# Introduction and Prerequisites

Welcome to Multivariable Calculus! This course will move quickly, and cover a lot of ground. For this reason, the instructors personal email and phone number will be available to students. We will begin with a discussion of functions of multiple variables, including differentiation and integration, and move on to more complicated matters associated with these functions, such as optimization with constraints, surface integrals, and the basics of differential geometry. It will cover the same material as a University-level Calculus 3 course, but we will begin with a review of concepts from Calculus 2 that the student should already be familiar with.

It is essential that the student have firm grasp of concepts from Calculus 1. This includes, but is not limited to,

• Limits of functions of a single variable, including L'Hôpitals rule,
• Differentiation of a function of a single variable,
• Integration of a function of a single variable, including integration by parts,
• The fundamental theorem of calculus
• Analytic geometry of the plane, including arc length calculations
• Infinite series, including the classic convergence tests and methods of calculation

The problems are absolutely essential to success in the course. It is by doing exercises that the student gains an understanding of a mathematical concept, not by rote memorization of a definition. Additionally, many peripheral concepts are discussed in the problems which are not touched upon, or only briefly discussed, in the main body of the lecture. Students are encouraged to do all the review exercises and all the of the main exercises, but those students who find the review material unnecessary may instead choose to complete the challenging exercises.

# Course Outline

### Lecture 1

• Introduction and Course Outline
• Review: Single variable calculus
• Geometry in Three Dimensions
• Scalar Functions in Two and Three Dimensions

### Lecture 2

• Vectors, and Vector Functions
• Coordinate Systems
• Limits of Multivariate Functions
• Partial Derivatives
• The Jacobian
• Double and Triple Integrals

### Lecture 3

• Directional Derivatives
• Parametric Curves and Surfaces
• Extrema of Multivariate Functions
• Optimization with Constraints: Lagrange Multipliers

### Lecture 4

• Velocity and Acceleration
• Tangents and Normals
• Binormals, Curvature, and Torison
• Surface Normals

### Lecture 5

• Introduction to Integrals in Multivariable Calculus
• Double and Triple Integrals
• Line Integrals
• Surface Integrals
• Integral Applications

### Lecture 6

• The Divergence
• The Curl
• Applications of the Divergence and Curl

### Lecture 7

• Greens Theorem
• Gauss' Theorem
• Stokes Theorem
• Applications

### Lecture 8

• Applications: The Maxwell Equations

### Lecture 9

• Review: Ordinary Differential Equations
• Linear Partial Differential Equations

### Lecture 10

• The Heat Equation

### Lecture 11

• The Wave Equation

### Lecture 14

• Course Review

# Single Variable Review

A continuous function and two discontinuous functions. For more on the third example, see Essential Discontinuity

This section is not intended to be a course in elementary calculus - the student is expected to understand this material before beginning the course. It is provided here as a reference for student.

We begin with a review of functions. Functions, in the most general sense, are rules which associate one thing with another. For example, there is a function which takes a place on Earth and returns its temperature - in that case, the "domain" of the function is the Earth, and the "range" are temperatures. In elementary calculus, the domain and the range tend to be subsets of the real numbers. In this course, the domain and range may be real numbers, or sets of vectors in higher dimensions, or matrix sets.

Functions which take a real numbers, and output a real number, are said to be continuous if, basically, they are connected. At right are illustrations of continuous and discontinuous functions. A strict definition of continuity is usually saved for a course in Real Analysis.

The limit of a function is defined so that $\lim_{x\rightarrow x_0} f(x) = L \$ if, for any $\epsilon>0 \$, there is an $\delta>0 \$ such that every number within $\delta \$ of $x_0 \$ is taken by the function to a number within $\epsilon \$ of $L \$.

The derivative of a function is the limit $\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} \$ and is the slope of the line tangent to the function's curve at $x \$. To write the derivative of a function $f \$ with respect to $x \$, we write $\frac{d}{dx}f, f'(x), \$ or $f_x(x) \$.

Some special derivative bear reviewing:

$\frac{d}{dx} x^n = nx^{n-1},$

$\frac{d}{dx} e^x = e^x,$

$\frac{d}{dx} \sin(x) = \cos(x),$

$\frac{d}{dx} \cos(x) = -\sin(x),$

$\frac{d}{dx} \log(x) = x^{-1}$

$\frac{d}{dx} f(g(x)) = f'(g(x))g'(x)$

$\frac{d}{dx} f(x)g(x) = f'(x)g(x) + f(x)g'(x)$

The indefinite integral or antiderivative of a function $f(x) \$ is any function $g(x) \$ such that $\frac{d}{dx}g(x) = f(x) \$.

The definite integral is a very important concept for students wishing to learn multivariate calculus, as we define many different kinds of definite integrals in this course.

As the division of the interval grows finer, the Riemann sum approaches a limit.

