Lecture Notes Introduction to Vector Analysis MATH 332

Lecture Notes Introduction to Vector Analysis MATH 332 Instructor: Ivan Avramidi Textbook: H. F. Davis and A. D. Snider, (WCB Publishers, 1995) New Mexico Institute of Mining and Technology Socorro, NM 87801 August 3, 2017 Author: Ivan Avramidi; File: vecanal332.tex; Date: November 21, 2017; Time: 9:00

Contents 1 Vector Algebra 1 1.1 LECTURE 1. Vector Spaces.................... 1 1.1.1 Vectors........................... 1 1.1.2 Vector Spaces........................ 2 1.1.3 Standard Basis....................... 4 1.2 LECTURE 2. Scalar Product.................... 6 1.2.1 Scalar Product....................... 6 1.2.2 Lines and Planes...................... 8 1.3 LECTURE 3. Vector Product.................... 12 1.3.1 Vector Product....................... 12 1.3.2 Triple Product....................... 15 1.4 LECTURE 4. Tensors and Vector Identities............ 16 1.4.1 Algebraic Vector Identities................. 22 2 Vector Functions 25 2.1 LECTURE 5. Differentiation.................... 25 2.1.1 Limits............................ 25 2.1.2 Derivatives......................... 26 2.2 LECTURE 6. Curves, Velocities and Tangents........... 27 2.2.1 Tangent Vectors....................... 27 2.2.2 Arc Length......................... 29 2.3 LECTURE 7. Acceleration and Curvature............. 30 2.3.1 Curvature.......................... 30 2.3.2 Normals........................... 30 2.3.3 Examples.......................... 31 I

II CONTENTS 3 Fields 35 3.1 LECTURE 8. Fields and Differential Operators.......... 35 3.1.1 Scalar and Vector Fields.................. 35 3.1.2 Gradient........................... 36 3.1.3 Divergence......................... 37 3.1.4 Laplacian.......................... 38 3.1.5 Curl............................. 38 3.2 LECTURE 9. Differential Vector Identities............ 39 3.3 LECTURE 10. Geometric Applications.............. 41 3.3.1 Directional Derivative................... 41 3.3.2 Flow Lines......................... 42 3.3.3 Geometrical Interpretation of Divergence......... 43 3.3.4 Geometrical Interpretation of Curl............. 44 3.3.5 Geometrical Interpretation of Laplacian.......... 45 3.4 LECTURE 11. Fluid Dynamics.................. 47 3.4.1 Flux............................. 47 3.4.2 Swirl............................ 48 3.4.3 Equations of Mathematical Physics............ 50 3.5 LECTURES 12-13. Curvilinear Coordinates........... 51 3.5.1 Cylindrical and Spherical Coordinates........... 51 3.5.2 General Curvilinear Coordinates.............. 53 4 Integrals 63 4.1 LECTURE 14: Line Integrals.................... 63 4.2 LECTURE 15: Simply Connected Domains............ 66 4.3 LECTURES 16-17: Conservative and Irrotational Vector Fields.. 68 4.3.1 Conservative Vector Fields are Irrotational......... 68 4.3.2 Line Integrals of Conservative Vector Fields........ 69 4.3.3 Irrotational Vector Fields in Simply Connected Domains. 71 4.4 LECTURE 18: Solenoidal Vector Fields.............. 73 4.4.1 Solenoidal Vector Fields and Vector Potential....... 73 4.4.2 Gauge Condition...................... 76 4.4.3 Two-dimensional Vector Fields.............. 76 4.5 LECTURE 19: Oriented Surfaces................. 77 4.5.1 Orientation of Surfaces................... 77 4.5.2 Parametrization....................... 78 4.6 LECTURE 20: Surface Integrals.................. 80 4.6.1 Surface Area........................ 80 vecanal332.tex; November 21, 2017; 9:00; p. 1

CONTENTS III 4.6.2 Surface Integrals...................... 81 4.7 LECTURE 21: Volume Integrals.................. 82 5 Integral Theorems 85 5.1 LECTURE 22: Divergence Theorem................ 85 5.2 LECTURE 23: Green s Functions................. 87 5.2.1 First Green Formula.................... 87 5.2.2 Second Green Formula................... 89 5.2.3 Delta Functions....................... 90 5.2.4 Third Green s Formula................... 91 5.2.5 Dirichlet Boundary Value Problem............. 93 5.3 LECTURE 23: Stokes Theorem.................. 94 5.3.1 Green s Theorem...................... 94 5.3.2 Stokes Theorem...................... 95 5.4 LECTURE 24:........................... 98 Bibliography 99 vecanal332.tex; November 21, 2017; 9:00; p. 2

IV CONTENTS vecanal332.tex; November 21, 2017; 9:00; p. 3

Chapter 1 Vector Algebra 1.1 LECTURE 1. Vector Spaces 1.1.1 Vectors Scalars, vectors, tensors,... Scalars. Physical quantities, like mass, energy, volume, temperature, density etc., that can be described by one number are called scalars. This number does not depend on the coordinate system; it is an invariant. Vectors. Vectors are physical quantities, like velocity, position, displacement, force, acceleration, electric field, magnetic field etc., that are described by three numbers. We will study vector analysis in the standard Euclidean space R 3. A vector is a quantity that has both direction and magnitude. It can be visualized as directed line segment. A line segment PQ is a portion of the line passing through the points P and Q. A directed line segment is a line segment when the endpoints are given a definite order. 1

2 CHAPTER 1. VECTOR ALGEBRA Two directed line segments are equivalent if they are parallel and have the same length. A vector is a collection of equivalent directed line segments. It is denoted by A or A. We will make a distinction between a directed line segment PQ and the corresponding vector PQ. The magnitude (or the length, or the norm) PQ of a vector PQ is the distance between its initial P and the terminal Q points. A line segment PP is just a point. The corresponding vector is called the zero vector. Zero vector has zero magnitude and does not have any direction, 0 = 0. A vector with magnitude equal to 1 is called a unit vector. Scalar is just a real number. Properties of Magnitude. 1. A 0, 2. A = 0 if and only if A = 0, 1.1.2 Vector Spaces The sum of the vectors A and B is the vector C = A + B obtained by attaching the vector B at the end of the vector A ; then the vector C has the initial point at the initial point of the vector A and the endpoint at the endpoint of the vector B. Properties of addition: addition is commutative, addition is associative, addition of the zero vector does not change any vector, every vector has the opposite vector. vecanal332.tex; November 21, 2017; 9:00; p. 4

