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  ***.  Three items are marked.  One will be on the exam.

Questions from [4.0-5]  and  [dual problem 1-3] and triple integrals will not be on midterm I.

Midterm I goes through acetate [97] and the textbook through section 20.4, except that section 20.3 will not be covered until midterm II.

Tools

Normal vectors to planes in three-dim space

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

 

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

Level surfaces of functions  [3, 6]

Domain of a function  [7-9]

Continuous function [10-11]

Partial derivatives  [15-16]

Directional derivatives  [20]

Unit vector [32]

Gradient  [32-33.0]

Tangent plane to a surface at a point on the surface [37]

Critical point [51]

Double integrals  [67]

Region  [61]

            Horizontally simple  [73.0-73.1, 76]

            Vertically simple [76]

 

            Simple  [76]

Geometry

            Related to partial derivatives  [15, 19.0-19.3]

            Related to directional derivatives [20-22, 30-31]

            Related to gradient  [38]

            Tangent vector to a curve in space  [34.2-37]

            Tangent plane to a surface at a point on the surface [33.1, 37]

Tangent vector in the tangent plane for a curve on a surface  [34.2-37]

Normal vector to a tangent plane of a surface  [39-40]

Gradient at a point is perpendicular to the level surface through the point [39-40]

Related to LaGrange multipliers [57.0-57.2]

Related to double integrals [63-66]

Double integral in polar coordinates  [85-89]

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

 

 

Examples

 

         Functions of several variables  [2]

 

            Partial derivatives [16-18]

 

            Directional derivatives  [33.0-33.1, 37.1-37.2]

 

            Chain rule  [44-46, 48-50]

 

            Implicit differentiation  [49]

 

            Max-min in several variables [54-57.0, 58-60.1]\

 

            Regions  [62, 73.0-73.1, 76]

 

            Double integral [69-72]

 

            Iterated integrals [74-75, 77, 79-83, 90-92]

 

 

 

Facts

 

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

 

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

 

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

 

Usually the order in which mixed partial derivatives are calculated makes no difference [18]

 

The tangent plane to the level surface of a function is perpendicular to the gradient.  [39-40]

 

Chain rule for several variables [43, 47]

 

Second derivative for max-min in two variables [53]

 

About double integrals [78-79]

 

cos2(theta) = (1/2)(1 + cos(2*theta)

 

sin2 (theta) = (1/2)(1 - cos(2*theta)

 

 

 

Theorems--with proofs

 

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

 

*** f(x, y) = xy  is continuous  [13-14]

 

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

Big delicate theorem

            Analogous result for functions of one variable [23-24]

            Motivation [22-24]

            ***Statement of the theorem [25]

            Proof [26-28]

            Generalization to three variables [29]

 

The directional derivative of  f(x, y)  in the direction of the unit vector u = ai + bj  is  afx(x, y) + bfy(x, y)   [31]

 

Calculations

 

            Distance formula in n-dimensional space  [throughut]

 

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

 

            Partial derivatives [16-18]

Normal vector to a tangent plane of a surface  [39-40]

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

about tangent planes of a surface [throughout]

gradient [32-33.0]

directional derivatives [31-33.0, 37.1-37.2]

chain rule for several variables  [34.0, 43, 47]

implicit partial differentiation  [49-50]

max-min in two variables

            second derivative test [53]

substitution  [55-56]

            Lagrange multipliers  [57.0-57.2]

Iterated integrals  [74-75, 79-83]

Applications of double integrals [82-84] (will be on midterm II)

Volumes as double integrals  [93-97]

Double integrals in polar coordinates  [85]
 

Notation

         for partial derivatives  [16]

            for gradient  [32-]

            for multiple integrals  [68]

            for iterated integrals  [72]