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]