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.