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]*