Review Sheet 1    Math H263

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

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

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

Bold face indicates material that will not be 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]

 

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]

 

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]

 

 

 

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)

Nothing

 

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]
 

Notation

         for partial derivatives  [14, 16]

            for multiple integrals  [65.0, 86]

            for iterated integrals  [68, 87.1]