the standard (x, y, z) coordinate three-space. Nota- tions for vectors. Geometric interpretation as vec- tors as displacements as well as vectors as points. Vector addition, the zero vector 0, vector subtrac- Things you need to know about tion, scalar multiplication, their properties, and linear algebra their geometric interpretation. Math 131 Multivariate Calculus We start using these concepts right away. In D Joyce, Spring 2014 chapter 2, we’ll begin our study of vector-valued functions, and we’ll use the coordinates, vector no- The relation between linear algebra and mul- tation, and the rest of the topics reviewed in this tivariate calculus. We’ll spend the first few section. meetings reviewing linear algebra. We’ll look at just about everything in chapter 1. Section 1.2. Vectors and equations in R3. Stan- Linear algebra is the study of linear trans- dard basis vectors i, j, k in R3. Parametric equa- formations (also called linear functions) from n- tions for lines. Symmetric form equations for a line dimensional space to m-dimensional space, where in R3. Parametric equations for curves x : R → R2 m is usually equal to n, and that’s often 2 or 3. in the plane, where x(t) = (x(t), y(t)). A linear transformation f : Rn → Rm is one that We’ll use standard basis vectors throughout the preserves addition and scalar multiplication, that course, beginning with chapter 2. We’ll study tan- is, f(a + b) = f(a) + f(b), and f(ca) = cf(a). We’ll gent lines of curves and tangent planes of surfaces generally use bold face for vectors and for vector- in that chapter and throughout the course. valued functions. So, here, a and b are vectors, that is, points, in Rn, while c is a scalar, that is, a Section 1.3. The length kak of a vector a (also real number in R, and f is a vector-valued function. called its norm). The triangle inequality. In multivariate calculus, we’ll look at nonlinear functions f : Rn → Rm from n-dimensional space ka − bk ≤ kak + kbk to m-dimensional space. We’ll restrict ourselves, however, to those that can be differentiated, or at b least continuous, just about everywhere. ¡HH ¡ H In order to study these nonlinear functions, we’ll HH ¡ HH use the linear functions from linear algebra. Thus, ¡ Hj 1 a you’ll need to know quite a bit about linear algebra. ¡  ¡   ¡  Topics for review. Here’s a list of the topics ¡ 0 we’ll review. The text covers them all in detail, and we’ll look at nearly all of them, but concentrate on Dot products. Dot products a · b of vectors (some- the most important ones. times called inner products and denoted ha|ib). We’ll use most of these concepts as early as chap- Law of cosines and angles. Dot product of two vec- ter 2, the chapter on derivatives of multivariate tors in terms of the angle between them functions. A few we won’t use until chapter 3, but we’ll need all of them by the end of the course. a · b = kak kbk cos θ.

Section 1.1. Vectors in two and three dimen- Two vectors are orthogonal, a ⊥ b, if and only if sions. The standard (x, y) coordinate plane, and their dot product is 0. The projection projab of the

1 vector b onto the vector a is point and a line. Distance between parallel planes. Distance between skew lines. a · b Occasionally, we’ll use these geometric concepts, projab = 2 a. kak especially in examples. Unit vectors. Normalization of vectors. The concept of length of a vector is very impor- Section 1.6. Summary of geometry in n-space tant, especially in the form kb − ak which gives the Rn. The Cauchy inequality and the triangle in- distance between two points b and a. In chapter equality. The standard basis {e1, e2,..., en} for 2 we’ll define the limit of a function f : Rn → Rm Rn. m × n matrices as descriptions of linear func- using this concept. In particular, we’ll define tions f : Rn → Rm. Composition of these functions justifies matrix multiplication. Row and column lim f(x) = L vectors. Properties of matrix operations. Hyper- x→a planes. Determinants, minors, and cofactors. A as saying that we can make kf(x) − Lk arbitrarily square matrix is invertible when its determinant is small (less than ) by keeping kx − ak sufficiently nonzero. small (less than δ). In chapter 2 we’ll define the derivative of a func- Later in chapter 2 we’ll see how directional tion f : Rn → Rm as the m × n matrix of partial derivatives can be found in terms of dot products, derivatives. Later in that chapter, we’ll see that the and we’ll find tangent planes (and hyperplanes) in chain rule describes the derivative of a composition terms of dot products. We’ll frequently use length of two functions as the product of the two matrices and dot products throughout the course. describing the derivatives of the two functions. In one section of chapter 3 we’ll study arc length Also in chapter 2 we’ll see how determinants are and curvature, and in that section we’ll see how unit used in the inverse and implicit function theorems. vectors and normalization can be used to simplify We’ll use matrices, vectors, and matrix products their analysis. beginning in chapter 3 when we study the Hessian of a function. Determinants are also used in chapter Section 1.4. Cross products. Cross products a× 3 when we look at curl. b of pairs of vectors in R3. Areas of parallelograms and triangles in terms of cross products. Matrices Section 1.7. Coordinate systems. Rectangular and determinants as related to cross products. The coordinates. Polar coordinates for planes. Conver- triple scalar product, volume of a parallelepiped. sion between rectangular and polar coordinates for Rotation and angular velocity. the plane. Cylindrical and spherical coordinates for Cross products and dot products together allow space. Conversion among the various coordinate us to study geometry in three dimensions. In chap- systems for space. ter 3 we’ll look at Kepler’s laws of planetary motion We’ll use these other coordinate systems in many using derivatives of cross products and dot products of the examples we examine throughout the course. of functions. Later in the chapter, we’ll study the gradient, divergence, and curl of scalar and vector Math 131 Home Page at fields. These concepts depend on cross products http://math.clarku.edu/~djoyce/ma131/ and dot products.

Section 1.5. Equations and distance in R3. Equations of planes in three space including para- metric equations of planes. Distance between a

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