Does your interest in math extend beyond bland proofs and endless theory? Well, say goodbye to convoluted explanations and hello to clear examples and practice exercises. In Multivariable Calculus (Calculus 3), you’ll find a teaching method meant to give advanced learners strong intuition about how calculus really works.
Calculus in Three (or More) Dimensions
So far, we’ve kept it simple. You’ve got a simple function – one thing depends on another. But in nature, there are too many things that depend on two or more different variables. At times those variables might depend on each other. Even the derivatives of some variables may appear in equations. These situations lead to a slightly different set of tricks, like partial differentiation, the chain rule, Laplace transforms, and more. As with derivatives and integrals, necessity and innovation were connected. The applications with multiple variables, however, far outnumber the simpler situations. Calculus advanced at a dizzying pace at times!
Partial derivatives allow us to take one variable at a time, even when several variables may be involved. Finding the slope on a three dimensional surface might depend on which direction we plan to walk. A partial derivative allows you to look at how z changes as you move in the x direction, or in the y direction. In two dimensions, calculus offered easy methods to find the maximum and minimum values of a function. There are tricks that allow us to find these values, even when a function depends on more than one variable.
How does this affect the price of a cup of coffee? It doesn’t, perhaps, but it might help us in studying that price. Let’s say that the price of a cup of coffee depends on multiple things that change predictably over time. We could choose wages, bean supply, and world population. The model might be imperfect, but give it a chance. It might be possible to predict when coffee prices might be more or less stable over time using calculus in ways where simple substitution might fall short. Partial differentiation, and the methods that come with it help us solve more complex problems; and most of the problems facing us are complex!
- Multivariable Functions
- Continuous Function and Multivariable Limit
- Partial Derivatives
- Differentials and the Total Differential
- Chain Rule for Multivariable Functions
- Directional Derivatives and Gradient Vectors
- Normal Lines and Tangent Planes
- Absolute Extrema and the Extreme Value Theorem
- Lagrange Multipliers
Multiple integrals allow us to integrate functions in complex situations, just as partial derivatives and the rules that come with them allow us to understand change in complex situations. Just as integrating began with addition of geometric shapes to find area, multiple integrals have many varied applications to finding volumes, areas, and surface areas.
Just one of the many applications is the ability to calculate the moment of inertia. Inertia is a resistance to change in motion; the moment of inertia is a resistance to a change in rotation. This depends on the shape of an object, and how the mass is distributed in that shape. Even thinking of a figure skater spinning suggests how complex this might get. As they pull in their arms, we are adding up the masses at different distances and in different directions. Multiple integrals can help us find solutions and make predictions in these kinds of situations.
The figure skater example is just one of many where a different set of coordinates simplifies a problem. Polar and cylindrical coordinates will be introduced, for dealing with situations where $x$, $y$, and $z$ (cartesian coordinates) aren’t the simplest way to describe what’s going on.
- Double Integrals
- Fubini’s Theorem
- Double Integrals in Polar Coordinates
- Surface Area
- Triple Integrals
- Applications of Integrals
- Triple Integrals in Cylindrical and Spherical Coordinates
- Jacobian (Change of Variables in Multiple Integrals)
Vectors will be the final topic in multiple variable calculus. Vectors are quantities with a direction attached to them. Without calculus, we can work with vectors, adding up forces in different directions, and other simple tasks. Applying calculus to vector problems allows us to do more. Derivatives can give us the slope at a point. Vector calculus can give us the gradient – the direction in which the slope is steepest, in addition to the value of that slope.
Doctors making artificial hearts can model blood flow with vector calculus. Divergence is the quantity being calculated here. We add vectors from all directions, using calculus methods, and determine where blood will flow. We also encounter curl, a measure of how vector quantities circulate around a point. Fluid mechanics puts this to use – just think of modeling a hurricane. Electromagnetism puts it to use as well, as circling electrical charges create magnetic fields.
Line integrals will close out this set of articles. A line integral can be applied to a curve, one of Newton’s “crooked lines” to calculate work. Integrating is repeated addition. A line integral adds up the little bits of work done as a particle moves along each tiny segment of the curve. If it helps you to think of the work ahead of you broken down into very small, manageable pieces, then keep this image fresh in your mind. It will add up, for sure.
On that note, let’s get to work!
- Vector Functions and Space Curves
- More Vector Differentiation
- Vector Integration
- Velocity and Acceleration
- Arc Length and Curvature of Smooth Curves
- Vector Fields
- Divergence and Curl of a Vector Field
- Line Integrals
- Conservative Vector Fields and Independence of Path
- Green’s Theorem
Math and Society
If you’ve gotten this far in the field of calculus, you probably know how important it is for society. It’s even a key part of the computer algorithms in the search engines that might have led you here. From these algorithms to public health studies, architecture, and more, life would be very different for all of us if it weren’t for calculus. In Multivariable Calculus (Calculus 3 by Example) you’ll get one step closer to understanding the many methods and applications of the field.
