Final Review Sheet    Math H263

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

Theory that might be on Final Exam  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] and  [186-2001]will not be on Final Exam.

Italics indicates material that was not on Midterm I or II, but will be on Final Exam.

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]

Surface integrals [150, 153-158]

Curl of a vector field [169]

Orientable surface  [170]

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]

 

Gauss' theorem [164-168]

 

Stoke's theorem  [180-185]

 

 

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) = xy is continuous

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

Careful statement of Green's theorem [126.0]

Gauss' theorem (divergence theorem)  [159-163]

Stoke's theorem  [171-179] 

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

Surfaces  z = f(x, y) [19.4]

Surfaces  f(x, y, z ) = c  [40]

Parametrically given surfaces  []

 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]

Surface integrals [150, 153-158]

Notation

         for partial derivatives  [14, 16]

            for multiple integrals  [65.0, 86]

            for iterated integrals  [68, 87.1]

            for Jacobian's [132]

         for gradient, divergence, curl [159]