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In this unit we will learn about derivatives of functions of several variables. Conceptually these derivatives are similar to those for functions of a single variable.
- They measure rates of change.
- They are used in approximation formulas.
- They help identify local maxima and minima.
As you learn about partial derivatives you should keep the first point, that all derivatives measure rates of change, firmly in mind. Said differently, derivatives are limits of ratios. For example,
Of course, we’ll explain what the pieces of each of these ratios represent.
Although conceptually similar to derivatives of a single variable, the uses, rules and equations for multivariable derivatives can be more complicated. To help us understand and organize everything our two main tools will be the tangent approximation formula and the gradient vector.
Our main application in this unit will be solving optimization problems, that is, solving problems about finding maxima and minima. We will do this in both unconstrained and constrained settings.
Part A: Functions of Two Variables, Tangent Approximation and Optimization
» Session 24: Functions of Two Variables: Graphs
» Session 25: Level Curves and Contour Plots
» Session 26: Partial Derivatives
» Session 27: Approximation Formula
» Session 28: Optimization Problems
» Session 29: Least Squares
» Session 30: Second Derivative Test
» Session 31: Example
» Problem Set 4
Part B: Chain Rule, Gradient and Directional Derivatives
» Session 32: Total Differentials and the Chain Rule
» Session 33: Examples
» Session 34: The Chain Rule with More Variables
» Session 35: Gradient: Definition, Perpendicular to Level Curves
» Session 36: Proof
» Session 37: Example
» Session 38: Directional Derivatives
» Problem Set 5
Part C: Lagrange Multipliers and Constrained Differentials
» Session 39: Statement of Lagrange Multipliers and Example
» Session 40: Proof of Lagrange Multipliers
» Session 41: Advanced Example
» Session 42: Constrained Differentials
» Session 43: Clearer Notation
» Session 44: Example
» Problem Set 6
» Practice Exam
» Session 45: Review of Topics
» Session 46: Review of Problems
» Exam Materials
Next »
This unit covers the basic concepts and language we will use throughout the course. Just like every other topic we cover, we can view vectors and matrices algebraically and geometrically. It is important that you learn both viewpoints and the relationship between them.
Part A: Vectors, Determinants, and Planes
» Session 1: Vectors
» Session 2: Dot Products
» Session 3: Uses of the Dot Product: Lengths and Angles
» Session 4: Vector Components
» Session 5: Area and Determinants in 2D
» Session 6: Volumes and Determinants in Space
» Session 7: Cross Products
» Session 8: Equations of Planes
» Problem Set 1
Part B: Matrices and Systems of Equations
» Session 9: Matrix Multiplication
» Session 10: Meaning of Matrix Multiplication
» Session 11: Matrix Inverses
» Session 12: Equations of Planes II
» Session 13: Linear Systems and Planes
» Session 14: Solutions to Square Systems
» Problem Set 2
Part C: Parametric Equations for Curves
» Session 15: Equations of Lines
» Session 16: Intersection of a Line and a Plane
» Session 17: General Parametric Equations; the Cycloid
» Session 18: Point (Cusp) on Cycloid
» Session 19: Velocity and Acceleration
» Session 20: Velocity and Arc Length
» Session 21: Kepler’s Second Law
» Problem Set 3
» Practice Exam
» Session 22: Review of Topics
» Session 23: Review of Problems
» Exam
Next »
Unit 3 Introduction
This unit starts our study of integration of functions of several variables. To keep the visualization difficulties to a minimum we will only look at functions of two variables. (We will look at functions of three variables in the next unit.)
Our main objects of study will be two types of integrals:
- Double integrals, which are integrals over planar regions.
- Line or path integrals, which are integrals over curves.
All integrals can be thought of as a sum, technically a limit of Riemann sums, and these will be no exception. If you make sure you master this simple idea then you will find the applications and proofs involving these integrals to be straightforward.
We will conclude the unit by learning Green’s theorem which relates the two types of integrals and is a generalization of the Fundamental Theorem of Calculus. Along the way we will introduce the concepts of work and two dimensional flux and also two types of derivatives of vector valued functions of two variables, the curl and the divergence.
