Math 2450 Dist. 01 Fall 2021 Professor: Eugenio Aulisa Office: Math 226 Phone: 806-834-6684 email: eugenio.aulisa@ttu.edu Meetings: TR at 11:30am-1:20pm on Zoom Office Hours: TR 9:00am-11:00am on Zoom or daily by e-mail from WebWork TA and Substitute Instructor: Bradley Vigil: bradley.vigil@ttu.edu

# Calculus III with applications

## Textbook

Calculus
K. Smith, M. Strauss and M. Toda, Kendall Hunt, 7th National Edition.

## Syllabus

The following link can be used to obtain a copy of the sylabus in Adobe Acrobat(.pdf) format

The following link provides a tutorial on how to use WeBWork

Introduction to WeBWork.pdf

## Course Description

This course covers Calculus of several variables. The concepts are extensions of the concepts from Calculus I. It is necessary to remind the students of those basic concepts, as the course progresses. Multivariable Calculus is an important tool in Science and Engineering. The instructor should emphasize the importance of all relevant concepts, including: curves and surfaces in Euclidean 3-space, length and curvature, area and volume; surfaces, partial derivatives, total differential, tangent planes to surfaces; gradient; vector-valued functions; path integral; Stokes' theorem, which should be stated, with an emphasis on its important particular cases, Green's Theorem and Divergence Theorem - followed by a few basic examples.

Homework is worth 20% of the final grade.
However in order to pass the class your overall grade in the HW at the end of the semester should be at least 50%. This may appear radical, but besides the exams, the HW system is a major tool the instructor has to asses your class performances. The instructor will check regularly your HW score and let you know if you are not on track.

Examinations:
 Exam #1: Thu, Sep 16 12:00-1:20pm Online worth 15% of the final grade Exam #2: Thu, Oct 14, 12:00-1:20pm Online worth 20% of the final grade Exam #3: Thu, Nov 11, 12:00-1:20pm Online worth 20% of the final grade Final Exam: Sat, Dec 4, 1:30-4:00pm Online worth 30% of the final grade

Overall grade: a perfect score in all tests and homeworks results in a overall grade of 105% .

Exam Policies

Students are expected to take the midterm exams and the final exam as scheduled. There are no make ups for the examinations, except for reasons of illness, stated in writing by a medical doctor, observance of a religious holiday, university justified field trips or work conflicts. Usually, no other reasons are accepted (events, plane tickets, weddings, ...).

See below some general exam information and rules:

1) All exams are taken on the WebWork system.
2) Exam 1 covers chapters 9 and 10 (HW01-HW03), Exam 2 covers chapter 11 (HW04-HW06), Exam 3 covers chapter 12 (HW07-HW09), the Final Exam is comprehensive and it covers chapters 9 to 13 (HW01-HW11).
3) In each exam there will be several multiple choice questions.
4) The correct answer is ALWAYS among the 5 choices: I never use the "None of the above" choice.
5) You will have only ONE ATTEMPT to get it right.
6) To submit your work, you need to press the "Grade Test" button at the very bottom of the page . After you press it, you cannot change any of your answers. Be sure to complete all your work before pressing it.
7) The "Grade Test" button should be pressed before the due date otherwise a 0% score will be recorded.
8) The test score and solution keys will be available after 2-3 days
9) If you experience any technical problem during the exam, DO NOT WAIT, but immediately notify me during or after the test. When you notify me, be sure to explain carefully what happened, and be ready to take action. There is no much to do for me if you contact me (OR IF YOU TAKE ACTION) after days have passed.