1.1. LECTURE 1. VECTOR SPACES 3 Triangle inequality A + B A + B The subtraction of the vectors is defined by adding the opposite vector. The scalar multiple of a vector A by the scalar s is the vector B = s A that has the same direction for s > 0 and the opposite direction for s < 0 and has the magnitude s A = s A The multiplication by scalars satisfies the following conditions: for any vectors u, v and scalars a and b, 1. a(bv) = (ab)v, 2. (a + b)v = av + bv, 3. a(u + v) = au + av, 4. 1 v = v 5. ( 1)v = v The collection of all vectors and all scalars with the vector addition and the scalar multiplication that satisfies the above properties is called a vector space. Examples. For any scalar s and any non-zero vector A, s 0 = 0 A = 0, A A = 1 A linear combination of a collection of vectors A = {e 1,..., e k } is a vector where {a 1,..., a n } are scalars. a 1 e 1 + + a k e k, A finite collection of vectors A = {e 1,..., e k } is linearly dependent if there exist scalars a 1,..., a k, not all zero, such that a 1 e 1 + + a k e k = 0 vecanal332.tex; November 21, 2017; 9:00; p. 5

4 CHAPTER 1. VECTOR ALGEBRA that is, if one vector is a linear combination of the other, e.g. if a k 0 e k = a 1 e 1 a k 1 a k A collection A of vectors is linearly independent if it is not linearly dependent. A collection A of vectors is linearly independent if no vector of A is a linear combination of a finite number of vectors from A. Two non-zero vectors u and v are linearly dependent if and only if they are parallel, u v, that is lie on the same line. Three non-zero vectors u, v and w, are linearly dependent if and only if they are coplanar, that is, lie in the same plane. Four non-zero vectors in R 3 are always linearly dependent. A collection B of three linearly independent vectors in R 3 forms a basis. If {e 1, e 2, e 3 } is a basis in R 3, then for every vector v there is a unique set of real numbers (v i ) = (v 1, v 2, v 3 ) such that a k e k v = 3 v i e i = v 1 e 1 + v 2 e 2 + v 3 e 3. i=1 The real numbers v i, i = 1, 2, 3, are called the components of the vector v with respect to the basis {e i }. It is customary to denote the components of vectors by superscripts, which should not be confused with powers of real numbers v 2 (v) 2 = vv,..., v n (v) n. 1.1.3 Standard Basis Cartesian coordinate system in R 3 is a one-to-one correspondence between the points in the space and ordered triples of real numbers. vecanal332.tex; November 21, 2017; 9:00; p. 6

1.1. LECTURE 1. VECTOR SPACES 5 The coordinates in R 3 are denoted by x, y, z or x 1, x 2, x 3 for short, or even x i, i = 1, 2, 3. The unit vectors i, j, k parallel to the x-axis, y-axis and z-axis pointing in the positive direction form a standard (canonical) Cartesian basis. To simplify notation we will also denote them by e 1, e 2, e 3. Every vector can be written as a linear combination A = A 1 i + A 2 j + A 3 k The real numbers A i, i = 1, 2, 3 are the Cartesian components of the vector A. The components are the orthogonal projections of the vector on the coordinate axes. The magnitute of the vector A is A = The vector addition in components Scalar multiplication in components A 2 1 + A2 2 + A2 3 A vector can be described by the magnitude and the direction The angles between the vector and the coordinate axes are the direction angles. They are determined by Pythagorean theorem. cos α i = A i, i = 1, 2, 3 A cos 2 α 1 + cos 2 α 2 + cos 2 α 3 = 1 There is a one-to-one correspondence between the components of the vector on the one side and its magnitude and the direction angles on the other side. vecanal332.tex; November 21, 2017; 9:00; p. 7

6 CHAPTER 1. VECTOR ALGEBRA Position vector R = x i + y j + z k Displacement vector. The vector represented by the line segment betweeen the points P 1 = (x 1, y 1, z 2 ) and P 2 = (x 2, y 2, z 2 ) (with tail at P 1 and the tip at P 2 ) is R 2 R 1 = (x 2 x 1 ) i + (y 2 y 1 ) j + (z 2 z 1 ) k 1.2 LECTURE 2. Scalar Product 1.2.1 Scalar Product First we prove that the cosine of the angle θ between any two nonzero vectors A and B can be computed In Cartesian coordinates by where Proof: We compute A B 2 = cos θ = A B A B, 0 θ π, A B = A 1 B 1 + A 2 B 2 + A 3 B 3 = 3 (A i B i ) 2 = i=1 3 A 2 i + i=1 3 A i B i i=1 3 B 2 j 2 j=1 3 A k B k On the other hand, from the triangle formed by the vectors A, B, A B, we have A B 2 = ( B sin θ) 2 + ( A B cos θ) 2 = A 2 + B 2 2 A B cos θ This implies that k=1 A B = 3 A k B k = A B cos θ k=1 This expression is called the scalar product (or the dot product) of the vectors A and B. vecanal332.tex; November 21, 2017; 9:00; p. 8

1.2. LECTURE 2. SCALAR PRODUCT 7 It is also called the inner product and denoted by ( A, B ). It is equal to the signed projection of one vector onto the other. The scalar square of a vector determines its norm A 2 = A A = A 2 Cauchy-Schwarz s Inequality. For any u, v E there holds The equality (u, v) u v. (u, v) = u v holds if and only if u and v are parallel. Maximum Principle. For a given vector A and a unit vector n the scalar product n A = A cos θ is maximal when n points in the same direction as A. Properties. For any vectors A, B, C and scalars a, b 1. A A 0, 2. A A = 0 if and only if A = 0, 3. A B = B A, 4. ( A + B ) C = A C + B C ), 5. (a A ) B = A (a B ) = a( A B ) For any u, v there holds u + v 2 = u 2 + 2(u, v) + v 2. or ( A B ) 2 = A 2 + B 2 2 A B vecanal332.tex; November 21, 2017; 9:00; p. 9

8 CHAPTER 1. VECTOR ALGEBRA Triangle Inequality. For any u, v E there holds u + v u + v. Parallelogram Law. For any two vectors u, v there holds u + v 2 + u v 2 = 2 ( u 2 + v 2) Therefore, the scalar product can be defined entirely in terms of the lengths of the vectors A B = 1 { A 2 + B 2 A B 2} 2 Two non-zero vectors u, v are orthogonal, denoted by u v, if (u, v) = 0. A basis {e 1, e 2, e 3 } is called orthonormal if each vector of the basis is a unit vector and any two distinct vectors are orthogonal to each other, that is, (e i, e j ) = δ i j, where For an orthonormal basis where δ i j = { 1, if i = j 0, if i j A = 3 A i e i, i=1 A i = e i A.. 1.2.2 Lines and Planes A collection of vectors forms a vector subspace if any linear combination of the vectors from this set is a vector from this set, or, alternatively, if any vector in this set can be written as a linear combination of vectors from this set. vecanal332.tex; November 21, 2017; 9:00; p. 10

