Math 242 C
Lecture Summaries and HW Assignements

Lecture 27, 12/4/06: Theoretical complements: Limit definitions of derivatives

I will use this lecture to return to some of the more theoretical concepts and definitions that have come up in earlier chapters, but which I had skipped over initially. I will focus on the definitions via limits of the various derivative-like concepts that have come up throughout this course.

• Review: Definition of ordinary derivative of a single variable function (3.1, (2), p. 102)
• Limit definition of derivative of a vector-valued function (11.5, formula (5), p. 807)
• Limit definition of partial derivatives of a function of several variables (12.4, formulas (3), (4), p. 870)
• Limit definition of directional derivative of a function of several variables (12.8, (3), p. 910)

What you need to know for the Final: All I expect you to know from this material are the definitions listed above. You won't be asked to give proofs or perform any calculations involving these definitions.

Lecture 26, 11/29/06: Section 13.8: Surface Area

• Formula for area of a surface in rectangular coordinates, z = f(x,y) (formula (9), p. 1000)

Examples: 2, 3 (using (9))

Homework: 2, 3, 9, 11

Skip: In addition to formula (9), p. 1000, the text contains several other formulas for the surface area, but you need not memorize those. In almost all situations arising in practice one can get by with (9).

Lecture 25, 11/27/06: Section 13.9: Change of variables in multiple integrals

• Jacobian of a transformation
• Change of variables formula for double and triple integrals ((7) and (11))

Examples: 1, 2, 5

Homework: 1, 3, 14

Note: Some of the examples and problems in this section (e.g., Ex. 2 and 4) involve transformations that are tricky (especially, when it comes to converting regions of integrations) and which appear contrived and pulled out of the air. You only need to be able to handle the simpler cases such as those in problem 14, where the transformation to be used is easy to spot or where it is given in the problem.

Lectures 23/24: 11/13/06 and 11/15/06: Sections 13.6/13.7

Section 13.6: Triple integrals

• Triple integrals in rectangular coordinates
• Volume by triple integrals
• Mass, mass density, and center of mass, of 3 dimensional solids
• Moments of inertia (3D case)

Examples: 1, 2

Homework: 5, 9

Section 13.7: Triple integrals in cylindrical and spherical coordinates

• Review (from 11.8): Definition and geometric interpretation of cylindrical and spherical coordinates, conversion formulas
• Volume element dV in cylindrical and spherical coordinates

Examples: 1, 2, 3, 4

Homework: 1, 7, 8, 9, 19, 21, 25, 33, 37, 39

Lecture 22, 11/8/06: Section 13.5: Applications of double integrals

• Mass, mass density, mass element dm
• Centroid (center of mass)
• Moments of inertia: I_x, I_y, I_₀ (polar moment of inertia)

Skip: First and second theorem of Pappus (p. 973 - 975), formulas for kinetic energy and radius of gyration ((9), (10), (11)).

Examples: 1, 2, 3, 9, 10

Homework: 7, 15, 27, 31, 33

Lecture 21, 11/6/06: Section 13.4: Double integrals in polar coordinates

• Area element dA in polar coordinates
• Computing double integrals in polar coordinates
• Converting double integrals from rectangular coordinates to polar coordinates
• Evaluation of the "Gaussian integral", the integral of exp(-x²) from -infinity to infinity (cf. Example 5)

Examples: 1, 2, 3, 4, 5

Homework: 3, 9, 13, 17, 27, 29, 34

Lecture 20, 11/1/06: Section 13.3: Area and volume by double integrals

• Computation of volume via double integrals
• Computation of area via double integrals

Examples: 1 - 4

Homework: 3, 7, 11, 15, 19, 27, 29, 35

Lecture 19, 10/30/06: Sections 13.1/13.2: Double integrals

• Definition of double integral via Riemann sums
• Computation of double integrals over rectangular regions (13.1)
• Computation of double integrals over general regions R
• Reversing order of integration

Skip: There won't be any quiz/exam problems on Riemann sums (p. 942/943), or the relation between double integrals and cross sections (p. 946 middle through end of 13.1). However, you should take a look at the pictures illustrating these concepts, as they help motivate the concept of a double integral and its application to the computation of volumes (which will come up in a later section).

Examples: Section 13.1: 2, 3, 4;
Section 13.2: 2, 3, 4

Homework: Section 13.2: 1, 11, 13, 15, 19, 31, 33

Lecture 18, 10/26/06: Section 11.8: Cylindrical and spherical coordinates

• Definition, geometric interpretation
• Conversion formulas between cylindrical, spherical, and rectangular coordinates
• Conversion of equations in rectangular coordinates to cylindrical and spherical coordinates
• Surfaces and regions in cylindrical and spherical coordinates: cylinders, spheres, cones, planes, half-planes

Skip: The last part of 11.8 (p. 844 through end), on longitudes and latitudes, presents an interesting real-world application of spherical coordinates, and is worth reading, but there won't be exam/quiz problems on this material.