## Review of Sections 9.1-9.4

Vector Basics

Vector Component Form

Scalar Multiplication

Vector Operations - Example 1

Standard Unit Vectors

Magnitude of a Vector

Magnitude of a Vector - Example 1

Unit Vector

How to Normalize a Vector

3D Vectors

Vector Dot Product

Dot Product - Example 1

Using Dot Product to Find the Angle Between Two Vectors

Finding Angles Using Dot Products - Example 1

Vector Projections

Vector Projections - Example 1

Vector Cross Product

Vector Cross Product - Example 1

Vector Cross Product - Extra Theory

## Section 9.5

Parametric Equations

Graphing Parametric Equations

Eliminating the Parameter

Eliminating the Parameter - Example 1

Differences in the Parametrization

How to Parametrize a Curve

Parametrization - Example 1

Lines in Space

Lines in Space - Example 1

Lines in Space - Example 2

Lines in Space - Symmetric Equations

Lines in Space - Parametric to Symmetric

Lines in Space - Symmetric to Parametric

Lines in Space - Are These Lines Parallel?

## Section 9.6

Equations of Planes in Space

Plane in Space - Extra Theory

Standard vs General Form of a Plane

Normal Vector of a Plane

Equation of a Plane - Example 1

Equation of a Plane - Example 2

Equation of a Plane - Example 3

Distance Between a Point and a Plane

Distance Between a Point and a Plane - Example 1

Distance Between a Point and a Line

Distance Between a Point and a Line - Example 1

Angle Between Two Planes

Angle Between Two Planes - Example 1

Line of Intersection of Two Planes

## Section 9.7

The Equation of the Sphere

Quadric Surface: The Hyperboloid of Two Sheets

Quadric Surface: The Hyperboloid of One Sheets

## Section 10.1

Introduction to Vector Valued Functions

The Domain of a Vector Valued Function

Determine a Vector Valued Function from the Intersection of Two Surfaces

Limits of Vector Valued Functions

## Section 10.2

The Derivative of a Vector Valued Function

Properties of the Derivatives of Vector Valued Functions

The Derivative of the Cross Product of Two Vector Valued Functions

Determining Where a Space Curve is Smooth from a Vector Valued Function

Determining Velocity, Speed, and Acceleration Using a Vector Valued Function

Indefinite Integration of Vector Valued Functions

Ex: Integrate a Vector Valued Function

Indefinite Integration of Vector Valued Functions with Initial Conditions

Ex: Find the Velocity and Position Vector Functions Given the Acceleration Vector Function

## Section 10.4

Determining the Unit Tangent Vector

Ex: Find a Unit Tangent Vector to a Space Curve Given by a Vector Valued Function

Determining the Unit Normal Vector

Arc Length Using Parametric Equations

Determining Arc Length of a Curve Defined by a Vector Valued Function

Ex: Determine Arc Length of a Helix Given by a Vector Valued Function

Determining Curvature of a Curve Defined by a Vector Valued Function

## Sections 11.1-11.3

Introduction to Functions of Two Variables

Level Curves of Functions of Two Variables

Limits of Functions of Two Variables

First Order Partial Derivatives

Implicit Differentiation of Functions of One Variable Using Partial Derivatives

Second Order Partial Derivatives

## Sections 11.4-11.6

Differentials of Functions of Two Variables

Applications of Differentials of Functions of Several Variables

The Chain Rule for Functions of Two Variable with One Independent Variable

Ex: Chain Rule - Function of Two Variables with One Independent Variable

Partial Implicit Differentiation

The Chain Rule for Functions of Two Variable with Two Independent Variables

Ex: Chain Rule - Function of Two Variables with Two Independent Variable

Ex: Chain Rule - Function of Two Variables with Three Independent Variable

Directional Derivatives

Ex: Find a Value of a Directional Derivative - f(x,y)=ln(x^2+y^2)

Ex: Find the Gradient of the Function f(x,y)=xy

Ex: Use the Gradient to Find the Maximum Rate of Increase of f(x,y)=(4y^5)/x from a Point

Determining a Unit Normal Vector to a Surface

Verifying the Equation of a Tangent Plane to a Surface

Determining the Equation of a Tangent Plane

Ex 1: Find the Equation of a Tangent Plane to a Surface

Ex 2: Find the Equation of a Tangent Plane to a Surface (Exponential)