1.2. LECTURE 2. SCALAR PRODUCT 9 Let A be a collection of vectors. The span of A, denoted by span A, is the set of all finite linear combinations of vectors from A of the form v = a 1 e 1 + + a k e k. We say that the subset span A is spanned by A. It is easy to see that the span of any subset of a vector space is a vector space. The span of a single vector u is the set of vectors of the form v = tu It defines a line passing through the origin in the direction of u. The span of a collection of two vectors u and v is the set of vectors of the form v = tu + sv. If the vectors u and v are nonparallel, then this defines a plane passing through the origin that is parallel to the vectors u and v. If the vectors u and v are parallel then they span a line parallel to one of the vectors. Equation of a Line. Let P 0 = (x 0, y 0, z 0 ) be a fixed point and v be a given non-zero vector. Let P = (x, y, z) be a point on a line passing through the point P 0 and parallel to the vector v = a i + b j + c k. Then for some scalar t. R R 0 = t v Therefore, the parametric equation of the line is R = R 0 + t v where t is a parameter that ranges in < t < (you may think of it as time). vecanal332.tex; November 21, 2017; 9:00; p. 11

10 CHAPTER 1. VECTOR ALGEBRA Since there is one parameter the line is a one-dimensional vector space spanned by v In components, the equation of the line is x = x 0 + at y = y 0 + bt z = z 0 + ct These gives 3 equations for 4 variables (x, y, z, t) leaving one parameter arbitrary. Alternatively, if we eliminate t (if a 0, b 0, c 0) we get the nonparametric equation of the line x x 0 a = y y 0 b = z z 0 c This gives two equations for 3 variables (x, y, z) leaving one arbitrary. Example. Find the equation of a line. Remarks. If a = 0 then the line is parallel to the yz-plane. The nonparametric equations of the line are x = x 0, y y 0 b = z z 0 c Example. Find the point of intersection of two lines R = 3 i + 2 j + (2 i + j + k )t or and x 3 2 = y 2 = z R = i 2 k + ( j + k )t x = 1, y = z + 2, Answer: P = (1, 1, 1). vecanal332.tex; November 21, 2017; 9:00; p. 12

1.2. LECTURE 2. SCALAR PRODUCT 11 Equation of a Plane. A plane can be described by specifying a fixed point P 0 = (x 0, y 0, z 0 ) on the plane and two nonzero nonparallel vectors A and B parallel to the plane. Let P = (x, y, z) be a point on the plane. Then the displacement vector R R 0 is parallel to the plane. Therefore, R = R 0 + t A + s B, where t, s are scalar parameters. This is the parametric equation of the plane. Since there are two arbitrary real parameters, a plane is a two-dimensional vector space spanned by the vectors A and B. To get the non-parametric equation of the plane we need to specify a vector N = a i + b j + c k, orthogonal to the plane. Then it is orthogonal to the displacement vector, therefore, N ( R R 0 ) = 0. or or where a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 ax + by + cz = d d = ax 0 + by 0 + cz 0. This gives one equation for 3 variables (x, y, z) leaving two variables arbitrary. Example. Let L be a line x x 1 a = y y 1 b = z z 1 c passing through a point P 1 and parallel to the vector N = a i + b j + c k. The equation of the plane passing through the point P 0 orthogonal to the line L is a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 vecanal332.tex; November 21, 2017; 9:00; p. 13

12 CHAPTER 1. VECTOR ALGEBRA 1.3 LECTURE 3. Vector Product 1.3.1 Vector Product Let { e 1, e 2 } be an orthonormal basis in the plane. We say that another basis { f 1, f 2 } is equivalent to the former basis (or has the same orientation) if it can be obtained from the first basis by a rotation and has the opposite orientation if it can be obtained from the first basis by a rotation and one reflection. We declare that the standard basis has positive (or right handed) orientation. Then there are just two types of bases, the ones with positive orientation (that have the same orientation as the standard one) and the one with negative (or left handed) orientation (that have the opposite orientation with the standard one). The orientation of the space is done similarly. We declare that the standard basis { i, j, k } has positive (or right-handed) orientation. Then all bases { e 1, e 2, e 3 } obtained by pure rotations from the standard one have positive orientation and those that need one reflection have negative orientation. Explain the right hand rule. The vector product (or cross product) of two non-zero vectors, A, B, is defined by A B = A B sin θ n, where n is the unit vector orthogonal to both vectors A and B and such that the triple { A, B, n } is right-handed. The magnitude of the cross product is equal to the area of the parallelogram formed by A and B. Properties. 1. A A = 0, 2. A B = B A, 3. ( A + B ) C = A C + B C, vecanal332.tex; November 21, 2017; 9:00; p. 14

1.3. LECTURE 3. VECTOR PRODUCT 13 4. (a A ) B = A (a B ) = a( A B ) 5. A B = 0 if and only if the vectors are zero or parallel, The vector product of orthogonal unit vectors, e 1, e 2, is a unit vector e 3 = e 1 e 2 such that the triple { e 1, e 2, e 3 } is a right-handed orthonormal triple. In particular, if { e 1, e 2, e 3 } is an right-handed orthonormal basis then e 1 e 2 = e 3, e 2 e 3 = e 1, e 3 e 1 = e 2. This can be written in the form 3 e i e j = ε ki j e k, i, j = 1, 2, 3, k=1 where ε i jk is the Levi-Civita symbol defined by ε 123 = ε 231 = ε 312 = 1 ε 213 = ε 321 = ε 132 = 1 and all other components are zero. This can be written in the form +1, if (i, j, k) is an even permutation of (1, 2, 3) ε i jk = 1, if (i, j, k) is an odd permutation of (1, 2, 3) 0, otherwise (1.3.1) The Levi-Civita symbol is completely anti-symmetric, that is, it changes sign under the permutation of any two indices, ε i jk = ε jik = ε k ji = ε ik j and does not change sign under the cyclic permutation of indices ε i jk = ε jki = ε ki j. vecanal332.tex; November 21, 2017; 9:00; p. 15

14 CHAPTER 1. VECTOR ALGEBRA Vector Product in Cartesian Coordinates. In an orthonormal basis 3 3 A B = A i e i B j e j That is, = i=1 3 i, j,k=1 ( A B ) k = j=1 ε ki j A i B j e k (1.3.2) 3 ε ki j A i B j This can be written in the form of a formal determinant i j k A B = ε i jk A j B k e i = A 1 A 2 A 3 B 1 B 2 B 3 Intersecting Planes. Consider two planes with the normals N 1 and N 2. If the vectors N 1 and N 2 are parallel then the plane are parallel and do not intersect. If the normals are not parallel, then the vector N 1 N 2 is non-zero and is parallel to the line L of the intersection of these planes. Then the parametric equation of the line L passing through the point P 0 is i, j R = R 0 + t N 1 N 2. Angular Velocity. Consider a rigid body rotating about a fixed axis with a constant angular speed ω. Then the velocity of the particle at the point R is v = ω R where ω is the angular velocity directed along the axis of rotation and with the magnitude ω = ω. The speed of the particle is v = ω R sin θ, where θ is the angle between R and the axis of rotation. vecanal332.tex; November 21, 2017; 9:00; p. 16