Examples: 1 - 7

Homework: 1, 5, 7, 9, 17, 23, 27, 29, 31, 33, 35, 39, 43, 45, 47, 49, 51 (a lengthy list, but most problems are quickies that shouldn't take more than a minute or so).

Lecture 16/17, 10/18/06 and 10/23/06: Section 12.9: Lagrange multipliers and optimization with constraints

• Method of Lagrange multipliers
• Application to optimization problems with constraints

Examples: 1, 2, 3

Homework: 1, 7, 13, 19, 21, 35

Lecture 15, 10/16/06: Sections 12.5/12.10

Section 12.5: Maxima and minima of functions of several variables, I. Critical points. First derivative test.

• maximum or minimum values occurring on the boundary
• maximum or minimum values occurring at points where partial derivatives do not exist (e.g., tip of a pyramid)

• Local/global maxima/minima of functions of several variables
• Critical points, first derivative test for functions of several variables

Examples: 1, 2, 3, 6

Homework: 3, 9, 13, 19, 31

Section 12.10: Maxima and minima of functions of several variables, II. Second derivative test.

• Discriminant
• Second derivative test for functions of two variables
• Classification of critical points as local maxima, minima, and saddle points.

Homework: 7, 11, 17

Lectures 13/14, 10/9/06 and 10/11/06: Section 12.8: Directional derivatives and gradients

• Directional derivative
• Rate-of-change interpretation of directional derivative
• Partial derivatives as directional derivatives
• Gradient of a function of several variables
• Geometric interpretation of the gradient
• Relation between the gradient and level curves/surfaces
• Computation of tangent planes and tangent lines via gradients

Examples: 1 - 6, 8

Homework: Section 12.8: 1, 9, 11, 15, 21, 31, 33, 45, 47, 49, 57, 61

Lecture 12, 10/4/06: Section 12.7: Multivariable chain rule

• Independent variables, dependent variables, intermediate variables
• Dependency diagram
• Multivariable chain rule
• Implicit partial differentiation

Examples: 1 - 8

Homework: Section 12.7: 3, 5, 13, 19, 23, 33, 40, 47, 49, 51

Lecture 11, 10/2/06: Section 12.6: Differentials and linear approximation

• Differential of a function of several variables
• Linear approximation to a function of several variables
• Connection to tangent planes (formula (11) from 12.4)
• Application to error estimates

Examples: 1, 2, 3, 5

Homework: Section 12.6: 1, 7, 15, 17, 25, 33, 35, 38. (For tangent plane problems, see Section 12.4, e.g. Problem 31.)

Lecture 10, 9/27/06: Section 12.4: Partial derivates

• Definition of partial derivatives, and notations
• Geometric interpretation
• Interpretation as rates of change
• Higher order partial derivatives
• Tangent plane to surface z = f(x,y)
• Normal vector to surface z = f(x,y)

Examples: 1 - 8

Homework: Section 12.4: 3, 5, 7, 31, 35, 55, 62, 63, 65, 71

Lecture 9, 9/25/06: Sections 12.1 - 12.3: Functions of several variables

• Function of two or more variables
• Graph of a function of two variables
• Level curves (for functions of two variables)
• Level surfaces (for functions of three variables)

Examples: 3 - 9

Homework: Section 12.2: 21, 23, 27, 31, 33, 37, 53, 55, 57

Lecture 8, 9/20/06: Section 11.7: Quadratic surfaces

• Equation of a sphere of given radius and center (p. 782 in 11.2)
• General equation of a quadratic surface
• Traces
• Special quadratic surfaces. Click on the above link for a summary of these special surfaces (a copy of the transparency that was shown in class).
  • Ellipsoid
  • Elliptic Paraboloid
  • Hyperbolic Paraboloid
  • Hyperboloid of one sheet
  • Hyperboloid of two sheet
  • Cylinder
  • Cone

• Find the equation of a sphere, given its center and its radius.
• Given the equation of a sphere, find its center and radius.
• Identity the type of a surface (from the above list), given its equation.
• Identify the type of surface (from the above list), given a 3-dimensional sketch.

Skip: Surfaces of revolution (p. 834, bottom, through middle of p. 835). (We'll cover this material when we get to cylindrical/spherical coordinates and triple integrals.)

Examples: 11.2, Examples 2 and 3; 11.7, Examples 1 - 5 and 8 - 13

Homework: Section 11.2: 19, 25; Section 11.7: 3, 9, 17, 23, 25, 27, 29, 41, 43, 45

Lectures 6/7, 9/13/06 and 9/18/06: Section 11.6: Curvature and Acceleration

• Arc length formula for a space curve (formula (2), p. 819)
• Unit tangent vector T
• Principal unit normal vector N
• Curvature K of space curves: Geometric interpretation (in terms of osculating circle), formulas in terms of T ((19), p. 823), in terms of r' and r'' ((27), p. 826)
• Curvature of a circe of radius R
• Curvature of a plane curves (formulas (12), (13) on p. 821 - don't memorize these)
• Tangential and normal components of acceleration: Formulas in terms of v and kappa ((23), (24), (25), p. 824) and in terms of r' and r'' ((26), p. 825, (28), p. 826)

Skip: The applications in the latter part of the section to Kepler's law and planetary motion, will be covered only lightly, but will not be on exams or quizzes.

