula of Michigan, U.S.A. -
Divergence and Curl
Intermediate Mathematics Divergence and Curl R Horan & M Lavelle The aim of this package is to provide a short self assessment programme for students who would like to be able to calculate divergences and curls in vector calculus. Copyright c 2004 [email protected] , [email protected] Last Revision Date: September 29, 2004 Version 1.0 Table of Contents 1. Introduction (Grad) 2. Divergence (Div) 3. Curl 4. Final Quiz Solutions to Exercises Solutions to Quizzes The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials. Section 1: Introduction (Grad) 3 1. Introduction (Grad) The vector differential operator ∇ , called “del” or “nabla”, is defined in three dimensions to be: ∂ ∂ ∂ ∇ = i + j + k . ∂x ∂y ∂z Note that these are partial derivatives! If a scalar function, f(x, y, z), is defined and differentiable at all points in some region, then f is a differentiable scalar field. The del vector operator, ∇, may be applied to scalar fields and the result, ∇f, is a vector field. It is called the gradient of f (see the package on Gradi- ents and Directional Derivatives). Quiz As a revision exercise, choose the gradient of the scalar field f(x, y, z) = xy2 − yz. (a) i + (2x − z)j − yk , (b) 2xyi + 2xyj + yk , (c) y2i − zj − yk , (d) y2i + (2xy − z)j − yk . Section 1: Introduction (Grad) 4 The vector operator ∇ may also be allowed to act upon vector fields. Two different ways in which it may act, the subject of this package, are extremely important in mathematics, science and engineering. -
Lectures on Vector Calculus
Lectures on Vector Calculus Paul Renteln Department of Physics California State University San Bernardino, CA 92407 March, 2009; Revised March, 2011 c Paul Renteln, 2009, 2011 ii Contents 1 Vector Algebra and Index Notation 1 1.1 Orthonormality and the Kronecker Delta . 1 1.2 Vector Components and Dummy Indices . 4 1.3 Vector Algebra I: Dot Product . 8 1.4 The Einstein Summation Convention . 10 1.5 Dot Products and Lengths . 11 1.6 Dot Products and Angles . 12 1.7 Angles, Rotations, and Matrices . 13 1.8 Vector Algebra II: Cross Products and the Levi Civita Symbol 18 1.9 Products of Epsilon Symbols . 23 1.10 Determinants and Epsilon Symbols . 27 1.11 Vector Algebra III: Tensor Product . 28 1.12 Problems . 31 2 Vector Calculus I 32 2.1 Fields . 33 2.2 The Gradient . 34 2.3 Lagrange Multipliers . 37 2.4 The Divergence . 41 2.5 The Laplacian . 41 2.6 The Curl . 42 2.7 Vector Calculus with Indices . 43 2.8 Problems . 47 3 Vector Calculus II: Other Coordinate Systems 48 3.1 Change of Variables from Cartesian to Spherical Polar . 48 iii 3.2 Vector Fields and Derivations . 49 3.3 Derivatives of Unit Vectors . 53 3.4 Vector Components in a Non-Cartesian Basis . 54 3.5 Vector Operators in Spherical Coordinates . 54 3.6 Problems . 57 4 Vector Calculus III: Integration 57 4.1 Line Integrals . 57 4.2 Surface Integrals . 64 4.3 Volume Integrals . 67 4.4 Problems . 69 5 Integral Theorems 70 5.1 Green's Theorem . -
Chapter 14. Vector Calculus. Section 14.5 Curl and Divergence. Curl. If
Chapter 14. Vector calculus. Section 14.5 Curl and divergence. Curl. If F~ = P~{ + Q~| + R~k is a vector field on R3 and the partial derivatives of P , Q, and R exist, then the curl of F~ is the vector field on R3 defined by @R @Q @P @R @Q @P curl F~ = − ~{ + − ~| + − ~k @y @z @z @x @x @y @ @ @ ~ Let r = @x~{ + @y~| + @z k be the vector differential operator. @f @f @f rf = ~{ + ~| + ~k @x @y @z Then ~{ ~| ~k @ @ @ @R @Q @P @R @Q @P r × F~ = = − ~{ + − ~| + − ~k = curl F:~ @x @y @z @y @z @z @x @x @y PQR Example 1. Find curl F~ if F~ (x; y; z) = xey~{ − ze−y~| + y ln(z)~k. 1 Theorem 1. If f is a function of three variables that has continuous second-order partial derivatives, then curl(rf) = ~0 Theorem 2. If F~ is a vector field defined on all on R3 whose component functions have continuous partial derivatives and curl F~ = ~0, then F~ is a conservative vector field. Example 2. Determine whether or not the vector field F~ = zx~{ + xy~| + yx~k is conservative. If it is conservative, find a function f such that F~ = rf. 2 The curl vector is associated with rotation. If for a vector field F~ curl F~ = ~0, then the field F~ is irrotational. Example 3. Let B be a rigid body rotation about the z-axis. The rotation can be described by the vector ~w = !~k, where ! is the angular speed of B, that is the tangential speed of any point P in B divided by the distance d from the axis of rotation.