Below are listed the approximate topics covered in class and recommended homework for Math 21a. The dates listed are for the Mon-Wed-Fri sections. Your section may differ slightly in topics, problems, and dates. The Tues-Thurs sections will cover the same topics each week, but the topics and the recommended problems will be divided differently. Weekly mandatory problem sets will generally be posted on Tuesday or Wednesday and will be due one week later.

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Feb 5: (Sections 1.1 and 1.3) Introduction to R² and R³. Points vs. vectors and perspectives on vectors as both n-tuples and as (equivalence classes of) directed line segments. Adding and scaling vectors, difference vector, {i, j, k} notation. Length (norm) of a vector, unit vectors, distance between points. Simple examples of curves in R² and surfaces in R³ defined by algebraic equations: lines, circles, spheres, cylinders.
Homework: 1.1/4,6,7,9,17,20,25,26 and 1.3/1,3,6,7.

Feb 7: (1.2, 1.4, and 1.5) Vector-valued functions - parametrized curves in R² and R³. Equation of a line and parametrization of a line segment between two points. Parametrization of circles, spirals, helices. Differentiation of vector valued functions - velocity and acceleration vectors. Speed and the calculation of arclength. Reversing direction and other parametrizations of a given curve. Examples with gravity and Newton’s law.
Homework: 1.2/1,3,4,7,8,15 and 1.4/3,4 and 1.5/15,17,21.

Problem Set #1 - dues Thurs, Feb 15 (TTh sections) or Fri, Feb 16 (MWF sections)
Solutions to Problem Set #1

Feb 9: (1.6 and 1.7) Dot product in R², R³, and Rⁿ. Scalar and vector projections. Equation of a plane in R³ through a given point and with a given normal vector. Law of Cosines and the relationship between the dot product, lengths, and angles. Perpendicularity. Parametrization of a plane using two parameters. Distance from a point to a line or a plane. Basic differentiation formulas involving the dot product.
Homework: 1.6/4,7,10,11,13,19,21,31,33 and 1.7/7,15,17.

Feb 12: (Sections 1.7, 1.8) The cross product in R³. Geometry and algebra of lines and planes and their intersection. Geometric definition of the cross product and the connection with the algebraic definition. Algebraic properties of the cross product. Using the cross product to find the normal to a plane, area of a parallelogram. The triple scalar product as the volume of a parallelepiped.   Homework: 1.7/5,6,7,11,12,17,18,24,32 and 1.8/2,9,10,15,16,18,21,26.

Feb 14: (Section 1.5) Finish discussion of the cross product and its applications. Further topics in parametrized curves, such as the unit tangent vector T and unit normal vector N for a parametrized curve, tangential and normal components of acceleration. Examples of equations of motion. [Note: Some sections may do these topics on different days or next week.]  Homework: 1.5/5,7,19,20,22.

Feb 16: (Section 2.1) Introduction to functions of several variables with emphasis on functions of two and three variables. Graph of a function of two variables. Level curves (contours) of a function of two variables. Level surfaces of a function of three variables. The distinction between the two ways of understanding a function of two variables, i.e. in terms of contours in the plane vs. graphs in space.   Homework: 2.1/2a-c,5,6,7,12,13,17,19.

Feb 20-23: (Sections 5.1 and 5.2 and previous material) Further examples of graphs, contours, and level surfaces. Examples of vector fields in R² and R³ from physics and (systems of) differential equations (as seen in Math 1b). Unit tangent vector T for a parametrized curve and the tangential component (and normal component) of a vector field along a parametrized curve. Integration of functions along parametrized curves. Work done by a variable force (vector field) along a parametrized curve. Other examples of integration along curves such as: arclength, mass of a wire, center of mass of a wire, "vorticity" of a fluid, flow of a vector field across a curve in R². Algebraic properties of line integrals. [Note: Line integrals will return after we discuss the concept of the gradient of a function and the idea of a potential function for a gradient (or conservative) vector field).] Homework: 5.1/1,3abc and 5.2/1,3,7ab.

Problem Set #2 - this was due Tues, Feb 27 (TTh sections) or Wed, Feb 28 (MWF sections)
Solutions to Problem Set #2

Feb 26: (Sections 2.2 and 2.3) Quick review of the definition of the derivative of a function of one variable. Partial derivatives of functions of several variables - definition, notations, and calculation. Linear approximation and tangent planes to graphs. Marginal rates in economics.
Homework: 2.2/1,5,7,11,17,19 and 2.3/4,11,15,19.

Feb 28: (Sections 2.3 and 2.4) Increments, differentials, and linear approximation. Rate of change of a function along a parametrized curve and the basic chain rule. Directional derivatives and the gradient vector. Geometry of the gradient vector field and its relation to level sets. Simple calculation of normal vectors. Note: A good reference for these topics may be found in Chapter 14 of the Stewart text from Math 1b.
Homework: 2.3/21 and 2.4/1,5,7,9,11,13. Try the last two problems of Problem Set #2 using directional derivatives and/or the Basic Chain Rule.

Problem Set #3 and Exam Study Guide - due no later than Wed, Mar 7 at 1:00pm (solutions will be posted then).
Solutions to Problem Set #3

Question Center: In addition to class, problem sessions, and office hours, the Mathematics Department operates the Math Question Center in Loker on Sunday, Monday, Tuesday, Wednesday, and Thursday evenings from 8pm to 10pm. The Question Center will be staffed by Course Assistants from Math 1a, 1b, 21a, and 21b and by graduate students and others. You are encouraged to use this resource as you do your homework and when questions arise. It is intended to supplement the office hours held by your Section Leader.