1.3. LECTURE 3. VECTOR PRODUCT 15 Orthogonal Decomposition. Let A and B be two nonzero vectors. Let n = A A be the unit vector in the direction of A. Then the vector B can be decomposed as the sum of two orthogonal vectors where B = B + B, B = n ( n B ) = n B cos θ is the signed projection of B onto A and is the orthogonal component. 1.3.2 Triple Product B = B B = ( n B ) n The triple product of three vectors A, B and C is defined by In tensor notation it is A ( B C ) = [ A, B, C ] = A ( B C ) 3 i, j,k=1 ε i jk A i B j C k = A 1 A 2 A 3 B 1 B 2 B 3 C 1 C 2 C 3 It is equal to the signed volume of a parallelepiped based on three vectors A i, B j, C k Properties. 1. [A, B, C] = [B, C, A] = [C, A, B] = [B, A, C] = [C, B, A] = [A, C, B], 2. [A, B, C] = 0 if and only if the vectors are coplanar, 3. [A, B, C] is linear in each argument, 4. for an orthonormal basis [ e 1, e 2, e 3 ] = 1 vecanal332.tex; November 21, 2017; 9:00; p. 17

16 CHAPTER 1. VECTOR ALGEBRA 1.4 LECTURE 4. Tensors and Vector Identities We will denote the Cartesian coordinates by x 1 = x, x 2 = y, x 3 = z and the unit vectors in the direction of positive axes (called the standard basis vectors) by e 1 = i, e 2 = j, e 3 = k This can be denoted simply by x i and e j, where i, j = 1, 2, 3. For the indices one usually uses the lowercase Latin letters i, j, k, l, m, n etc. (do not confuse with i, j, k). If you run out of letters, you can use any other letters. The convention is though that the indices are denoted by small (versus capital) Latin (versus Greek) letters, and take values 1, 2, 3. Greek indices are used in four-dimensional space-time in special relativity, where they take values 0, 1, 2, 3, with x 0 = t denoting time. The scalar products of the basis vectors are: e i e j = { 1, if i = j 0, if i j One says that they form an orthonormal system. This can be written in a compact form by defining so called Kronecker symbol δ i j { 1, if i = j δ i j = 0, if i j This can also be represented by the unit 3 3 matrix Then (δ i j ) = 1 0 0 0 1 0 0 0 1 e i e j = δ i j vecanal332.tex; November 21, 2017; 9:00; p. 18

1.4. LECTURE 4. TENSORS AND VECTOR IDENTITIES 17 Physical quantities, like mass, energy, volume, temperature, density etc., that can be described by one number are called scalars. This number does not depend on the coordinate system; it is an invariant. Vectors are physical quantities, like velocity, position, displacement, force, acceleration, electric field, magnetic field etc., that are described by three numbers. A tensor is a geometric object that requires for its full description more than just one number, as scalar, and even more than three numbers, as a vector. Examples of tensors include: stress tensor, strain tensor, inertia tensor, energymomentum tensor, tensor of the electromagnetic field, metric tensor, curvature tensor etc. These numbers are called the components of the tensor. The components of a tensor are labeled by indices, for example, δ i j, ε i jk, T i j, B i j, σ i j, R i i jk A tensor whose all components are zero is called a zero tensor. The tenswors with upper indices are called contravariant, and the ones with lower indices are called covariant. If a tensor has both types of indices then it is of mixed type. The total number of indices is called the rank of the tensor. A tensor that has p upper indices and q lower indices T i 1...i p j1... j q is called a tensor of type (p, q); it has the rank r = p + q. So, a scalar is a tensor of rank 0, a vector is a tensor of rank 1 etc. The actual numerical values of the components of a tensor do depend on the coordinate system. If one changes the coordinate system, for example, rotates it, then the components of a tensor will change. If one goes from the Cartesian coordinate system to a curvilinear coordinate system, for example, a system of spherical or cylindrical coordinates, then the components of a tensor will also change. It is this transformation law of the components of the tensor that makes a collection of numbers a tensor. We will not give the formal definition of a tensor, rather we give here a very short review of tensor analysis in Cartesian coordinates along with some very useful formulas and rules that enable one to deal with tensors. In any tensor equation an index can appear only once (single index) or twice (repeated index). vecanal332.tex; November 21, 2017; 9:00; p. 19

18 CHAPTER 1. VECTOR ALGEBRA A pair of repeated indices cannot appear more than once. Einstein Summation Convention. One always encounters the sums over the indices that appear twice in an equation. According to the standard convention, called Einstein summation convention, one has agreed to sum over repeated indices and omit the summation signs. For example, δ i j A i B j = 3 i=1 3 δ i j A i B j j=1 A i B i = δ i i = 3 ε i jk A j C k B i = i=1 3 A i B i i=1 3 δ i i i=1 3 3 ε i jk A j C k B i j=1 k=1 One can add tensors of the same type. The result is a tensor of the same type. One can multiply tensors by scalars. The result is a tensor of the same type. If one multiplies a tensor of rank r with a tensor of rank k, one gets a new tensor of rank r + k. More precisely, if one multiplies a tensor of type (p, q) with a tensor of type (r, s), then one gets a new tensor of type (p + r, q + s). For example, A i B j = C i j, T mn σ i j = R mn i j Note: one just multiplies the components of the tensors without any summation. Given a tensor of type (p, q) (that is of rank r = p + q) one may select a pair of indices, of which one should be an upper index and another an lower index, and replace them by two identical (repeated indices), summation over the latter being implied by the summation convention. This process is called contraction. As a result one gets a new tensor of type (p 1, q 1) of rank r 2 = p + q 2. For example, A i i, R i j ki, C ik k (1.4.3) vecanal332.tex; November 21, 2017; 9:00; p. 20

1.4. LECTURE 4. TENSORS AND VECTOR IDENTITIES 19 Clearly, δ i i = δ1 1 + δ2 2 + δ3 3 = 3 (1.4.4) A tensor of rank 2 is said to be symmetric if A i j = A ji (1.4.5) and anti-symmetric (or skew-symmetric) if A i j = A ji (1.4.6) Any tensor A i j of second rank can be decomposed A i j = A (i j) + A [i j] (1.4.7) into its symmetric A (i j) = 1 2 (A i j + A ji ) (1.4.8) and anti-symmetric parts A [i j] = 1 2 (A i j A ji ) (1.4.9) One can also symmetrize a tensor of rank 3 over three indices: B (i jk) = 1 6 (B i jk + B jki + B ki j + B ik j + B jik + B k ji ) (1.4.10) Correspondingly, the anti-symmetrization of a tensor of rank 3 is defined by B [i jk] = 1 6 (B i jk + B jki + B ki j B ik j B jik B k ji ) (1.4.11) What one does here is one sums over all possible permutations of indices and changes sign if the permutation is odd. Remark. The contraction of symmetric and an anti-symmetric tensors is equal to zero. Let a tensor A i j be symmetric and B i j be antisymmetric. Then A i j B i j = A ji B i j = A ji B ji = A i j B i j, (1.4.12) and, therefore, A i j B i j = 0. (1.4.13) vecanal332.tex; November 21, 2017; 9:00; p. 21

20 CHAPTER 1. VECTOR ALGEBRA The scalar products of the basis vectors e i define a symmetric second rank tensor called the metric tensor e i e j = g i j. The contravariant components of the metric tensor are defined by the inverse matrix g i j = (g i j ) 1 In Cartesian coordinates the components of the metric tensor are given by Kronecker delta symbol g i j = δ i j, g i j = δ i j. The metric tensor can be used to raise and lower indices of tensors. For example, if A i are contravariant components of a vector then its covariant components are A i = g i j A j Conversely, A i = g i j A j This operations, called raising and lowering indices can be applied to any tensor. The scalar product of two vectors is A B = g i j A i B j. Remark. In Cartesian coordinates the covariant and contravariant components are equal A i = A i. In Cartesian coordinates the position of the tensor indices (up or down) does not make any difference. Therefore, δ i j A i B j = A i B i = A B vecanal332.tex; November 21, 2017; 9:00; p. 22

1.4. LECTURE 4. TENSORS AND VECTOR IDENTITIES 21 Levi-Civita symbol ε i jk is defined by ε i jk = +1, if (i, j, k) is an even permutation of (1, 2, 3) 1, if (i, j, k) is an odd permutation of (1, 2, 3) 0, otherwise If one raises the indices then one sees that in Cartesian coordinates one obtains the same symbol, so that ε i jk = ε i jk (1.4.14) The Levi-Civita symbol defines a tensor of rank 3 (strictly speaking it is a pseudo-tensor density of weight 1), called a Levi-Civita tensor. It describes the signed volume of a parallelepiped based on three displacement vectors A i, B j, C k V = ε i jka i B j C k The Levi-Civita symbol defines a completely antisymmetric tensor. It changes sign under the permutation of any two indices. ε i jk = ε jik = ε k ji = ε ik j ε i jk = ε jki = ε ki j As a consequence its contraction vanishes ε i j j = 0, also, for any vector A i ε i jk A j A k = 0 Further, one can show that the product of two Levi-Civita symbols can be expressed in terms of the Kronecker symbols ε i jk ε mnl = 6δ m [i δn j δl k] = δ m i δn j δl k + δm j δn k δl i + δm k δn i δl j δ m i δn k δl j δm j δn i δl k δm k δn j δl i vecanal332.tex; November 21, 2017; 9:00; p. 23

22 CHAPTER 1. VECTOR ALGEBRA By contracting the indices k and l we get ε i jk ε mnk = 2δ m [i δn j] = δ m i δn j δm j δn i further, by contracting the indices j and n we obtain ε i jk ε m jk = 2δ m i and, finally, by contracting the indices i and m we have ε i jk ε i jk = 6 The vector product D = B C in tensor notation is given by The triple product is then D i = ( B C ) i = ε i jk B j C k [ A, B, C ] = A ( B C ) = ε i jk A i B j C k Note that the position of indices (up versus down) in Cartesian coordinates is not important. However, it is still more clear, when you see one index up and the same index down then you should immediately notice that this is a contraction and there is a summation over this index from 1 to 3. We repeat once again that the name of such repeated indices is not important, they are dummy indices; one can rename them to any other letter if needed (make sure that there are no other indices with that name in the given tensor equation!). 1.4.1 Algebraic Vector Identities Tensor notation is very useful in vector analysis, in particular when manipulating the multiple vector products and vector identities. By using the properties of Levi-Civita symbol and Kronecker symbol one can derive now all vector identities. For example, A ( B C ) = B ( A C ) C ( A B ) ( A B ) ( C D ) = [ D, A, B ] C [ C, A, B ] D ( A B ) ( C D ) = (A C)(B D) (A D)(B C) A ( B C ) + B ( C A ) + C ( A B ) = 0 vecanal332.tex; November 21, 2017; 9:00; p. 24

1.4. LECTURE 4. TENSORS AND VECTOR IDENTITIES 23 Proofs. [( A B ) ( C D )] i = ε i jk ( A B ) j ( C D ) k = ε i jk ε jmn A m B n ε kpq C p D q = (δ p i δq j δp j δq i )ε jmn A m B n C p D q = (δ p i εqmn δ q i εpmn )A m B n C p D q = ε qmn A m B n C i D q ε pmn A m B n C p D i = [ D, A, B ]C i [ C, A, B ]D i vecanal332.tex; November 21, 2017; 9:00; p. 25

24 CHAPTER 1. VECTOR ALGEBRA vecanal332.tex; November 21, 2017; 9:00; p. 26

Chapter 2 Vector Functions 2.1 LECTURE 5. Differentiation 2.1.1 Limits A vector-valued function of a single variable t is a rule that associates a vector A (t) to each real number t in some interval t [a, b]. Example. We say that a vector-valued function A (t) has the limit A 0 at t 0 and write lim A (t) = A 0, or A (t) t t 0 A 0 t t0 if the real-valued function f (t) = A (t) A 0 has limit 0 as t t 0, lim A (t) A 0 = 0, t t0 which means that for any ε > 0 there is δ > 0 such that if t t 0 < δ then A (t) A 0 < ε Remarks. This means that the length and the direction of the vector A (t) approaches the length and the direction of the vector A 0. This also means that the components of the vector A (t) approach the components of the vector A 0, lim A i (t) = A i 0. t t0 25

26 CHAPTER 2. VECTOR FUNCTIONS We say that a vector-valued function A (t) is continuous at t 0 if lim A (t) = A (t 0 ) t t0 This means that the components (and the length) of the vector A (t) are continuous at t 0, lim t t0 A i (t) = A i (t 0 ), 2.1.2 Derivatives lim A (t) = A (t 0 ). t t0 We say that a vector-valued function A (t) is differentiable at t 0 if there exists the limit A (t 0 ) = d A dt (t 1 0) = lim h 0 h { A (t 0 + h) A (t 0 )} This means that the components of the vector A (t) are differentiable at t 0 ; then d A dt (t 0) = da i dt (t 0) e i Properties. Proof. 1. d dt ( A + B ) = d dt A + d dt B 2. d dt ( f A ) = ( d dt f ) A + f d dt A 3. d dt ( A B ) = ( d dt A ) B + A d dt B 4. d dt ( A B ) = ( d dt A ) B + A d dt B Proposition. The derivative of the magnitude is equal to d dt A 2 = 2 A d dt A Corollary. The magnitude of a non-zero non-constant vector-valued function is constant if and only if it is orthogonal to its derivative. Examples. vecanal332.tex; November 21, 2017; 9:00; p. 27

2.2. LECTURE 6. CURVES, VELOCITIES AND TANGENTS 27 2.2 LECTURE 6. Curves, Velocities and Tangents 2.2.1 Tangent Vectors A curve is described by a vector-valued function R (t) = x i (t) e i, t [a, b] that is, x i = x i (t), a t b. Parametrization Orientation Circle. Circle of radius ρ centered at R 0 lying in the plane spanned by the orthonormal vectors e 1, e 2, Example. Circle in the plane R (t) = R 0 + ρ cos t e 1 + ρ sin t e 2. x + y + z = 1 passing through the points (1, 0, 0), (0, 1, 0), (0, 0, 1). It has the center R 0 = 1 3 ( i + j + k ) and the radius 6 ρ = 3 The orthonormal basis can be chosen to be e 1 = 1 2, e 2 = 1 6 ( i 2 j + k ) Helix. Let e i be a right-handed orthonormal basis. The helix of radius ρ and pitch 2π a with axis passing through R 0 and parallel to e 3 is described by R (t) = R 0 + ρ cos t e 1 + ρ sin t e 2 + at e 3. The helix is right-handed for a > 0 and left-handed for a < 0. vecanal332.tex; November 21, 2017; 9:00; p. 28

28 CHAPTER 2. VECTOR FUNCTIONS Non-parametrix equation of the helix is x = ρ cos(z/a), y = ρ sin(z/a). The velocity of a moving particle with position R (t) is given by the derivative v = d R dt Its speed is determined by the magnitude of the velocity v = v(t) The unit tangent vector is given by T = v v A curve is smooth if there is a parametrization R (t), t [a, b], satisfying: 1. the velocity is continuous, 2. there are no self-crossings, 3. the velocity is non-zero. Remark. It is allowed that the curve is a closed loop, that is, R (a) = R (b). A curve is regular if it consists of a finite number of smooth arcs joined togetrher without self-crossings. Examples. A curve is oriented if a direction is specified along it. The orientation is described by the direction of increasing parameter. Example. vecanal332.tex; November 21, 2017; 9:00; p. 29

2.2. LECTURE 6. CURVES, VELOCITIES AND TANGENTS 29 2.2.2 Arc Length The length of a smooth curve is given by or L = b a L = d R dt b a ds dt where ds = d R dt dt = d R = dx 2 + dy 2 + dz 2 is called the line element. The length of a regular curve is equal to the sum of the lengths of the smooth arcs. The length of the arc from t = a to t is t s(t) = d R (τ) dτ dτ Therefore, the speed is given by d R dt a = ds dt The arc length s(t) can be used as a parameter, called the natural parameter. In natural parametrization the speed is equal to 1 d R ds = 1, and the velocity is equal to the unit tangent Example. T = d R ds. vecanal332.tex; November 21, 2017; 9:00; p. 30

30 CHAPTER 2. VECTOR FUNCTIONS 2.3 LECTURE 7. Acceleration and Curvature 2.3.1 Curvature The acceleration of a moving particle with position R (t) is given by the second derivative a = d v = d2 R dt dt 2 The magnitude of the acceleration is a = a = d 2 R dt 2 The curvature of the curve is determined by the rate of change of the unit tangent κ = d T ds = 1 d T v dt The radius of curvature is defined by ρ = 1 κ. Examples. For the straight line the curvature is zero and the radius of curvature is infinite. For the circle the radius of curvature is equal to the radius of the circle. 2.3.2 Normals The principal normal to the curve is defined by N = 1 d T d T dt dt The derivative of the unit tangent d T = d T dt dt N = κ d R dt N = κv N vecanal332.tex; November 21, 2017; 9:00; p. 31

2.3. LECTURE 7. ACCELERATION AND CURVATURE 31 In natural parametrization The binormal is defined by d T ds = κ N B = T N The triple { T, N, B } forms a right-handed orthonormal basis. The torsion is defined by Frenet Formulas. τ = B d N ds = N d B ds d T ds d N ds d B ds = κ N = κ T + τ B = τ N Any two curves with the same curvature and torsion are congruent, that is, they can be obtained from each other by rotations and translations. 2.3.3 Examples Orthogonal decomposition of acceleration. So, a = d dt (v T ) = dv dt T + v d dt T = dv dt T + κv2 N (2.3.1) a = a t T + a n N, vecanal332.tex; November 21, 2017; 9:00; p. 32

32 CHAPTER 2. VECTOR FUNCTIONS where a t = dv dt, a n = κv 2 are the tangential and the normal acceleration. The magnitude of the acceleration is a 2 = a 2 = a 2 t + a 2 n. Proposition. The curvature is equal to Proof. We compute Example. Circle. κ = v a v 3 v a = v T (a t T + a n N ) = va n B = κv 3 B (2.3.2) R = R 0 + r cos(ωt) e 1 + r sin(ωt) e 2 Tangent T = sin(ωt) e 1 + cos(ωt) e 2 Principal normal N = cos(ωt) e 1 sin(ωt) e 2 Binormal B = e 3 Curvature κ = 1 r Radius of curvature Torsion ρ = r τ = 0 vecanal332.tex; November 21, 2017; 9:00; p. 33

2.3. LECTURE 7. ACCELERATION AND CURVATURE 33 Example. Helix. Velocity Speed Tangent Arc length Acceleration R = R 0 + r cos(t) e 1 + r sin(t) e 2 + at e 3 v = r sin(t) e 1 + r cos(t) e 2 + a e 3 T = r r 2 + a sin(t) e 2 1 + v = r 2 + a 2 r r 2 + a 2 cos(t) e 2 + s = vt a r 2 + a 2 e 3 a = r cos(t) e 1 r sin(t) e 2, a = r Derivative of the tangent Principal normal d T dt Decomposition of the acceleration = r v cos(t) e 1 r v sin(t) e 2 d T dt = r v N = cos(t) e 1 sin(t) e 2 a t = 0, a n = kv 2 = r Binormal Curvature B = a r 2 + a 2 sin(t) e 1 κ = a r 2 + a 2 cos(t) e 2 + r r 2 + a 2 r r 2 + a 2 e 3 vecanal332.tex; November 21, 2017; 9:00; p. 34

34 CHAPTER 2. VECTOR FUNCTIONS Radius of curvature Derivative of the principal normal ρ = r + a2 r Torsion d N dt = sin(t) e 1 cos(t) e 2 τ = a r 2 + a 2 vecanal332.tex; November 21, 2017; 9:00; p. 35

Chapter 3 Fields 3.1 LECTURE 8. Fields and Differential Operators 3.1.1 Scalar and Vector Fields We use the following notation x 1 = x, x 2 = y, x 3 = z and denote the partial derivatives by i =, i = 1, 2, 3. xi Then we have i x j = δ j i We also use the standard orthonormal basis e 1 = i, e 2 = j, e 3 = k and introduce a formal vector (called nabla (or del)) = e i i Let R = x i e i and R = R = x 2 + y 2 + z 2 35

36 CHAPTER 3. FIELDS A scalar field is a rule that assigns a number ϕ(x, y, z) = ϕ( R ) to each point P = (x, y, z) of a region in space. It is a scalar-valued function of three variables. A scalar field ϕ( R ) = f (R), that only depends on R is called spherically symmetric. Example. ϕ( R ) = 1 R A vector field is a rule that assigns a vector F (x, y, z) = F ( R ) = F i ( R ) e i to each point P = (x, y, z) of a region in space. It is a vector-valued function of three variables. Example. F ( R ) = R R 3.1.2 Gradient A differential operator L is a rule that assigns a (scalar or vector) field Lϕ to every given (scalar or vector) field ϕ. A differential operator is linear if for any two fields ϕ, psi and a constant c, L(ϕ + ψ) = Lϕ + Lψ L(cϕ) = clϕ Problem: How to define (first-order) partial differential operators on scalar and vector fields? The partial derivatives of scalar and vector fields define the following tensors i ϕ, i F j. What scalar and vector fields can one form from these tensors by using only the Kronecker and the Levi-Civita symbols, δ i j, ε i jk? vecanal332.tex; November 21, 2017; 9:00; p. 36

3.1. LECTURE 8. FIELDS AND DIFFERENTIAL OPERATORS 37 The gradient is a differential operator that assigns the vector field to a scalar field ϕ, or Gradient is a linear operator. Examples. grad ϕ = i ϕ e i = ϕ ( grad ϕ) i = i ϕ. grad R = R R The gradient of a spherically symmetric scalar field has the form In particular, and 3.1.3 Divergence grad f (R) = f R R R grad R n = nr n 2 R grad 1 R = R R 3 The divergence is a differential operator that assigns a scalar field to a vector field F. Divergence is a linear operator. Examples. We have Therefore, More generally, In particular, div F = i F i = F i x i = δ i i = 3. div R = 3 div ( f (R) R ) = R f R + 3 f. ( ) R div = 0 R 3 vecanal332.tex; November 21, 2017; 9:00; p. 37

38 CHAPTER 3. FIELDS 3.1.4 Laplacian The Laplacian is a differential operator that assigns a scalar field ϕ = div grad ϕ = i i ϕ = 2 ϕ to a scalar field ϕ. Laplacian is a linear operator. A scalar field ϕ such that ϕ = 0 is called harmonic. Examples. The Laplacian of a spherically symmetric scalar field is f (R) = 2 f R + 2 1 f 2 R R. Therefore, R n = n(n + 1)R n 2, that means, in particular, that the field 1/R is harmonic, 1 R = 0. 3.1.5 Curl The curl is a differential operator that assigns a vector field curl F = ε i jk j F k e i = F to a vector field F, or ( curl F ) i = ε i jk j F k This can also be written as a formal determinant e 1 e 2 e 3 curl F = 1 2 3 F 1 F 2 F 3 Curl is a linear operator. vecanal332.tex; November 21, 2017; 9:00; p. 38

3.2. LECTURE 9. DIFFERENTIAL VECTOR IDENTITIES 39 Examples. More generally, curl R = 0 curl ( f (R) R ) = 0. Proposition. For any scalar field ϕ and any vector field F curl grad ϕ = 0 div curl F = 0 Proof. Proposition. Fort any vector field F curl curl F = grad div F F Proof. 3.2 LECTURE 9. Differential Vector Identities Recall that curl grad ϕ = 0 div curl F = 0. (3.2.1) and curl curl F = grad div F F By using the tensor methods we can prove that grad (ϕψ) = ( grad ϕ)ψ + ϕ grad ψ, div (ϕ F ) = ( grad ϕ) F + ϕ div F curl (ϕ F ) = ( grad ϕ) F + ϕ curl F div ( F G ) = ( curl F ) G F curl G, In particular, div ( grad ϕ grad ψ) = 0 vecanal332.tex; November 21, 2017; 9:00; p. 39

40 CHAPTER 3. FIELDS Also, if B is constant then grad ( B R ) = B Let A be a vector field. We define a new operator A = A by A = A = A i i Then A ϕ = A ϕ = A grad ϕ We also define its action on vector fields by [ A F ] j = [( A ) F ] j = A i i F j Then ( A ) R = A More generally, one can show that curl ( F G ) = ( F ) G + ( G ) F ( div F ) G + F ( div G ) grad ( F G ) = ( F ) G + ( G ) F + F curl G + G curl F Example. Let A be a constant vector. Compute div ( A R ), curl ( A R ) div curl ( A grad 1 ) R ( A grad 1 ) R vecanal332.tex; November 21, 2017; 9:00; p. 40

3.3. LECTURE 10. GEOMETRIC APPLICATIONS 41 3.3 LECTURE 10. Geometric Applications 3.3.1 Directional Derivative Sets of points on which a scalar field remains constant ϕ(x, y, z) = C defines a family of equipotential (or isotimic) surfaces. Examples. Planes, spheres, cones, cylinders. Equipotential surfaces do not intersect. Let P 0 = (x 0, y 0, z 0 ) be a point and u = u i e i be a unit vector. Let R (s) = R 0 + s u be the line L passing through P 0 in the direction of u in the natural parametrization. The values of the scalar field along the line define the scalar function of a single variable f (s) = ϕ( R (s)). The derivative u ϕ( R 0 ) = d ds ϕ( R (s)) s=0 is called the directional derivative of ϕ at P 0 in the direction of u. Partial derivatives i = = x i ei are the directional derivatives along the unit vectors e i. More generally, let R (s) be a curve in the natural parametrization and be the unit tangent. u = d R ds vecanal332.tex; November 21, 2017; 9:00; p. 41

42 CHAPTER 3. FIELDS Then the directional derivative is Formally, Properties of the gradient. u ϕ( R ) = d ϕ ϕ( R (s)) = ui = u grad ϕ. ds x i u = dxi ds = d R i ds 1. The component of the gradient grad ϕ in a given direction gives the directional derivative in that direction. 2. Maximum principle. The largest possible value of the directional derivative is obtained in the direction of the gradient. 3. The gradient points in the direction of the maximum rate of increase of the scalar field. 4. The maximum value of the directional derivative is equal to the magnitude of the gradient (achieved in the direction of the gradient) and the minimum of the directional derivative is equal to the negative of the magnitude of the gradient (achieved in the opposite direction). Proposition. At every point R 0 where the gradient of a scalar field is not equal to zero, grad ϕ( R 0 ) 0, there is an equipotential surface ϕ( R ) = C passing through R 0 such that the gradient grad ϕ( R 0 ) is normal to this surface at this point. Proof. Let R (s) be a curve such that R (0) = R 0 and u = d R ds (0). Then the set of all curves such that u ϕ = 0, that is the set of all curves with the tangents orthogonal to the gradient grad ϕ forms a surface ϕ( R ) = C with the normal N = grad ϕ. Examples. 3.3.2 Flow Lines Let F be a vector field. A point R is a regular point of the vector field F if it is smooth at this point and F ( R ) 0; otherwise, it is called a singular point. vecanal332.tex; November 21, 2017; 9:00; p. 42