Calculus 3: What’s Inside
Chapter 1 of the Multivariable Calculus book starts with a review of materials found in previous books in the series. Material from Calculus 1 in particular is a big focus in this chapter, and is of great importance for keeping up in the rest of the book. If you’re not completely comfortable with the concepts of derivatives and integrals found in Calculus 1, be sure to brush up on them here. Beyond the review of Calculus 1 topics, the first chapter also applies these topics to vector functions and space curves.
Chapter 2 of Multivariable Calculus then takes a look at partial differentiation. It includes sections on multivariable functions, limits and continuity, the chain rule, and related topics, finishing off with Lagrange multipliers. In Chapter 3 you’ll learn about multiple integrations, the differences between double and iterated integrals, and change of variables. This section is very thorough and includes integrals in various coordinate systems. These include the typical Cartesian (rectangular) system as well as the spherical and cylindrical systems.
The fourth and last chapter of Multivariable Calculus delves into vector calculus and vector fields. Similar to material from Chapter 1, vector calculus involves inputting numbers to get an output. But instead of putting in one number and getting a vector, you input multiple numbers to get several vectors. You’ll also find important theorems in this chapter such as Green’s Theorem, and of course the crucial Fundamental Theorem of Line Integrals. Also known as the Gradient Theorem, this theorem has a powerful role in calculus.
Always Lead by Example
Because math can be abstract and hard to mentally picture, it helps to have visual guides and hands-on practice. Thus, each of these chapters in Multivariable Calculus covers applications of the topics and provides thorough explanations. They also have many detailed diagrams as well as examples to aid comprehension. You’ll often find that typical Calculus 3 textbooks don’t always contain these types of helpful tools. So, this book serves well as a supplement for classroom texts that are heavy in proofs and theory. If used together with drier classroom materials, or even as material for those doing independent study, it will help students gain a more in-depth understanding of the ideas.
What You’ll Learn in Vector-Valued Functions
- Curves Defined by Vector Functions
- Limits and Continuity
- The Derivative of a Vector Function
- Higher-Order Derivatives
- Rules of Differentiation
- Integration of Vector Functions
- Arc Length
- Smooth Curves
- Arc Length Parameter
- Radius of Curvature
- Velocity, Acceleration, and Speed
- Motion of a Projectile
- The Unit Normal
- Tangential and Normal Components of Acceleration
- Derivation of Kepler’s First Law
- Multiplication and Division of Complex Numbers
In this course, we begin by studying vector-valued functions. These are functions of one variable whose output is a vector. We cover the calculus of vector-valued functions by considering limits, continuity, differentiation, and integration of vector functions.
After that introduction to vector functions, we examine the distance traveled along the graph of a vector function (arc length) and curvature. Next, we model motion along the path of a vector function in three dimensions. We study velocity, acceleration, and speed. This modeling leads us to a discussion of the tangential and normal components of acceleration. In particular, we examine the unit normal and unit tangent vectors of a vector function.
In the end, we discuss Kepler’s Laws of planetary motion.
What You’ll Learn in Functions of Several Variables
- Functions of Two Variables
- Graphs of Functions of Two Variables
- Computer Graphics
- Level Curves
- Functions of Three Variables
- Level Surfaces
- An Intuitive Definition of a Limit
- Continuity of a Function of Two Variables
- Continuity on a Set
- Functions of Three or More Variables
- A Formal Definition of a Limit
- Partial Derivatives with Two Variables
- Computing Partial Derivatives
- Implicit Differentiation
- Partial Derivatives with More Variables
- Higher-Order Derivatives
- The Total Differential
- Error in Approximating
- Differentiability with Two Variables
- Differentiability and Continuity
- Functions of Three or More Variables
- The Chain Rule for Functions Involving One Independent Variable
- The Chain Rule for Functions Involving Two Independent Variables
- The General Chain Rule
- Implicit Differentiation
- The Gradient of a Function of Two Variables
- Properties of the Gradient
- Geometric Interpretation of the Gradient
- Tangent Planes and Normal Lines
- Using the Tangent Plane to Approximate a Surface
- Relative and Absolute Extrema
- Critical Points: Candidates for Relative Extrema
- The Second Derivative Test for Relative Extrema
- Finding the Absolute Extremum Values of a Continuous Function on a Closed Set
- Constrained Maxima and Minima
- The Method of Lagrange Multipliers
- Optimizing a Function Subject to Two Constraints
In this exciting course, we study functions of several variables. For most students taking a calculus course, this is their first look at functions of more than one variable. So before we begin with the calculus, we explain what multivariable functions are and how to graph them using level curves, level surfaces, and computer graphics. After that, we introduce calculus. We give a thorough introduction to the limits of multivariable functions and discuss continuity as well.
One of the most exciting parts of calculus is learning partial derivatives. We illustrate what partial derivatives are both formally and graphically. We then work through many examples, including implicit differentiation, higher-order partial derivatives, and Clairaut’s Theorem. After that, we examine differentials and the total derivative. These topics lead us to understand what differentiability means for multivariable functions; in particular, we discuss tangent planes.
Now enters the chain rule for multivariable functions. We work through different stages and go through several examples demonstrating how the chain rule works, including the Implicit Differentiation Theorem. Next, partial derivatives are generalized by studying the directional derivative and gradient vectors. This study includes a discussion using the tangent plane to approximate a surface locally.
