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]

cos^{2}(theta) = (1/2)(1 + cos(2*theta)

sin^{2} (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** = a**i** + b**j** is af_{x}(x, y) + bf_{y}(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]