# Divergence Theorem, II

Friday, Apr 20, 2012

## Problems

16.9: 17, 18, 19, 22

# Divergence Theorem, I

Thursday, Apr 19, 2012
Also called Gauss's theorem. The divergence theorem connects the flux across a closed surface to the divergence of the vector field on the enclosed region. We begin by re-doing some of our earlier surface integrals using the divergence theorem

## Problems

16.9: 1, 5, 7, 11, 12, 13

# Stokes' Theorem, II

Wednesday, Apr 18, 2012
A couple more examples of Stokes' theorem.

## Problems

16.8: 17, 18, 19, 22

# Stokes' Theorem, I

Monday, Apr 16, 2012
Stokes' theorem is a generalization of Green's theorem to 3 dimensions.

16.8: 1,2,3,5,9

# Surface integrals, III

Friday, Apr 13, 2012
More examples.

## Problems

16.7: 27, 31, 45, 47, 49

# Surface integrals, II

Thursday, Apr 12, 2012
Some more examples and the surface area

## Problems

16.7: 23, 25, 28, 29

# Surface integrals, I

Wednesday, Apr 11, 2012
Restate the definition of the surface integral and work a couple of examples.

## Problems

16.7: 7,9,15,19,40

# EXAM 3

Monday, Apr 09, 2012
16.1 -- 16.5, need to know the definition of conservative field, vector field, incompressible flow, irrotational flow. Need to know the statements of Green's theorem in both forms and to understand the terms simple, closed, postively oriented for curves

# Flux, surface integral definition

Friday, Apr 06, 2012
We look at the notion of flux and use it to motivate the defintion of a surface integral. We then set about giving a computational method for evaluating surface integrals over parameterized surfaces. Next time we will do a bunch of computation

# Surfaces

Thursday, Apr 05, 2012
We will parametrize some simple surfaces. This is the 2d analogue of a parametrized curve. Tomorrow we will talk about surface integrals and their relation to the flux of a vector field

## Problems

16.6: 3, 13, 19, 21, 23, 24, 26, 25, 33, 37, 41, 44, 45, 48

# Curl and Divergence and a digression about Navier Stokes.

Wednesday, Apr 04, 2012
The velocity field of a fluid is called incompressible if the divergence is 0. It is called irrotational if the curl is 0.

# Curl and Divergence

Monday, Apr 02, 2012
The divergence of a vector field measures it's rate of expansion, while the curl measures its rotation. We defined these two operations on vector fields, and the way that the curl can be used to check whether a field is conservative. Next time we will look at the relation between curl and divergence and the start on surface integrals.

## Problems

1, 5, 7, 9, 11, 12, 14, 15, 17, 21, 22

# Green's Theorem version II

Friday, Mar 30, 2012
The second version of Green's theorem related the line integral of the vector field and the outward normal to a certain double integral over the enclosed region. This is the divergence-flux version of the theorem. Next time we will define curl, divergence and write Green's theorem in the most elegant way

## Problems

page 1089: 13, 17, 18, 19

# Green's Theorem version I

Thursday, Mar 29, 2012
Green's theorem relates the line integral of a vector field F along a closed curve to a certain double integral over the enclosed region. Today we look at the first form of Green's theorem, which we can call the *work version*, or tangential version.

## Problems

page 1089: 1, 5, 6, 9, 11

# Conservation vector fields

Wednesday, Mar 28, 2012
We give a partial converse to the test for conservative vector fields and discuss simply-connected regions

# Fundamental theorem of line integrals, III

Monday, Mar 26, 2012
Basic tests for conservative fields. Some examples of non-conservative fields and the notion of simple connectivity.

## Problems

page 1082: 11, 13, 19, 23, 24, 25, 36

# Fundamental theorem of line integrals, II

Friday, Mar 23, 2012
Examples of conservative fields and computations using the fundamental theorem.

## Problems

page 1082: 1, 2, 3, 5, 7, 9, 15, 30

# Fundamental theorem of line integrals

Thursday, Mar 22, 2012
Conservative vector fields play a significant role in physical applications. The computation of line integrals for such fields is simplified and depends only on the endpoints of the path and the potential function.

## Problems

page 1082: 29, 31, 33, 35

# Line integrals of vector fields, II

Wednesday, Mar 21, 2012
More examples

## Problems

32(a), 33, 41, 42, 49, 50, 51, 52

# Line integrals of vector fields, I

Monday, Mar 19, 2012
Line integrals of vector fields allow us to compute the work done by a force field in moving a particle along a given path.