» Session 47: Definition of Double Integration
» Session 48: Examples of Double Integration
» Session 49: Exchanging the Order of Integration
» Session 50: Double Integrals in Polar Coordinates
» Session 51: Applications: Mass and Average Value
» Session 52: Applications: Moment of Inertia
» Session 53: Change of Variables
» Session 54: Example: Polar Coordinates
» Session 55: Example
» Problem Set 7
Part B: Vector Fields and Line Integrals
» Session 56: Vector Fields
» Session 57: Work and Line Integrals
» Session 58: Geometric Approach
» Session 59: Example: Line Integrals for Work
» Session 60: Fundamental Theorem for Line Integrals
» Session 61: Conservative Fields, Path Independence, Exact Differentials
» Session 62: Gradient Fields
» Session 63: Potential Functions
» Session 64: Curl
» Problem Set 8
» Session 65: Green’s Theorem
» Session 66: Curl(F) = 0 Implies Conservative
» Session 67: Proof of Green’s Theorem
» Session 68: Planimeter: Green’s Theorem and Area
» Session 69: Flux in 2D
» Session 70: Normal Form of Green’s Theorem
» Session 71: Extended Green’s Theorem: Boundaries with Multiple Pieces
» Session 72: Simply Connected Regions and Conservative
» Problem Set 9
Unit 4 Introduction
In our last unit we move up from two to three dimensions. Now we will have three main objects of study:
- Triple integrals over solid regions of space.
- Surface integrals over a 2D surface in space.
- Line integrals over a curve in space.
As before, the integrals can be thought of as sums and we will use this idea in applications and proofs.
We’ll see that there are analogs for both forms of Green’s theorem. The work form will become Stokes’ theorem and the flux form will become the divergence theorem (also known as Gauss’ theorem). To state these theorems we will need to learn the 3D versions of div and curl.
» Session 74: Triple Integrals: Rectangular and Cylindrical Coordinates
» Session 75: Applications and Examples
» Session 76: Spherical Coordinates
» Session 77: Triple Integrals in Spherical Coordinates
» Session 78: Applications: Gravitational Attraction
» Problem Set 10
Part B: Flux and the Divergence Theorem
» Session 79: Vector Fields in Space
» Session 80: Flux Through a Surface
» Session 81: Calculating Flux; Finding ndS
» Session 82: ndS for a Surface z = f(x, y)
» Session 83: Other Ways to Find ndS
» Session 84: Divergence Theorem
» Session 85: Physical Meaning of Flux; Del Notation
» Session 86: Proof of the Divergence Theorem
» Session 87: Diffusion Equation
» Problem Set 11
Part C: Line Integrals and Stokes’ Theorem
» Session 88: Line Integrals in Space
» Session 89: Gradient Fields and Potential Functions
» Session 90: Curl in 3D
» Session 91: Stokes’ Theorem
» Session 92: Proof of Stokes’ Theorem
» Session 93: Example
» Session 94: Simply Connected Regions; Topology
» Session 95: Stokes’ Theorem and Surface Independence
» Session 96: Summary of Multiple Integration
» Problem Set 12
» Practice Exam
» Exam
» Session 97: Curl and Physics
» Session 98: Maxwell’s Equations
» Session 99: Unit 1 Review
» Session 100: Unit 2 Review
» Session 101: Unit 3 Review
» Session 102: Unit 4 Review
Introduction
Prerequisites
The prerequisite to this course is 18.01 Single Variable Calculus.
Course Overview
This course covers vector and multi-variable calculus. At MIT it is labeled 18.02 and is the second semester in the MIT freshman calculus sequence. Topics include vectors and matrices, parametric curves, partial derivatives, double and triple integrals, and vector calculus in 2- and 3-space.
As its name suggests, multivariable calculus is the extension of calculus to more than one variable. That is, in single variable calculus you study functions of a single independent variable
y=f(x).
In multivariable calculus we study functions of two or more independent variables, e.g.,
z=f(x, y) or w=f(x, y, z).
These functions are interesting in their own right, but they are also essential for describing the physical world.
Many things depend on more than one independent variable. Here are just a few:
- In thermodynamics pressure depends on volume and temperature.
- In electricity and magnetism, the magnetic and electric fields are functions of the three space variables (x,y,z) and one time variable t.
- In economics, functions can depend on a large number of independent variables, e.g., a manufacturer’s cost might depend on the prices of 27 different commodities.
- In modeling fluid or heat flow the velocity field depends on position and time.
Single variable calculus is a highly geometric subject and multivariable calculus is the same, maybe even more so. In your calculus class you studied the graphs of functions y=f(x) and learned to relate derivatives and integrals to these graphs. In this course we will also study graphs and relate them to derivatives and integrals. One key difference is that more variables means more geometric dimensions. This makes visualization of graphs both harder and more rewarding and useful.
By the end of the course you will know how to differentiate and integrate functions of several variables. In single variable calculus the Fundamental Theorem of Calculus relates derivatives to integrals. We will see something similar in multivariable calculus and the capstone to the course will be the three theorems (Green’s, Stokes’ and Gauss’) that do this.
