Multivariable Calculus

Multivariable Calculus in a Nutshell

In a nutshell, multivariable calculus extends the familiar concepts of limits, derivatives, and integrals to functions with more than one independent variable.

Multivariable calculus is much more than just a repeat of single-variable calculus, however. It's a rich subject with its own unique puzzles and surprises. It introduces new tools that solve important problems in machine learning, neural networks, engineering, quantum computing, and astrophysics, to name just a few.

This course engages you with expertly designed problems, animations, and interactive three-dimensional visualizations all prepared to help you hone your multivariable calculus skills.

This unit, in particular, sets the stage for our first chapter, which provides a compact introduction to the essential ideas of multivariable calculus.

Multivariable Calculus in a Nutshell

Vectors play an essential role in multivariable calculus. For now, we can think of vectors as arrows in space, like the blue one in the 3D visualization below. A vector is defined by its direction and its length (or magnitude).

You have control over the vector's length \( l\) as well as the two angles \( \theta \) and \( \phi \) setting the vector's direction. Can you adjust these sliders so that the tip of the arrow sits exactly on the point in space?


Hint: By touch interaction, you can adjust your viewing perspective on the vector and the point. You can also zoom in and out. You might want to consider a perspective where the point is centered in the viewing screen to figure out \( \theta \) first.

               

Multivariable Calculus in a Nutshell

Later in the course, we'll discover that vectors can also be thought of as collections of numbers, making them ideal building blocks for multivariable functions.

For example, the point in the last problem sits in space and (as we'll learn) locating it requires three numbers called coordinates. The vector whose tip sits at the point can also be described with these same three numbers!

Spherical coordinates diagram taken from “Coordinates in 3D”

Spherical coordinates diagram taken from “Coordinates in 3D”

Looking at it from a different perspective, we can use the two angles specifying the direction of the vector and its length to locate the point in space. This is the essential idea behind spherical coordinates, a topic covered in detail in the Coordinates in 3D unit.

Multivariable Calculus in a Nutshell

Calculus truly is the mathematics of limits. Without limits, we couldn't define derivatives or integrals, the two pillars of our subject. This is true no matter how many independent variables we have.

A single-variable limit can often be done with the help of continuity. Mathematically, continuity at a point \( a \) means \[ \lim\limits_{x \to a } f(x) = f(a).\] Intuitively, it means that the graph of the function has no holes or jumps or breaks.

Later in the course, we'll learn precisely what it means to take a multivariable limit. We'll find continuity to be a huge help in this setting, too.

The graph below represents a function of two variables we'll soon encounter. Use only intuition to determine if this function is continuous everywhere or discontinuous at some points.

               

Multivariable Calculus in a Nutshell

The integral was originally designed to solve planar area problems. Similarly, multiple integrals are very useful in solving volume problems in higher dimensions.

We can start thinking about volumes of simple objects in higher dimensions even though we don't know how to integrate yet or even how to properly visualize them with our 3D minds. We can do this by analogy.

Spheres in \(n\) dimensions are characterized by a radius. The circle is the lowest dimensional sphere familiar to you. If it has radius \( r,\) its area is \( \pi r^2.\) Also, the sphere in 3D has volume \( \frac{4}{3} \pi r^3 \) if it has radius \( r.\)

Arguing by analogy, complete the statement

A sphere of radius \( r \) in \(n \) dimensions has volume proportional to \(\text{__________}.\)

               

Multivariable Calculus in a Nutshell

It may seem silly to consider volumes that are more than three-dimensional, but they play important roles in probability and physics where there could be thousands, millions, or even billions of variables.

Mathematically, if \( f(x) > 0 \) on \( [a,b],\) then the area between the graph and the lines \( y = 0, x =a, x = b \) is \[ \int\limits_{x=a}^{x=b} f(x)\, dx.\] When we generalize to multiple variables, we'll have an integral sign for each new variable, or dimension. For example, the \(n\)-dimensional sphere \[ x_{1}^2+ x_{2}^2 + \dots + x_{n}^2 = r^2 \] has volume \[ \int \ldots \int_{ x_{1}^2+ x_{2}^2 + \dots + x_{n}^2 \leq r } 1 \, dx_{1} \ldots\, dx_{n} \propto r^{n}. \] If this expression doesn't make sense yet, don't worry: it will soon! The upcoming 3D volumes unit will set us on the right path by introducing two-variable integrals through the Riemann sum.

Multivariable Calculus in a Nutshell

One of the greatest applications of calculus (specifically derivatives) is finding the maximum and minimum values of a function.

The upcoming Finding Extreme Values unit walks us through how this extends to a function with many variables. Before we get there, let's get a sense of what optimizing a function of two variables is like.

Let's say \( x \) and \( y \) are any two real numbers that obey the inequality \( x^2+y^2 \leq 1. \) Geometrically, this means that \( (x,y) \) sits inside (or on) the unit circle centered at the origin.

Let's also define the rule \( f(x,y) = 2+3x^2+3y^2, \) which outputs a single number for a pair of input values. For example, \[ f(\textcolor{red}{-1},\textcolor{blue}{2}) = 2 + 3 ( \textcolor{red}{-1})^2 + 3 ( \textcolor{blue}{2})^2 = 17.\] Select all of the options that apply to this function if we only consider input satisfying \( x^2+y^2 \leq 1. \)

               

Select one or more

Multivariable Calculus in a Nutshell

The example \[ f(x,y) = 2 + 3 \big(x^2+y^2 \big) , \ x^2+y^2 \leq 1 \] was chosen since it could be optimized without the help of multivariable calculus.

We'll encounter many new problems in our course where algebra and single-variable calculus simply won't be enough. Our next unit dives deeper into multivariable optimization. There, we'll uncover a powerful new tool and our first truly multivariable concept: the partial derivative.

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