## Sections 11.7 and 11.8

Determining the Relative Extrema of a Function of Two Variables

Applications of Extrema of Functions of Two Variables I

Applications of Extrema of Functions of Two Variables II

Applications of Extrema of Functions of Two Variables III

Absolute Extrema of Functions of Two Variables

Lagrange Multipliers - Part 1

Lagrange Multipliers - Part 2

Maximize a Function of Two Variable Under a Constraint Using Lagrange Multipliers

## Sections 12.1-12.2

Introduction to Double Integrals and Volume

Fubini's Theorem

Ex: Evaluate a Double Integral to Determine Volume (Basic)

Use a Double Integral to Find the Volume Under a Paraboloid Over a Rectangular Region

Double Integrals and Volume over a General Region - Part 1

Double Integrals and Volume over a General Region - Part 2

Evaluating Double Integrals

Ex: Double Integrals - Describe a Region of Integration (Triangle)

Ex: Double Integrals - Describe a Region of Integration (Quadric)

Ex: Double Integrals - Describe a Region of Integration (Advanced)

Evaluate a Double Integral Over a General Region - f(x,y)=xy^2

Evaluate a Double Integral Over a General Region with Substitution - f(x,y)=e^(x/y)

Setting up a Double Integral Using Both Orders of Integration

Double Integrals: Changing the Order of Integration - Example 1

Double Integrals: Changing the Order of Integration - Example 2

## Sections 12.3-12.4

Introduction to Double Integrals in Polar Coordinates

Double Integrals in Polar Coordinates - Example 1

Double Integrals in Polar Coordinates - Example 2

Area Using Double Integrals in Polar Coordinates - Example 1

Area Using Double Integrals in Polar Coordinates - Example 2

Double Integrals in Polar Form - Volume of a Half Sphere Over a Circle

Surface Integrals with Explicit Surface Part 1

Surface Integrals with Explicit Surface Part 2

## Sections 12.5 and 12.8

Triple Integrals and Volume - Part 1

Triple Integrals and Volume - Part 2

Triple Integrals and Volume - Part 3

Changing the Order of Triple Integrals

Introduction to Cylindrical Coordinates

Triple Integrals Using Cylindrical Coordinates

Triple Integral and Volume Using Cylindrical Coordinates

Rewrite Triple Integrals Using Cylindrical Coordinates

Introduction to Spherical Coordinates

Triple Integral and Volume Using Spherical Coordinates

Double Integral: Change of Variables Using the Jacobian

Triple Integral: Change of Variables Using the Jacobian

## Sections 13.1 and 13.2

Introduction to Vector Fields

The Divergence of a Vector Field

The Curl of a Vector Field

Defining a Smooth Parameterization of a Path

Line Integrals in R^2

Line Integrals in R^3

Line Integral of Vector Fields

Line Integrals in Differential Form

## Sections 13.3-13.7

Determining the Potential Function of a Conservative Vector Field

The Fundamental Theorem of Line Integrals - Part 1

The Fundamental Theorem of Line Integrals - Part 2

Fundamental Theorem of Line Integrals - Closed Path/Curve

Green's Theorem - Part 1

Green's Theorem - Part 2

Determining Area using Line Integrals

Surface Integral with Explicit Surface Part 1

Surface Integral with Explicit Surface Part 2

Surface Integral of a Vector Field - Part 1

Surface Integral of a Vector Field - Part 2

Stoke's Theorem - Part 1

Stoke's Theorem - Part 2

The Divergence Theorem - Part 1

The Divergence Theorem - Part 2

Class Lecture Notes

Lecture Notes for sections 9.1-9.5

Lecture Notes for section 9.6

Lecture Notes for sections 9.7

