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Vector Calculus Math  Course Notes

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Vector Calculus Math  Course Notes Vector Calculus Math 213 Course Notes Cary Ross Humber November 28, 2016 Humber ma 213 course notes ii Preface iii Humber ma 213 course notes iv Contents Preface iii 1 Linear Algebra Primer 1 1 Vectors....................................... 1 §1.1.1 Vector Operations............................. 2 §1.1.2 Some Geometric Concepts........................ 3 §1.1.3 Linear Independence, Bases, other definitions............. 9 §1.1.4 Projection................................. 12 2 A little about matrices.............................. 13 §1.2.1 Determinant Formulas.......................... 15 §1.2.2 Determinant Geometry.......................... 17 §1.2.3 Cross product, triple product...................... 20 §1.2.4 Lines and Planes............................. 22 2 Multivariable functions 27 §2.0.1 Level Sets................................. 29 §2.0.2 Sections................................... 29 1 Derivatives..................................... 33 §2.1.1 What’s wrong with partial derivatives?................. 40 §2.1.2 Directional derivatives.......................... 43 v Humber ma 213 course notes 2 Tangent vectors and planes............................ 44 §2.2.1 Surfaces in R3 ............................... 44 3 Coordinates.................................... 46 4 Parametric Curves................................. 49 5 Parametric surfaces................................ 50 6 Practice problems................................. 53 3 Exterior Forms 57 1 Constant Forms.................................. 58 §3.1.1 1-Forms.................................. 58 2 2-Forms....................................... 61 §3.2.1 Wedge (Exterior) product........................ 61 3 k-forms....................................... 63 §3.3.1 Wedge product again........................... 63 4 Vector fields.................................... 66 5 Differential forms................................. 70 §3.5.1 Exterior derivative............................ 71 §3.5.2 Pullbacks/Change of coordinates.................... 76 6 Practice problems................................. 79 4 Integration and the fundamental correspondence 85 1 The correspondence between vector fields and differential forms...... 85 2 Flux integrals................................... 87 §4.2.1 Line integrals and work......................... 96 §4.2.2 Orientations................................ 102 §4.2.3 Integration of 3-forms.......................... 109 3 Practice problems................................. 112 vi Humber ma 213 course notes 5 Stokes’ Theorem 115 1 Surfaces with boundary.............................. 115 2 The generalized Stokes’ Theorem........................ 116 §5.2.1 Stokes’ Theorem for 1-surfaces..................... 116 §5.2.2 Stokes’ Theorem for 2-surfaces..................... 123 §5.2.3 Stokes’ Theorem for 3-surfaces..................... 126 A Coordinate representations 131 B Some applications of differential forms and vector calculus 135 1 Extreme values................................... 135 §2.1.1 Constrained Extrema........................... 138 2 Maxwell’s equations................................ 141 vii Humber ma 213 course notes viii Chapter 1 Linear Algebra Primer §1 Vectors The majority of our calculus will take place in 2-dimensional and 3-dimensional space. Occasionally, we may work in higher dimensions. For our purposes, a vector is like a point in space, along with a direction. Other information, such as magnitude or length of a vector, can be determined from this point and direction. We visualize a vector as an arrow emanating from the origin, which we often denote as O, and ending at this point. The space (so called vector space) R2 = (x ;x ) x ;x R f 1 2 j 1 2 2 g consists of pairs of real numbers. Such a pair, which we often denote by a single letter (bold, hatted, arrow on top), is a vector in R2. The convention taken for these notes is to denote vectors by bold letters. It is typical to express a vector x in column form x x = 1 x2 ! on a chalkboard/whiteboard, or whenever space is not a concern. Whenever space is at a premium, it is just as typical to denote the same vector x in row form x = (x1;x2). The space R3 consists of 3-tuples of real numbers, or real 3-component vectors. Just as with R2, we can express R3 as the set R3 = (x ;x ;x ) x ;x ;x R : f 1 2 3 j 1 2 3 2 g Each vector x R3 consists of three components, each of which is a real number. 