## Problems

5, 13, 15, 17, 19, 29

# Vector fields, II

Friday, Mar 09, 2012
More examples, conservative fields, and streamlines

# Vector Fields

Thursday, Mar 08, 2012
We will now move on to the subject vector fields. This is the main focus of our course and we will spend the rest of the semester discussin vector fields. Today we will discuss examples and tomorrow we will consider the notion of a streamline from fluid dynamics.

## Problems

page 1061: 1, 5, 11, 13, 15, 18, 22, 26, 29, 34, 35, 36

# Exam 1

Wednesday, Mar 07, 2012
Double and triple integrals. There will be one problem from each of the following sections: 15.3, 4, 6, 7, 8, 9.

# Review for Exam I

Monday, Mar 05, 2012
Review for exam I and practice integrals

# Triple integrals in spherical coordinates

Friday, Mar 02, 2012
Examples of triple integrals in spherical coordinates

# Triple integrals in cylindrical coordinates

Thursday, Mar 01, 2012
Examples of triple integrals in cylindrical coordinates

# Triple integrals in cartesian coordinates

Wednesday, Feb 29, 2012
Examples of triple integrals in cartesian/rectangular coordinates

# Triple integrals

Monday, Feb 27, 2012
Introduction to triple integrals

## Problems

page 1025: 9, 14, 15, 18, 19, 23, 33, 41, 42, 54

## Reminders

• Do the first two webassigns on triple integrals

# Cylindrical and Spherical coordinates

Friday, Feb 24, 2012

## Reminders

• Read the definition of cylindrical and spherical coordinates in chapter 15.8 and 15.9

# Double integrals in polar coordinates (15.4)

Thursday, Feb 23, 2012

## Reminders

• Webassign on this topic is due 2/24 **Change in date**
• quiz 5 is tomorrow

# Double integrals in polar coordinates (15.4)

Wednesday, Feb 22, 2012

## Problems

page 1002: 23, 24, 30, 31, 36

## Reminders

• Webassign on this topic is due 2/24 **Change in date**

# Double integrals in polar coordinates (chapter 10.3 and 15.4)

Friday, Feb 17, 2012

## Problems

page 662: 7,9,17,22,25; page 1002: 1,5,9,11,17,22,25

## Reminders

• Webassign on this topic is due 2/22
• Read the derivation of $r dr d\theta$ in 15.4. We will talk about this on Wednesday.

# Double integrals over general regions (chapter 15.3)

Thursday, Feb 16, 2012

## Problems

page 995 : 30, 47, 49, 51, 60, 63, 67

## Reminders

• Webassign on this topic is due 2/21

# Double integrals over general regions (chapter 15.3)

Wednesday, Feb 15, 2012

## Problems

page 995 : 3, 5, 9, 21, 22, 26, 27

## Reminders

• Webassign on this topic is due 2/21
• Quiz coming up on Friday 2/17

# Iterated integrals (chapter 15.2)

Monday, Feb 13, 2012

## Problems

p.987: 3,7,13,17,23,25,35,37

## Reminders

• Webassign on this topic is due 2/14
• Quiz coming up on Wednesday 2/15

# Iterated integrals (chapter 15.2)

Friday, Feb 10, 2012

## Problems

p.987: 3,7,13,17,23,25,35,37

## Reminders

• Webassign on this topic is due 2/14
• Quiz coming up on Wednesday

# Double integrals over rectangles (chapter 15.1)

Thursday, Feb 09, 2012

## Problems

p.981: 5,6,9,12,13

## Reminders

• Webassign on this topic is due 2/14
• Quiz coming up

# Exam 1

Wednesday, Feb 08, 2012

# Review day

Monday, Feb 06, 2012
Review for the exam 1

## Reminders

• Exam 1 is coming up

# Maxima and minima

Friday, Feb 03, 2012
We will look at the test for determining the location of local maxima and minima.

## Problems

p.953: 1,3,9,12,16

## Reminders

• Exam 1 is coming up

# Tangent planes and linearization

Thursday, Feb 02, 2012

## Problems

p.922: 6,15(see Thm 8), 21,22,24,34,39

## Reminders

• Exam 1 is coming up
• Practice exam will be posted on Friday night, maybe Saturday (most likely, actually)

# Directional derivatives, gradients, rates of change practice

Wednesday, Feb 01, 2012
More problems. Pokemon was chosen as the source of pseudonyms.

# Directional derivatives, gradients, rates of change practice

Monday, Jan 30, 2012

## Problems

p.943: 5,32,38,42,43,54 Proof of Theorem 15*

# Rates of changes and properties of the gradient

Friday, Jan 27, 2012
We will describe three key properties of the gradient and prove them.

## Problems

p.943: 5,32,38,42,43,54 Proof of Theorem 15*

# Directional derivatives and the gradient

Thursday, Jan 26, 2012
Today we discover the connection between the directional derivative and the gradient.