Course Goals
After completing this course, students should have developed a clear understanding of the fundamental concepts of multivariable calculus and a range of skills allowing them to work effectively with the concepts.
The basic concepts are:
- Derivatives as rates of change, computed as a limit of ratios
- Integrals as a ‘sum,’ computed as a limit of Riemann sums
The skills include:
- Fluency with vector operations, including vector proofs and the ability to translate back and forth among the various ways to describe geometric properties, namely, in pictures, in words, in vector notation, and in coordinate notation.
- Fluency with matrix algebra, including the ability to put systems of linear equation in matrix format and solve them using matrix multiplication and the matrix inverse.
- An understanding of a parametric curve as a trajectory described by a position vector; the ability to find parametric equations of a curve and to compute its velocity and acceleration vectors.
- A comprehensive understanding of the gradient, including its relationship to level curves (or surfaces), directional derivatives, and linear approximation.
- The ability to compute derivatives using the chain rule or total differentials.
- The ability to set up and solve optimization problems involving several variables, with or without constraints.
- An understanding of line integrals for work and flux, surface integrals for flux, general surface integrals and volume integrals. Also, an understanding of the physical interpretation of these integrals.
- The ability to set up and compute multiple integrals in rectangular, polar, cylindrical and spherical coordinates.
- The ability to change variables in multiple integrals.
- An understanding of the major theorems (Green’s, Stokes’, Gauss’) of the course and of some physical applications of these theorems.
Course Structure
This course, designed for independent study, has been organized to follow the sequence of topics covered in an MIT course on Multivariable Calculus. The content is organized into four major units:
- Vectors and Matrices
- Partial Derivatives
- Double Integrals and Line Integrals in the Plane
- Triple Integrals and Surface Integrals in 3-Space
Each unit has been further divided into parts (A, B, C, etc.), with each part containing a sequence of sessions. Because each session builds on knowledge from previous sessions, it is important to progress through the sessions in order. Each session covers an amount you might expect to complete in one sitting.
Within each unit you will be presented with sets of problems at strategic points, so you can test your understanding of the material. At the end of each unit, there is a comprehensive exam that covers all of the topics you learned in the unit.
MIT expects its students to spend about 150 hours on this course. More than half of that time is spent preparing for class and doing assignments. It’s difficult to estimate how long it will take you to complete the course, but you can probably expect to spend an hour or more working through each individual session.
Lecture Video
Most sessions include video clips from lectures of Professor Denis Auroux teaching 18.02 recorded live on the MIT campus in the fall of 2007. The video was carefully segmented by the developers of this OCW Scholar course to take you step-by-step through the content. The lecture video clips are accompanied by reproductions of the materials presented on chalkboards used during the class.
Recitation Video
This OCW Scholar course includes dozens of Recitation Videos – brief problem solving sessions taught by an experienced MIT Recitation Instructor – developed and recorded especially for you, the independent learner. Meet the recitation instructors and learn more about how to benefit from this help by watching their introductory video.
Readings, Activities and Exams
The readings in each session are adapted from materials that have been used for years to teach multivariable calculus at MIT. They provide a slightly different perspective on the material as well as a written record of the topics covered in lecture.
“Examples” present problems with their solutions. Take a minute to think about how you would solve each problem before you read the solution. Reading these solutions carefully will prepare you to solve similar problems on your own.
In some sessions you will have the opportunity to work with a Mathlet. These interactive learning tools will help you visualize multi-dimensional objects and observe their changes as you manipulate different variables.
“Problems and Solutions” let you test your mastery of the content of each session. Solve the problems on your own and check your work against the solutions before proceeding to the next session.
At the end of each Part is a Problem Set. As you start each part familiarize yourself with the problems in its problem set; this will enable you to work on each problem as you gain the knowledge you need to solve it. Once you have completed the problem set you can check your answers against the solutions provided. (The problem sets are carefully selected from a longer list of questions available to you. Do not hesitate to work any problem that piques your interest.)
Each unit ends in an exam. To prepare for an exam first work through the sample exam then check that you are proficient in each of the topics discussed in the exam review lecture. You should also review the solutions to any problems you found difficult in that unit. Allow yourself one hour to work each exam and three hours to complete the final. The exams are quite challenging; do not be surprised if you are unable to answer all of the questions correctly in the time allowed.
Textbook
This OCW Scholar course is self-contained and no textbook is required. If you have access to a multivariable calculus text it will probably serve as a useful companion to this course, although you might have to deal with slight differences in terminology and notation.
Technical Requirements
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