2 1 Humber ma 213 course notes In general, the space Rn consists of n-tuples of real numbers, or real n-component vectors, Rn = (x ;:::;x ) x R;j = 1;:::;n : f 1 n j j 2 g The higher the dimension, the more space is preserved by using row form x = (x1;:::;xn). §1.1.1 Vector Operations There are two basic vector operations, that of vector addition and scalar multiplication. Both operations are defined component-wise. Given two vectors a;b Rn with component 2 forms a = (a1;a2;:::;an) and b = (b1;b2;:::;bn), the vector sum a + b is the vector obtained by adding the components of a to those of b, a + b = (a1 + b1;a2 + b2;:::;an + bn): Similarly, if α R is a scalar, the scalar multiple αa is obtained by multiplying each 2 component of a by α, αa = (αa1;αa2;:::;αan): In what follows, whether we are discussing R2;R3 or Rn, in general, we denote the zero vector by 0, which is simply the vector with 0 in every component. With respect to vector addition and scalar multiplication, the following conditions are satisfied for all α;β R 2 and a;b;c Rn 2 V1) a + b = b + a V2) (a + b) + c = a + (b + c) V3) a + 0 = a V4) a + ( a) = 0 − V5) 1a = a V6) α(βa) = (αβ)a V7) (α + β)a = αa + βa V8) α(a + b) = αa + αb: Example 1.1 Let a = (2;1) and b = ( 3;4). Then, − a + b = (2 3;1 + 4) = ( 1;5): − − The vector a + b is the diagonal of the parallelogram with sides a and b as depicted in Figure 1.1. ȴ 2 Humber ma 213 course notes y a + b b a x Figure 1.1: The vector a + b is the diagonal of the parallelogram with sides a;b. §1.1.2 Some Geometric Concepts Rn Given two vectors x = (x1;:::;xn) and y = (y1;:::;yn) in , we define their inner product by n x;y = x y : h i j j Xj=1 The term inner product is synonymous with scalar product. If we input two vectors, the output is a scalar (real number). This particular inner product is often called the dot product in vector calculus texts. So, the same formula may be denoted k x y = x y : · j j Xj=1 Soon, we will see what the inner product tells us about the geometric relationship between two (or more) vectors. Another important scalar quantity is the length or magnitude of a vector. This is a scalar associated with a single vector, whereas the inner product is a scalar associated with two vectors. However, these quantities are related. The norm (in particular, Euclidean norm) 3 Humber ma 213 course notes of the vector x Rn is 1 2 n =2 1=2 2 x = x;x = xj : k k h i 0 1 BXj=1 C B C 2 B C In other words, the quantity x is the inner product@B AC of x with itself. Geometrically, the norm x represents the lengthk k of x. k k The Cauchy-Schwarz inequality gives another relationship between the norm and inner product, namely a;b a b (1.1) jh ij ≤ k kk k for any a;b Rn. Though simple, the Cauchy-Schwarz inequality is very powerful. Another powerful2 inequality is the triangle inequality a b a b a + b : (1.2) k k − k k ≤ k ± k ≤ k k k k Theorem 1.2 (Properties of the inner product) . Let α;β be real numbers and let a;b;c Rn. 2 1. αa;b = α a;b h i h i 2. a;βb = β a;b h i h i 3. a + b;c = a;c + b;c h i h i h i 4. a;b + c = a;b + a;c h i h i h i These properties highlight why it is often preferrable to work with inner products, rather than norms. With inner products, scalars factor out of both arguments. In contrast, the analogous property for norms is αa = α a ; k k j jk k only the absolute value factors out, in general. Since inner products have nicer properties, whenever it makes sense to do so, we will often square norms so that they become inner products. Definition 1.3. A vector a Rn is called a unit vector if a = 1. From any vector b Rn 2 k k 2 we can obtain a unit vector by normalizing it. The vector u = b= b has norm 1. k k 2 Let a = (a1;a2),b = (b1;b2) be two vectors in R . We want to determine an expression for the angle, ', between the vectors a and b. Let 'a;'b be the angles between the positive x-axis (e1-axis) and a;b, respectively. To each vector there corresonds a right triangle, 4 Humber ma 213 course notes y a b a2 b2 ϕ ϕa ϕb x a1 b1 Figure 1.2: The angle ' between a and b whose side lengths correspond to the components of the vector and hypotenuse is the norm, as depicted in Figure 1.2. This gives the following trigonometric relations a b sin' = 2 ; sin' = 2 a a b b k k k k a b cos' = 1 ; cos' = 1 a a b b k k k k a2 b2 tan'a = ; tan'b = : a1 b2 If ' is the angle between a and b, then ' = ' ' . Thus, a − b cos' = cos(' cos' ) = cos' cos' + sin' sin' a − b a b a b a b a b = 1 1 + 2 2 a b a b k k k k k k k k a b + a b = 1 1 2 2 ; a b k kk k where the numerator in the last expression is a;b . Note that this analysis holds in h i higher dimensions, as well.
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