## Problems

p.943: 1,9,11,17,20,21,33,34

# Partial derivatives and the gradient

Wednesday, Jan 25, 2012
Today we will look at the gradient of a function of two variables.

# Level curves and partial derivatives

Monday, Jan 23, 2012
Level curves, or contour plot, provide a lot of information about a function of two-variables. Some approximations can be made from the level curves. In particular it is possible to tell whether a function is increasing or decreasing in a particular direction. We also talked about the notion of a partial derivative. Tomorrow we will talk more about this topic and do some calculations.

## Problems

page 911: 3,10,11,15,19,31,33,39,53

## Reminders

• If you haven't passed the gateway you should come by during EI to take it again
• Quiz thursday on arc-length and line integrals of scalar functions

# Multivariable functions

Friday, Jan 20, 2012
In calc I, II we've encountered functions whose domains are subsets of the real numbers. The codomains are also generally subsets of the real numbers. Parametrized curves introduce the possibility that the codomain can be two- or three-dimensional. In calculus III we will deal with functions of several variables

## Problems

page 888: 1,2,9,13,15,25,29,35,36,37,53,54

# Motion in space

Thursday, Jan 19, 2012
Continuing the projectile motion calculations form yesterday

## Problems

p.870: 10,16,17(a),18(a),22,23; p.870: 24,26,27,45p.1072: 3,9,10

## Reminders

• Read Ex. 5 p. 864

# Motion in space

Wednesday, Jan 18, 2012
We will define, position, speed, velocity and acceleration. We will then use calculus to derive Newton's laws of motion and examine the motion of a projectile.

## Problems

p.870: 10,16,17(a),18(a),22,23; p.870: 24,26,27,45p.1072: 3,9,10

## Reminders

• Read Ex. 5 p. 864
• WebAssign homework is due tonight

# Line integrals of scalar-valued functions

Friday, Jan 13, 2012
The line integral of a scalar-valued function allows, for example, to compute the mass of a wire of non-uniform density. The formula requires that we integrate with respect to the arc-length s. The parametrization of a curve in terms of its arc-length is intrinsic. In general every curve has infinitely many parametrizations.

p.1072: 3,9,10

# Gateway quiz. Arc-length.

Thursday, Jan 12, 2012
We talked about arc-length, distance traveled and speed.

p.860: 1,2,12,65

# Review of calculus II: vectors

Wednesday, Jan 11, 2012
Review vectors: length, dot and cross product, angles.

## Problems

p.846: 17; p.852: 5

## Reminders

• There will be a gateway quiz tomorrow. **No calculators**. You must score a 90% to pass. Topics: basic differentation, integration, vectors, planes, parametric curves

# Review of calculus II: differentiation and integration

Tuesday, Jan 10, 2012
Today we will review basic differentiation and integration really quick. There will be a quiz on Thursday.

## Problems

p.790: 31,33; p.798: 25,29; p.806: 39,49; p.814: 19,27; p.824: 9, 25,27

## Reminders

• Set up WebAssign account
• Study for gateway quiz

# Quizzes

• Wed, Jan 11, 2012
• Topic: Gateway quiz based on Calculus II
• Thu, Jan 26, 2012
• Topic: arc-length and line integrals of scalar functions
• Mon, Jan 30, 2012
• Topic: projectile motion
• Wed, Feb 15, 2012
• Topic: Double integrals over rectangles and iterated integrals (15.1 and 15.2)
• Fri, Feb 17, 2012
• Topic: Double integrals over general regions (15.3)
• Fri, Feb 24, 2012
• Topic: Double integrals in polar coordinates (15.4)
• Fri, Mar 23, 2012
• Topic: Vector fields and line integrals (16.1, 16.2)
• Wed, Apr 18, 2012
• Topic: Surface integrals (16.6, 16.7)

# Exams

• Exam 1
• Wednesday, Feb 08, 2012
• 13.3, 16.2, 13.4, 14.1, 14.3, 14.6, 14.4, 14.5
• Review Monday, 2/6/2012, in class.
• Practice Exam
• Exam 2
• Wednesday, Mar 07, 2012
• Double and triple integrals of all kinds (15.1 -- 15.5, 15.7 -- 15.9).
• Review Monday, 03/05/2012, in-class and EI on 03/06/2012 at 2000
• Exam 3
• Monday, Apr 09, 2012
• Vector field basics (16.1 -- 16.5)
• Review Sunday 4/8/2012, 2000 -- 21??
• Exam 4
• Friday, Apr 27, 2012
• Integral theorems (16.6 -- 16.9)