Lecture Notes for sections 10.1

Lecture Notes for section 10.2

Lecture Notes for section 10.4

Lecture Notes for section 11.1

Lecture Notes for sections 11.2 and 11.3

Lecture Notes for section 11.4

Lecture Notes for sections 11.5 and 11.6

Lecture Notes for sections 11.6 and 11.7

Lecture Notes for section 11.7

Lecture Notes for section 11.8

Lecture Notes for section 12.1

Lecture Notes for of section 12.2

Lecture Notes for of section 12.3

Lecture Notes for of section 12.4

Lecture Notes for of section 12.5

Lecture Notes for of section 12.7

Lecture Notes for of section 12.8

Lecture Notes for of section 13.1

Lecture Notes for of section 13.2

Lecture Notes for of section 13.3

Lecture Notes for of section 13.4

Lecture Notes for of section 13.5

Lecture Notes for of section 13.6

Lecture Notes for of section 13.7

Lecture Notes for Final Exam Review 1

Lecture Notes for Final Exam Review 2

## from Eugenio Aulisa

Vectors in 2D and 3D reference system, Sec 9.1-9.2

Vectors in 3D, dot product and cross product, Sec 9.2-9.4

The equation of the line in 3D and parametric equations, Sec 9.5

The equation of the plane and quadric surfaces, Sec 9.6-9.7

Quadric surfaces and vector value functions, Sec 9.7-10.1

Vector value functions, limit, continuity, derivative, and integration, Sec 10.1-10.2

Unit tangent vector, principal unit normal vector, arch length, curvature, Sec 10.4

Function of several variables, limit and continuity, Sec 11.1 and 11.2

Partial Derivatives, Sec 11.3

Tangent plane, incremental approximation, total differential and differentiability, Sec 11.4

Multivariable Chain Rule, Sec 11.5 (missing lecture! To be added)

Directional derivative, gradient and relative extrema, Sec 11.6-11.7

Absolute extrema, theorem of Lagrange multiplier, Sec 11.7-11.8

Double integral on rectangular region, Sec 12.1

Double integral on non-rectangular region and in polar coordinates, Sec 12.2-12.3

Double integral in polar coordinates and surface area, Sec 12.3 - 12.4

Surface area and triple integrals, Sec 12.4-12.5

Triple integrals in cylindrical coordinates, Sec 12.7

Integration in spherical coordinates and Jacobian, Sec 12.7 - 12.8

Vector fields, Sec 13.1

Line integrals, Sec 13.2

Conservative vector fields and Green's theorem, Sec 13.3-13.4

Surface and flux integrals, Sec 13.5

Stokes's and divergence theorem, Secs 13.6-13.7

## from Magdalena Toda

TTU Math2450 Calculus3 Sec 9.5-9 6

TTU Math2450 Calculus3 Sec 9.7

TTU Math2450 Calculus3 Sec 10.1

TTU Math2450 Calculus3 Sec 10.2 and 10.4 part 1

TTU Math2450 Calculus3 Sec 10.2 and 10.4 part 2

TTU Math2450 Calculus3 Sec 10.2 and 10.4 part 3

TTU Math2450 Calculus3 Sec 11.1 and 11.2

TTU Math2450 Calculus3 Sec 11.2 and 11.3

TTU Math2450 Calculus3 Sec 11.4 part 1

TTU Math2450 Calculus3 Sec 11.4 and 11.5

TTU Math2450 Calculus3 Sec 11.5 and 11.6

TTU Math2450 Calculus3 Sec 11.6 and 11.7

TTU Math2450 Calculus3 Sec 11.7 and 11.8

TTU Math2450 Calculus3 Sec 12.1 and 12.2

TTU Math2450 Calculus3 Sec 12.3 (large board)

TTU Math2450 Calculus3 Sec 12.3 (Substitute)

TTU Math2450 Calculus3 Sec 12.4

TTU Math2450 Calculus3 Sec 12.5

TTU Math2450 Calculus3 Sec 12.6 - 12.7

TTU Math2450 Calculus3 Sec 12.7 - 12.8

TTU Math2450 Calculus3 chap. 10-11-12 review

TTU Math2450 Calculus3 Sec 13.1 - 13.2

TTU Math2450 Calculus3 Sec 13.2 - 13.3

TTU Math2450 Calculus3 Secs 13.3

TTU Math2450 Calculus3 Secs 13.4 -13.5

TTU Math2450 Calculus3 Secs 13.6 - 13.7