Review Sheet 3 Math H263

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

*Italics indicate material that could be on
the final exam but that was not on the midterms.*

*Final exam goes through acetate [215]. *

**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

**Properties of
line integrals [127-128]**

**Differentiation
of integrals [142-143]**

**Green's Theorem
[140]**

**Dot products and
cross products of vectors [throughout]**

**Determinants
[107-109]**

**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]

**Triple integrals [98]**

**Cylindrical coordinates [102, 108]**

**Spherical coordinates [104-106, 109]**

**Line integral [119]**

**Scalar field [126]**

**Vector field [126]**

**Conservative field and its characterizations
[125,129-130, 137-140]**

**Jacobian of a mapping [159-160]**

**Linear mapping [164]**

**Surface integral [168-170]**

*Flux density [188]*

*Divergence of a vector field [191]*

**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]

**Triple integrals in cylindrical and spherical
coordinates [102-109]**

** **

**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]

**Triple
integrals [99-101, 103, 105-109]**

**Work
as motivation for line integrals [118-119, 124-125]**

** **

** Line
integral [122-123] ***and
[200.0-200.1]*

** **

** Vector
field [136.1]**

** **

**Simple closed curves [141.1 on 0506]**

** **

**Green’s theorem [148-149] ***and [201]*

** **

**Green’s theorem for general regions [146, 147.1]**

** **

**Change
of variables and Jacobians [151, 153, 161-165, 180]**

** **

*Surface
integrals [172-174]*

** **

*Conservative
vector fields [211, 215]*

** **

*Surface integral [172-174]*

** **

*Closed orientable surfaces [193]*

* *

*Divergence theorem [198-199, 204]*

* *

*Stoke's Theorem [212-214]*

** **

**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)

**Line
integrals [136.0, 137]**

** **

**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 and statement of the theorem [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]

**Proofs of the
characterizations of conservative fields [131-135, 140]**

**Green’s theorem
[140, 143-146]**

*Gauss'
Theorem(Divergence Theorem) [192, 194-197]*

*Stoke's Theorem
[205-210]*

**Calculations**

Distance formula in n-dimensional space [throughout]

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 and triple integrals
[82-84, 100-101] ***and [202]*

Volumes as double integrals [93-97]

Double integrals in polar coordinates [85]

**Triple integrals [99-101]**

**Triple integrals in cylindrical coordinates
[102-103, 108]**

**Triple integrals in spherical coordinates
[104-106, 109]**

**Surface area**

** As
a double integral [110-114, 116-117.4]**

** For
surfaces given parametrically [183-186]**

**Line integrals [120-121]**

**Using Green's Theorem [148-149]**

**Change of variables (substitution) in integrals
in dimensions greater than one. [157-163]**

**Change of variables and Jacobians [150-156,
175-176, 178]**

**Parametric representations of surfaces [166-167]**

**Determinants [107-109, 176-177]**

*Surface integrals [171, 187] *

*Fluid flow across a surface [188-190, 192]*

**Notation**

** **for partial derivatives [16]

for gradient [32]

for multiple integrals [68]

for iterated integrals [72]

**for
triple integrals [98]**

** for
line integrals [119, 121]**

**for
Jacobians [107-109]**

** ***for surface integrals [171]*

** ***use of the 'del'-operator [203, 209]*