Examples: 1,2,4,5,6,7

Homework: Section 11.6: 1, 9, 11, 21, 23, 33, 38 (using the curve defined in 33), 43 (curve defined in 33), 46, 50

Lecture 5, 9/11/06: Section 11.5: Curves and Motion in Space

• Vector-valued functions: Definition and geometric interpretation as motion in space
• Derivatives of vector-valued functions: algebraic definition (componentwise differentiation) and geometric interpretation (tangent vector)
• Differentiation rules (Theorem 2, p. 808)
• Integrals of vector-valued functions
• Motion in space: position, velocity, speed, acceleration, scalar acceleration
• Application: motion of projectiles (covered lightly)

Examples: 1 - 8

Homework: Section 11.5: 5, 8, 15, 17, 23, 27, 35, 42, 55

Lecture 4, 9/5/06 and 9/6/06: Section 11.4: Lines and Planes in Space

• Vector equation
• Parametic equations
• Symmetric equations

• Normal vector to plane
• Vector equation
• Scalar equation (linear equation)

• Given a vector equation of a line, find parameter equations and (if possible) symmetric equations. Conversely. given parameter equations of a line, find a direction vector for the line, and a vector equation.
• Find the equation of a line given (a) a point on the line and a direction vector, (b) two points on the line.
• Given two lines, determine whether they (a) are parallel or coincide, (b) intersect at a single point, (c) are skew (do not intersect and are not parallel). (Hints: (1) Compare direction vectors; (2) compute intersection points (if any) by writing both lines in parameter form (using different parameters, say t and s), then equating the x, y, and z coordinates for the two lines, and solving the resulting system.)
• Given a vector equation of a plane, find a scalar equation. Conversely, given a scalar equation of a plane, find a normal vector and a vector equation.
• Find the equation of a plane given (a) a point on the plane and a normal vector, (b) a point on the plane and two vectors on the plane, (c) three points on the plane
• Given two planes, determine whether they (a) coincide, (b) are parallel (but do not coincide), (c) intersect at an angle; in the latter case determine the angle of intersection (Hint: Compare normal vectors.)
• Given a line and a plane, determine whether the line (a) lies entirely in the plane, (b) is parallel to the plane (but does not lie in it), or (c) intersects the plane at a single point. In the latter case determine the point of intersection. (Hint: Compare the normal vector of the plane and the direction vector of the line.)

Examples: 1 - 6

Homework: Section 11.4: 3, 7, 9, 15, 17, 23, 29, 31, 33, 35, 37, 39

Lecture 3, 8/30/06: Section 11.3: The cross product

• 2 by 2 and 3 by 3 determinants
• Definition of cross product in terms of 3 by 3 determinants
• Geometric interpretation of cross product
• Cross products of parallel vectors
• Cross products of basic unit vectors
• Properties (formulas (12) - (15) on p. 794, skip (16))
• Scalar triple product

• Computing vectors perpendicular to two given vectors
• Area of parallelograms and triangles
• Volume of parallelepipeds and pyramids
• Testing if three given vectors (or four given points) are coplanar (i.e., lie in the same plane)

Examples: 1 - 8

Homework: Section 11.3: 1, 7, 13, 15, 17, 19

Lecture 2, 8/28/06: Section 11.2: Three-dimensional vectors. The dot product

• 3-dimensional coordinate system, rectangular coordinates, coordinate axes, coordinate planes, octants
• Distance formula
• 3-dimensional vectors: length (magnitude), addition/subtraction, multiplication by scalar, unit vectors, unit basis vectors i, j, k

• Definition in terms of components
• Geometric interpretation
• Properties ((9) on p. 784)

• Testing perpendicularity of vectors
• Computing angle between vectors
• Computing angles in triangle
• Direction angles
• Scalar projection of a onto b (component of a along b, comp_ba)
• Work

Deferred: Equation for sphere (p. 782). (This is easy material, but fits better with 11.8, so we will cover it when we get to this section.)

Examples: 4 - 11 (The first three examples are skipped since they deal with equations of spheres)

Homework: Section 11.2: 39, 41, 43, 45, 49, 53, 55

Lecture 1, 8/23/06: Section 11.1: Vectors in the Plane

• Vectors in the plane. Definition and notations (angle brackets, and boldface/arrow). Geometric interpretation as arrow in plane.
• Position vector.
• Length (or magnitude) of a vector (formula and geometric meaning).
• Equality of two vectors.
• Addition of two vectors. Definition and geometric interpretation.
• Multiplication of a vector by a scalar. Definition and geometric interpretation.
• Unit vectors. Normalizing a given vector (i.e., forming a unit vector in the same direction as the given vector).
• Unit basis vectors i and j.
• Representation of a vector as linear combination of basis vectors.

Examples: 1 - 4 (you can skip 5 and 6)

Homework Section 11.1: 1, 5, 9, 17, 21, 33, 51.