Towards the end of the course, we explore some applications, including finding the extrema of multivariable functions and using Lagrange multipliers. In both cases, we work through many examples and provide formal verification of the theorems.
What You’ll Learn in Multiple Integrals
- An Introductory Example
- Double Integrals Over General Regions
- Volume of a Solid Between a Surface and a Rectangle
- The Double Integral Over a Rectangular Region
- Properties of Double Integrals
- Iterated Integrals Over Rectangular Regions
- Fubini’s Theorem for Rectangular Regions
- Iterated Integrals Over Nonrectangular Regions
- Polar Rectangles
- Double Integrals Over Polar Rectangles
- Double Integrals Over General Regions
- Mass of a Lamina
- Moments and Center of Mass of a Lamina
- Moments of Inertia
- Radius of Gyration of a Lamina
- Area of a Surface
- Area of Surfaces with Equations
- Triple Integrals Over a Rectangular Box
- Triple Integrals Over General Bounded Regions in Space
- Evaluating Triple Integrals Over General Regions
- Volume, Mass, Center of Mass, and Moments of Inertia
- Triple Integrals in Cylindrical Coordinates
- Triple Integrals in Spherical Coordinates
- Change of Variables in Double Integrals
- Change of Variables in Triple Integrals
We begin this course with a thorough introduction to double integrals. This approach closely resembles the definite integral from calculus 1. We then discuss double integrals over rectangular regions and work through several examples. At the end of this first lesson, we scrutinize the properties of the double integral.
As we have seen in the first lesson, evaluating double integrals can be cumbersome. So, in the next lesson, we study iterated integrals. We explain Fubini’s Theorem for rectangular and nonrectangular regions and demonstrate when this theorem fails.
Double integrals in polar coordinates are then studied as well as some applications of double integrals. We examine the mass, the moments and center of mass, and the gyration radius of a lamina. We also apply our new knowledge of double integrals to finding surface area, including parametrically.
Next, we begin to explore triple integrals. We motivate their meaning, formalize their definition, and work through examples. We generalize Fubini’s theorem and work through some of the applications concerning lamina again, but with an additional variable. We also discuss probability density functions.
As we studied double integrals in polar coordinates, we now learn triple integrals in cylindrical and spherical coordinates. For both cases, we introduce the coordinate system and then work through several examples. In the exciting conclusion to this course, we explore changing the variables of integration in multiple integrals. This technique involves producing a coordinate system. The strategies and processes involved are detailed.
What You’ll Learn in Vector Analysis
- Vector Field in Two-Dimensional Space
- Vector Field in Three-Dimensional Space
- Conservative Vector Fields
- Line Integrals
- Line Integrals w.r.t Coordinate Variables
- Line Integrals in Space
- Line Integrals of Vector Fields
- Work Done on a Particle by a GF
- Fundamental Theorem for Line Integrals
- Line Integrals Along Closed Paths
- Independence of Path and Conservative Vector Fields
- Determining Whether a Vector Field Is Conservative
- Finding a Potential Function
- Conservation of Energy
- Green’s Theorem for Simple Regions
- Green’s Theorem for More General Regions
- Why We Use Parametric Surfaces
- Finding Parametric Representations of Surfaces
- Tangent Planes to Parametric Surfaces
- Area of a Parametric Surface
- Surface Integrals of Scalar Fields
- Oriented Surfaces
- Surface Integrals of Vector Fields
- Parametric Surfaces
- The Divergence Theorem
- Interpretation of Divergence
- Understanding Stokes’ Theorem
- Interpretation of Curl
- Summary of Line and Surface Integrals
- Summary of Major Theorems Involving Line Integrals and Surface Integrals
This course is the exciting conclusion to the calculus 1, calculus 2, and calculus 3 subjects. Here we bring together limits and continuity and the theories of differentiation and integration into some fantastic applications.
We begin moving from vector functions of one variable to vector functions of several variables, which we call vector fields. We study different types of vector fields and operations on them, including the divergence, curl, and gradient of a vector field.
Next, we have a detailed examination of line integrals and then surface integrals. In both cases, we motivate the underlying concepts and work through many theorems and examples. In particular, we derive the Law of Conservation of Energy.
After a detailed study of parametric surfaces, we study surface integrals of vector fields. In the end, we provide an overwhelming and thorough summary of the beauty that lies in the Fundamental Theorem of Line Integrals, Green’s Theorem, Divergence Theorem, and Stoke’s Theorem. Finally, we elaborate on and illustrate Maxwell’s equations for electromagnetism.
Gain expertise in Calculus 3 with these articles and videos that explain everything (step-by-step) on calculus in three dimensions.
Calculus 3 is a high-school or undergraduate mathematics course. This course is the third in a series that introduces students to calculus and usually begins with vector functions and functions of several variables. After that, instructors consider limits and continuity of functions. Then students study partial derivatives and multiple integrals. In particular, Lagrange multipliers and integration using a transformation between coordinate systems are favorites. Finally, in the end, students explore the most beautiful theorems in all of calculus, including Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem.