VECTOR CALCULUS Syafiq Johar [email protected] Contents 1 Introduction 2 2 Vector Spaces 2 n 2.1 Vectors in R ......................................3 2.2 Geometry of Vectors . .4 n 2.3 Lines and Planes in R .................................9 3 Quadric Surfaces 12 3.1 Curvilinear Coordinates . 15 4 Vector-Valued Functions 18 4.1 Derivatives of Vectors . 18 4.2 Length of Curves . 22 4.3 Parametrisation of Curves . 23 4.4 Integration of Vectors . 28 5 Multivariable Functions 29 5.1 Level Sets . 30 5.2 Partial Derivatives . 31 5.3 Gradient . 34 5.4 Critical and Extrema Points . 41 5.5 Lagrange Multipliers . 44 5.6 Integration over Multiple Variables . 45 6 Vector Fields 52 6.1 Divergence and Curl . 54 7 Line and Surface Integrals 56 7.1 Line Integral . 56 7.2 Green's Theorem . 62 7.3 Surface Integral . 65 7.4 Stokes' Theorem . 72 1 1 Introduction In previous calculus course, you have seen calculus of single variable. This variable, usually denoted x, is fed through a function f to give a real value f(x), which then can be represented 2 as a graph (x; f(x)) in R . From this graph, as long as f is sufficiently nice, we can deduce many properties and compute extra information using differentiation and integration. Differentiation df here measures the rate of change of the function as we vary x, that is dx is the infinitesimal rate of change of the quantity f. In this course, we are going to extend the notion of one-dimensional calculus to higher dimensions. We are going to work over vector spaces over the real numbers R. In higher dimensional spaces, we have a distinction between two objects which are called scalars and vectors. Scalars are quantities that only have magnitudes or size, which can be described by a real number. Vectors on the other hand are quantities that have a magnitude and direction. Vectors can be described with an array of numbers, as we shall see later. This array of numbers is reminiscent of the array (x; f(x)) we have seen above, but of course, in higher dimensions, we would have even move numbers. In higher dimensions, it can be difficult to have an explicit graphical representation of the vectors, so we will restrict our attention to 2 and 3 dimensions most of the time. Even so, there is a rich amount of mathematics available in these low dimensions. There are also various applications of vector calculus in physics, engineering, economics, geophysical sciences, meteorology, astronomy, and optimisation. It is also used as a tool and generalised in further studies of pure mathematics, such as topology and differential geometry. 2 Vector Spaces In order to study objects in higher dimensions, we first define vector spaces. Abstractly, a real vector space is a collection of points (or can be seen as arrows) with two operations: addition and scalar multiplication. More concretely: Definition 2.1 (Vector spaces). A vector space V over the field F is a non-empty set V together with the addition map V × V ! V such that (u; v) 7! u + v and a scalar multiplication map F × V ! V such that (λ, v) 7! λv satisfying the vector space axioms: 1. addition is commutative: u + v = v + u, 2. addition is associative: (u + v) + w = u + (v + w), 3. existence of additive identity: there exists an element 0 such that 0 + v = v + 0 = v, 4. existence of additive inverse: for every v 2 V , there exists a u 2 V such that v + u = u + v = 0, 5. distributivity of scalar multiplication over addition: λ(u + v) = λu + λv, 2 6. distributivity of scalar multiplication over field addition: (λ + µ)u = λu + µu, 7. compatibility of scalar multiplication with field multiplication: (λµ)u = λ(µu), 8. existence of scalar multiplication identity: there exists an element 1 2 F such that 1v = v. 2.1 Vectors in Rn However, for concreteness and practical applications, we are mostly interested in the vector n n space R over the field R. In this vector space, every element v 2 R can be written as the list (v1; v2; : : : ; vn) where vi 2 R for all i = 1; 2; : : : ; n. Sometimes, these numbers are arranged in a column. In matrix notation, this is written as the transpose (v1; v2; : : : ; vn)|. We can also write n the vectors in terms of the standard basis of R : n n Definition 2.2 (Standard basis of R ). The standard basis of R is given by the collection of vectors fe1; e2;:::; eng such that ei = (0;:::; 0; 1; 0;:::; 0) where the number 1 appears in the i-th position and 0 appears everywhere else. This basis is also called the Cartesian coordinate system. In this system, we can express a vector v = (v1; v2; : : : ; vn) as the sum v = v1e1 + v2e2 + 2 ::: + vnen. Note that this is simply a generalisation of the Cartesian plane R we have seen in high school where the standard basis is simply the unit vector in the x and y axes. Thus, 2 vectors in R are written as the pair (x; y) = xe1 + ye2. The difference in higher dimensions is just that we have more components to specify a point. e3 v3 v v1 v2 e1 e2 Figure 1: A vector v and its coordinates v = v1e1 + v2e2 + v3e3 = (v1; v2; v3). With these concrete expression of vectors, we can define the addition and scalar multiplica- tions explicitly. Suppose that u = (u1; u2; : : : ; un) and v = (v1; v2; : : : ; vn), then: u + v = (u1; u2; : : : ; un) + (v1; v2; : : : ; vn) = (u1 + v1; u2 + v2; : : : ; un + vn); λu = λ(u1; u2; : : : ; un) = (λu1; λu2; : : : ; λun): Of course, the zero vector 0 in this expression is given by (0; 0;:::; 0). With these defined, one can easily check that all the vector space axioms are satisfied. Remark 2.3. In engineering or physics, where one usually works in low dimensional vector 2 3 2 3 spaces like R or R , the standard bases of R and R are written as fi = (1; 0); j = (0; 1)g and fi = (1; 0; 0); j = (0; 1; 0); k = (0; 0; 1)g respectively. 3 2.2 Geometry of Vectors Geometrically, we can view vectors as the position of a point relative to the 0 vector (called the origin). This type of vector is called the position vector. The scalars vi are called coordinates (or components) of the vector (or point) v with respect to the standard basis. Two position vectors are called parallel if they are scalar multiples of each other. Another interpretation of vectors is that it represents a movement or direction from a specific point. For example, starting from a point w = (w1; w2; : : : ; wn) if we move in the direction v = (v1; v2; : : : ; vn), we would end up at the point (w1 + v1; w2 + v2; : : : ; wn + vn) = w + v. The vector v is called the translation vector. Therefore, if we start at a point u and wishes to end up at the point w, we have to move in the direction w − u. e3 e3 w + v u w v w e e e1 e2 1 2 w − u (a) Vector addition. (b) Vector subtraction. Figure 2: Vector addition and subtraction. Since vectors represent geometrical objects, we can deduce some geometrical properties such as lengths and angles. n Definition 2.4 (Length). The length or magnitude of a vector v = (v1; v2; : : : ; vn) 2 R is the non-negative real number jvj defined by: q 2 2 2 jvj = v1 + v2 + ::: + vn; which measures the distance of the point v from the origin 0. n From the above definition, the distance between any two point u and v in R is given by the magnitude of the vector v − u, that is: p 2 2 2 dist(u; v) = jv − uj = (v1 − u1) + (v2 − u2) + ::: + (vn − un) : Associated to any vector v is the unit vector v^ which is defined as the vector parallel to v with unit length. This definition is useful in denoting the direction a vector is pointing, without regards to its magnitude. More concretely, it is given by the following: n Definition 2.5 (Unit vector). Let v 2 R be a vector. Then, the unit vector in the direction of v is given by: 1 v^ = v: jvj Dividing a non-zero vector by its magnitude is an operation called normalising. 4 n Proposition 2.6 (Properties of magnitude). Suppose that u; v 2 R and λ 2 R. Then: 1. juj ≥ 0 with equality if and only if u = 0, 2. jλuj = jλjjuj, 3. triangle inequality: ju + vj ≤ juj + jvj with equality if and only if u = λv for some λ ≥ 0. 4. reverse triangle inequality: ju − vj ≥ jjuj − jvjj. Proof. The first two assertions are clear. To prove the third assertion, suppose that u = (u1; u2; : : : ; un) and v = (v1; v2; : : : ; vn). The inequality is trivial if v = 0. So assume that v 6= 0. Then for any real number x 2 R, we have: n n 2 X 2 2 2 2 X 0 ≤ ju + xvj = (ui + xvi) = juj + x jvj + 2x uivi; (1) i=1 i=1 which is a quadratic expression in x. Since the quadratic expression is non-negative, its dis- criminant must be non-positive: !2 n n X 2 2 X 2 uivi − 4juj jvj ≤ 0 ) uivi ≤ jujjvj: i=1 i=1 Hence, we compute: n n 2 2 2 X 2 2 X 2 2 2 ju+vj = juj +jvj +2 uivi ≤ juj +jvj +2 uivi ≤ juj +jvj +2jujjvj = (juj+jvj) ; (2) i=1 i=1 which, upon taking the square root on both sides, implies the desired inequality.