3.3. LECTURE 10. GEOMETRIC APPLICATIONS 43 A flow line (or a stream line, integral curve, characteristic curve) of a vector field F is a curve such that F is tangent to the curve at all regular points. The flow lines are only defined where the vector field is not equal to zero. No flow line passes though a singular point. The flow lines do not intersect. The equation of a flow line is where ψ is a scalar field, or d R ds = ψ F dx i ds = ψfi. If all components are non-zero then the equation can be written in the form dx = dy = dz. F 1 F 2 F 3 If F 3 = 0 in some region, then the flow line lies in the plane z = const. Example. F = R R 2 Flow lines are the lines through the origin. Example. F = y i x j Flow lines are the circles in the planes z = const. 3.3.3 Geometrical Interpretation of Divergence Let us fix a small cubic box with vertices at the points (0, 0, 0), (a, 0, 0), (a, a, 0), (0, a, 0), (0, 0, a), (a, 0, a), (a, a, a), (0, a, a), vecanal332.tex; November 21, 2017; 9:00; p. 43

44 CHAPTER 3. FIELDS The volume of the box and the surface area of the faces are V = a 3, S = a 2 The flux of a vector field v through the faces is defined by where n is the outside normal. We use the Taylor series v n S, v i (x, y, z) = v i (0) + x 1 v i (0) + y 2 v i (0) + z 3 v i (0) +... We compute the total net outflux v n ds = a 3 { 1 v 1 (0) + 2 v 2 (0) + 3 v 3 (0)} +... S Geometric interpretation of the divergence. The divergence is equal to the net outflux per unit volume div v = lim V 0 v n S V 3.3.4 Geometrical Interpretation of Curl The curl of a vector field describes its tendency to swirl. The true meaning of the curl will become more clear later after the discussion of the Stokes theorem. To illustrate this idea, we consider a small square C with the vertices (0, 0, 0), (a, 0, 0), (a, a, 0), (a, 0, 0), oriented couterclockwise. The total swirl of a vector field v around that square is a a v d R = v 1 (x, 0, 0)dx + v 2 (a, y, 0)dy C 0 a 0 v 1 (x, a, 0)dx 0 a 0 v 2 (0, y, 0)dy vecanal332.tex; November 21, 2017; 9:00; p. 44

3.3. LECTURE 10. GEOMETRIC APPLICATIONS 45 The unit normal to the square and its surface area are We use Taylor series near the origin n = k, S = a 2. v i (x, y, 0) = v i (0) + x 1 v i (0) + y 2 v i (0) +... Then the total swirl per unit area is 1 v d R = a 2 1 v 2 (0) 2 v 1 (0) = k curl v(0) C Geometric interpretation of curl. Thus the n-component of the curl v is equal to the total swirl per unit surface area of a small disk bounded by a circle C orthogonal to n -axis, n curl v = lim S 0 1 (total swirl) = lim S S 0 1 S 3.3.5 Geometrical Interpretation of Laplacian C v d R Let D be a small ball of a sufficiently small radius r centered at the origin. The boundary of the ball D is the sphere S of radius r centered at the origin. We introduce spherical coordinates x = R sin θ cos ϕ, y = R sin θ sin ϕ, z = R cos θ. Recall that the surface area element in spherical coordinates is ds = R 2 sin θdθdϕ and, therefore, the surface area of the sphere of radius R is S (R) = 4πR 2 The average of a scalar field f over the sphere S is defined by the integral f = 1 f ( R ) ds = 1 π 2π f ( R ) sin θdθdϕ 4πR 2 4π S 0 0 vecanal332.tex; November 21, 2017; 9:00; p. 45

46 CHAPTER 3. FIELDS The average of a spherically symmetric field f ( R ) = f (R) is f = f. Proposition. The average of polynomials x i = 0 x i x j = 1 3 δ i jr 2 Proof. We have under the transformation x i x i, x i = x i = 0 By the same reason, if i j, x i x j = 0. Further, by symmetry we have x 2 = y 2 = z 2 = 1 3 x 2 + y 2 + z 2 = 1 3 R2 The result follows. By using the Taylor series we compute f ( R ) = f (0) + x i i f (0) + 1 2 xi x j i j f (0) +... = f (0) + 1 2 x i x j i j f (0) +... = f (0) + 1 6 R2 f (0) +... Therefore, the Laplacian is determined by the average of the difference 6 f (0) = lim R 0 R f ( R ) f (0) 2 More generally, for any point we have 6 f ( R 0 ) = lim R 0 R f ( R ) f ( R 0) 2 vecanal332.tex; November 21, 2017; 9:00; p. 46

3.4. LECTURE 11. FLUID DYNAMICS 47 3.4 LECTURE 11. Fluid Dynamics 3.4.1 Flux Fluid is modelled by a collection of particles moving in space. The number of particles per unit volume defines the scalar field ν = ν(x, y, z) called the particle density. The number of particles in a small box of volume V = x y z is ν V Let m be the mass of a single particle. Then the mass density is so that the mass of a small box is µ = mν mν V Let q be the charge of a single particle. Then the charge density is ρ = qν so that the charge of a small box is qν V The velocity of particles defines the vector field v. The mass flow rate density is then F = mν v and the charge flow density (current) is j = qν v. vecanal332.tex; November 21, 2017; 9:00; p. 47

48 CHAPTER 3. FIELDS We fix a small planar patch of area S with unit normal n. The particles that cross this patch in time t will be the particles in the slanted cylinder with base S (with sides made of v t) with slant height v t. The volume of this cylinder is V = S ( n ( v t)) The number of such particles is ν V = ν v n S t Thus, the mass flow rate through S in time t is mν V = mν v n S t = F n S t The mass flow rate through S per unit time defines the flux of the vector field F through S F n S = F S = mν V t where S = n S. Physical interpretation of the divergence. The divergence is equal to the net outflux per unit volume 3.4.2 Swirl div F = lim V 0 F n S V The curl of a velocity field of a fluid describes its tendency to swirl. To illustrate this idea, we consider a small paddle wheel immersed in the fluid. vecanal332.tex; November 21, 2017; 9:00; p. 48