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Multivariable and Vector Calculus Multivariable and Vector Calculus Lecture Notes for MATH 0200 (Spring 2015) Frederick Tsz-Ho Fong Department of Mathematics Brown University Contents 1 Three-Dimensional Space ....................................5 1.1 Rectangular Coordinates in R3 5 1.2 Dot Product7 1.3 Cross Product9 1.4 Lines and Planes 11 1.5 Parametric Curves 13 2 Partial Differentiations ....................................... 19 2.1 Functions of Several Variables 19 2.2 Partial Derivatives 22 2.3 Chain Rule 26 2.4 Directional Derivatives 30 2.5 Tangent Planes 34 2.6 Local Extrema 36 2.7 Lagrange’s Multiplier 41 2.8 Optimizations 46 3 Multiple Integrations ........................................ 49 3.1 Double Integrals in Rectangular Coordinates 49 3.2 Fubini’s Theorem for General Regions 53 3.3 Double Integrals in Polar Coordinates 57 3.4 Triple Integrals in Rectangular Coordinates 62 3.5 Triple Integrals in Cylindrical Coordinates 67 3.6 Triple Integrals in Spherical Coordinates 70 4 Vector Calculus ............................................ 75 4.1 Vector Fields on R2 and R3 75 4.2 Line Integrals of Vector Fields 83 4.3 Conservative Vector Fields 88 4.4 Green’s Theorem 98 4.5 Parametric Surfaces 105 4.6 Stokes’ Theorem 120 4.7 Divergence Theorem 127 5 Topics in Physics and Engineering .......................... 133 5.1 Coulomb’s Law 133 5.2 Introduction to Maxwell’s Equations 137 5.3 Heat Diffusion 141 5.4 Dirac Delta Functions 144 1 — Three-Dimensional Space 1.1 Rectangular Coordinates in R3 Throughout the course, we will use an ordered triple (x, y, z) to represent a point in the three dimensional space. The real numbers x, y and z in an ordered triple (x, y, z) are respectively the x-, y- and z-coordinates which, by convention, are defined according to the following diagram: z c (a, b, c) b a y (a, b, 0) x Figure 1.1: Rectangular coordinates system in 3-space Notation We will use the notation R3 to denote the entire three dimensional space. Any point on the x-axis has the form (x, 0, 0), i.e. y = 0 and z = 0. Similarly, points on the y-axis are of the form (0, y, 0), and points on the z-axis are of the form (0, 0, z). The three coordinate axes meet at a point with coordinates (0, 0, 0) which is called the origin. A vector in R3 is an arrow which is based at one point and is pointing at another point. If a vector v is based at (x0, y0, z0) and points toward (x1, y1, z1), then the vector is written as: v = (x1 − x0)i + (y1 − y0)j + (z1 − z0)k. For example, the vector based at (3, 2, −1) pointing at (5, 2, 0) is expressed as (5 − 3)i + (2 − 2)j + (0 − (−1))k = 2i + k. 6 Three-Dimensional Space i Consequently, any two vectors that are pointing at the same direction and have the same length are considered to be equal, even though they may have different base points. For instance, a vector v based at (1, 2, 3) pointing at (4, 3, 2), i.e. v = (4 − 1)i + (3 − 2)j + (2 − 3)k = 3i + j − k, is considered to be equal to the vector w based at (0, 0, 0) pointing at (3, 1, −1). In other words, we can write w = v. An alternative notation for a vector is the angle bracket ha, b, ci (to save the hassle of writing down i, j and k: Notation ha, b, ci = ai + bj + ck. In this course, we make very little conceptual distinction between a point (x, y, z) and a vector based at (0, 0, 0) pointing at the point (x, y, z). However, speaking of notations, one should use hx, y, zi or xi + yj + zk to denote a vector and (x, y, z) to denote a point so as to avoid confusion. Vector additions can scalar multiplications are defined as follows: Definition 1.1 — Vector Additions and Scalar Multiplications. Let a = ha1, a2, a3i and b = 3 hb1, b2, b3i be two vectors in R , and c be a real scalar, then: a + b = ha1 + b1, a2 + b2, a3 + b3i (vector addition) ca = hca1, ca2, ca3i (scalar multiplication) The negative of a vector is defined as: −a = (−1)a. The difference between vectors is defined as a − b = a + (−b). Geometrically, these vector operations can be represented by the following diagrams: a a + b b b a a − b Figure 1.2: Geometric representations of various vector operations Property Vector additions and scalar multiplications have the following algebraic properties 