Review Sheet 2    Math H263

[n] means the topic is found on acetate n.

Theory that might be on Midterm II  is marked by asterisks  ***.  One item is marked.  It WILL be on the exam.

Questions from [25.0-31]  and  [82-85] and [142-145.1] will not be on midterm II.

Surface integrals will not be on Midterm II.

Bold face indicates material that was not on Midterm I, but will be on Midterm II.

Tools

Normal vectors to planes in three-dim space  [5]

How to tell if planes are parallel or at right angles by considering their normal vectors  [5]

Triangle inequality  [6]

Properties of line integrals [114]

Differentiation of integrals [120.1-120.3]

Green's Theorem [121-124.0]

Cross product of vectors [131-132]

Determinants [132, 139-141]

 

Definitions (know them and be able to write them correctly)

Level surfaces of functions  [4]

Domain of a function  [8-10]

Continuous function [11]

Partial derivatives  [13-16]

Directional derivatives  [32, 35.0]

Unit vector [32]

Gradient  [35.0]

Double integrals  [63-65.4]

Region  [64]

            Horizontally simple  [67.0-68]

            Vertically simple [67.0-68]

            Simple  [67.0-68]

Triple integrals  [86]

Iterated integrals  [66.0-66.3, 68-70, 87.0-88.1completed]

Line integral [104]

Scalar field [109]

Vector field [109]

Gradient of a scalar field [110]

Conservative field and its characterizations [111-119.1]

Jacobian of a mapping [132-133.2]

Linear mapping [136-138.1]

 

Geometry

            Related to partial derivatives  [17-18]

            Tangent plane to a surface at a point on the surface [19.0-19.2]

Tangent vector in the tangent plane for a curve on a surface  [19.3]

Normal vector to a tangent plane of a surface  [19.4]

Gradient at a point is perpendicular to the level surface through the point [37]

Double integral in polar coordinates  [77-78]

Triple integrals in cylindrical and spherical coordinates [79-94]

Be able to sketch figures in 2 and 3 dimensions [throughout]

 

 

Examples

 

         Functions of several variables  [2]

 

Loci [3]

 

            Paraboloid of revolution  [2]

 

f(x, y) = x + y is continuous  [12]

 

Work as motivation for line integrals [103.0-103.1]

 

 

Facts

 

         z = f(x, y)  defines a surface in three-dimensional space [2]

 

            f(x, y) = c  (constant) defines a curve in the plane  [2]

 

            f(x, y, z) = c  (constant) defines a surface in three-dim space  [4]

 

 

 

Theorems--with proofs

 

*** f(x, y) = x + y is continuous  [12]

 

Theorems--no proofs (know what they mean and be able to state and use them correctly)

Careful statement of Green's theorem [126.0]

 

Calculations

 

            Distance formula in n-dimensional space  [6]

 

            Use of the dot product to find angles in n-dim space  [7]

 

            Partial derivatives  [15]

Normal vector to a tangent plane of a surface  [19.4]

Equation for a tangent plane of a surface  [19.5, 40-42]

about tangent planes of a surface [20-24]

gradient [35.0--37]

directional derivatives [35.0-35.2, 47.0-47.1]

chain rule for several variables  [43-46, 48.0-48.2]

implicit partial differentiation  [48.2-49]

max-min in two variables

            second derivative test [50-55]

            Lagrange multipliers  [56-62.3]

Iterated integrals  [66.0-66.3, 68-70, 73-76]

Volumes as double integrals  [71-72.1]

Double integrals in polar coordinates  [79-81]

Triple integrals in cylindrical coordinates [89.0-90]

Triple integrals in spherical coordinates [91.0-94]

Surface area

         As a double integral [95-102.3]

         For surfaces given parametrically [147-148]

Line integrals [105-108]

Using Green's Theorem [124.1-125.1]

Change of variables (substitution) in integrals in dimensions greater than 1. [129-133.1]

Change of variables and Jacobian's [133.0-138.1]

Parametric representations of surfaces [145-147]


 

Notation

         for partial derivatives  [14, 16]

            for multiple integrals  [65.0, 86]

            for iterated integrals  [68, 87.1]

            for Jacobian's [132]