MATH 251 – LECTURE 1 JENS FORSGÅRD http://www.math.tamu.edu/~jensf/ This week: 11.1–3 webAssign: 11.1–2, due 1/25 11:55 p.m. Next week: 11.4–7 webAssign: 11.3–6, opens 1/25 12 a.m. Help Sessions: yet to be posted Office Hours: BLOC 641C M 12:30–2:30 p.m. W 2–3 p.m. or by appointment. Rectangular coordinates Choose a point O to be the origin, and introduce the three coordinate axes. Rectangular coordinates Definition 1. The coordinate system is positively oriented if it fulfills the right hand rule. Points in 3-dimensional space We represent a point P ∈ R3 by its coordinates P = P (a, b, c). Points in 3-dimensional space Exercise 2. What surface in R3 is represented by the equation y = 2? Points in 3-dimensional space Exercise 3. What surface in R3 is represented by the equation x2 + y 2 = 1? Points in 3-dimensional space Exercise 4. What surface in R3 is represented by the equation y + 2x = 2? Distance in R3 Let P1(x1, y1, z1) and P2(x2, y2, z2) be two points in R3. The distance |P1P2| is given by the formula p |P1P2| = (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2. Distance in R3 The sphere with radius R and center C(a, b, c) consists of all points with distance R from C(a, b, c). Local maximum and minimum Exercise 5. Find the equation of a sphere that has center C(0, 1, 2) and contains the origin.