1. commutative rule: a + b = b + a 2. associatative rule: (a + b) + c = a + (b + c) 3. distributive rules: (l + m)a = la + ma and l(a + b) = la + lb 1.2 Dot Product 7 1.2 Dot Product There are two types of products for vectors in R3, namely the dot product and the cross product. The former outputs a scalar whereas the latter outputs a vector. In this section, We first talk about the dot product. Definition 1.2 — Dot Product. Let a = ha1, a2, a3i and b = hb1, b2, b3i, then dot product between the vectors a and b are defined as: a · b = a1b1 + a2b2 + a3b3. It is important to note that the dot product of a vector a = ha1, a2, a3i by itself is given by: 2 2 2 a · a = a1 + a2 + a3 which is incidently the square of the length of the vector a (by the Pythagoreas’ Theorem in R3). Notation Let a = ha1, a2, a3i. We denote the length of a vector a by jaj, which is given by: q 2 2 2 jaj = a1 + a2 + a3. It is important to note that a · a = jaj2. The length of a vector, i.e. jaj, is also called the norm, or the magnitude, of the vector. Property It can easily be verified that the dot product satisfies the following algebraic properties: 1. a · b = b · a. 2. (a + b) · c = a · c + b · c. 3. (la) · b = l(a · b) 4. 0 · a = a · 0 = 0. The following theorem gives the geometric meaning of the dot product: 3 Theorem 1.1 Let a = ha1, a2, a3i and b = hb1, b2, b3i be two vectors in R , and q be the angle between these two vectors. Then we have: a · b = jaj jbj cos q. (1.1) Proof. The proof uses the Law of Cosines. Consider the triangle in the diagram below: b a − b q a The side opposite to the angle is represented by the vector a − b. Using the Law of Cosines: ja − bj2 = jaj2 + jbj2 − 2 jaj jbj cos q (a − b) · (a − b) = a · a + b · b − 2 jaj jbj cos q a · a − a · b − b · a + b · b = a · a + b · b − 2 jaj jbj cos q X X a ·a − 2a · b + b ·XXb = a ·a + b ·XXb − 2 jaj jbj cos q −2a · b = −2 jaj jbj cos q a · b = jaj jbj cos q. 8 Three-Dimensional Space One immediate consequence of Equation (1.1) is that it allows us to use the dot product to find the angle between two vectors. Precisely, the angle q between two vectors a and b is given by: a · b q = cos−1 . jaj jbj The most important case is that the two vectors a and b are perpendicular, also known p as orthogonal. The angle between the vectors is 2 and so we have the following important fact: Corollary 1.2 Two non-zero vectors a and b are orthogonal if and only if a · b = 0. This corollary is particularly useful to determine whether two vectors are perpendicular. Example 1.1 Show that any triangle which is inscribed in a circle and has one of its side coincides the diameter of the circle must be a right-angled triangle. Solution Let O be the center of the circle. Define vectors a and b as in the diagram below. b −a O a We would like to show the vectors in red and blue are orthogonal to each other. By basic vector additions and subtractions: Red vector = −a − b Blue vector = a − b. Their dot product equals to (−a − b) · (a − b) = −a · a + a ·b − b ·a + b · b = − jaj2 + jbj2 (recall v · v = jvj2) Since both a and b represent the radii of the circle, they have the same magnitude. Therefore jaj = jbj and we have (−a − b) · (a − b) = 0. This shows the red and blue vectors are orthogonal. 1.3 Cross Product 9 1.3 Cross Product The cross product is another important vector operations. In contrast to the dot product, the cross product outputs a vector instead of a scalar. A vector is characterized by its length and direction, we define the cross product by declaring these two attributes: Definition 1.3 — Cross Product. Given two vectors a = a1i + a2j + a3k and b = b1i + b2j + b3k in R3 with angle q between them, the cross product between a and b, denoted by a × b, is defined as the vector such that: • the length is given by ja × bj = jaj jbj sin q, i.e. the area of the parallelogram formed by vectors a and b; • the cross product a × b is orthogonal to both a and b; • the direction of a × b is determined by the right-hand grab rule illustrated by the Figure 1.3. Figure 1.3: right-hand grab rule From the right-hand grab rule, we can clearly see that a × b and b × a are vectors with the same length but in opposite direction, i.e.
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