Let $f(x) \$ be any function associating real numbers with real numbers, and let the closed interval $[a,b] \$ be in the domain of the function. Let's divide this interval into $n \$ smaller intervals, so that the first one is $[a,q_1] \$ and the last one is $[q_{n-1},b] \$, and for convenience we'll call $a = q_0, b=q_n \$. Furthermore, let's pick a point in each of these intervals, say $x_j \in [q_{n-1},q_n] \$. Now we form the sum

$\sum_{j=1}^n f(x_j) (q_j-q_{j-1}) \$

This is called a Riemann sum, named after the famous 19th century mathematician Bernhard Riemann. If this sum approaches a limit as $n \rightarrow \$$\infty \$ no matter how we choose our $q_j \$'s or our $x_j \$'s, then $f \$ is said to be Riemann integrable and that limit is called the definite integral of $f \$ from $a \$ to $b \$, and this is written $\int_a^b f(x)dx \$

The fundamental theorem of calculus states that $\int_{x_o}^x \frac{d}{dx}f(x)dx = f(x) \$.

The material just described should be so familiar to the student that it can be recited as easily as multiplication tables or trigonometric values on the unit circle.

# Geometry in Three Dimensions

The right-handed xyz coordinate system.

In three dimensions, we most often work with the xyz coordinate system. This system is called "right handed," because if we take our right palm, with our knuckles pointing in the positive x direction, and bend our fingers so that they point in the positive y direction, our thumbs automatically point in the positive z direction. See illustration.

We are familiar with the basic geometry of the plane already, but a brief review will help illustrate the concepts we encounter in the next dimension.

Any line in the xy plane can be described as $y=ax+b \$, where $a \$ is the "slope" and $b \$ is the y-intercept. When we raise the number of dimensions, we see how similar the equation of a plane is in three-space: $z=ax+by+c \$, where $a \$ is the slope along the x-axis, etc.

Describing a line in three dimensions is trickier, and requires what is called a parametric equation. This method of describing lines relies on some variable which isn't a coordinate, like time, to describe a point as it travels through space. For example, if a point is at some location $(x_0, y_0, z_0) \$ at time $t=0 \$, and then over 1 second travels $a \$ feet in the x direction, $b \$ feet in the y direction, and $c \$ feet in the z direction, we can describe its location at any time $t \$ as $(x_0+at, y_0+bt, z_0+ct) \$. As it travels, this point traces out a line.

Other geometric equations from two dimensions will find familiar analogues in three dimensions. For example, recall the formula for a circle centered at the origin in the plane, $x^2 + y^2 = r^2 \$, where $r \$ is the radius of the circle. It should come as no shock to learn the formula for a sphere in three dimensions is $x^2 + y^2 + z^2 = r^2 \$. Other familiar geometric curves from the plane find analogues in three dimensions as well: just as the shape described by $y=x^2 \$ in the plane is called a parabola, so the surface described by $z=x^2 + y^2 \$ is called a parabaloid. At this time, it is not necessary to discuss all of the various surfaces which appear often - the basic forms are below, and the exercises here and in future lectures will be sufficient to introduce the student to them.

Some common surfaces found in three dimensional space. In the equations given, multiplying any coordinate by a constant yields a similar surface.

# Scalar Functions

Scalar functions are the kinds of functions you're used to, but in more dimensions. The key property of a scalar function is that it returns a number. The example given earlier of temperature at a point on the Earth is a scalar function, because even though the inputs are not numbers (they are locations, e.g., New York), the output is a temperature (e.g., 35).

Frequently in multivariate calculus, we encounter functions which take as their inputs points in the plane or points in three dimensional space, but return numbers. These are scalar functions. A typical example would be a function which describes the pressure at any point in a fluid. However, a function which takes a point in a fluid and returns the direction of the flow at that point would NOT be scalar function because the information returned it not a number - we will examine these kinds of functions in the next lecture.

Level surfaces of a scalar function in three dimensions.

Scalar functions on the plane are easily visualized. For example, a scalar function on the plane which takes a point $(x,y) \$ and returns a number $z \$ can be visualized by constructing a surface consisting of points in three dimensional space $(x,y,z) \$. See illustration at right.

Scalar functions which take points in space as the input are not so easily visualized, but not less useful. For example, while we might visualize the temperature at a point on a hot plate by constructing a rubber surface above it which is raised at the hottest points, visualizing the temperature at points in our atmosphere, which differs at different altitudes, is a horse of another color, but absolutely essential to weather prediction and climatology.

One way of visualizing scalar functions in three dimensions is to construct level surfaces. Level surfaces are three-dimensional generalization of level curves, which are not unlike a topographic map. Level surfaces are a set of surfaces in space which are constructed so that at each point on the surface, the value of the scalar function is precisely the same as it is everywhere else on the surface. See illustration, right.

# Problems

## Review Problems

1. Using the definition of "limit" given above, show that $\lim_{x \rightarrow x_0} ax = x_0 \$, assuming $a \$ is a constant.

2. Under the same assumptions as above, show that $\lim_{x \rightarrow x_0} ax^n = ax_0^n \$

3. Use mathematical induction to show that $\lim_{x \rightarrow x_0} a_n x^n + ... + a_1 x + a_0 = a_n x_0^n + ... + a_0 \$.

You may assume that for any functions with valid limits $A, B, \$

$\lim_{x \rightarrow x_0} (A(x)+B(x)) = \lim_{x \rightarrow x_0} A(x)+ \lim_{x \rightarrow x_0} B(x) \$

and $\lim_{x \rightarrow x_0} (A(x)B(x)) = \lim_{x \rightarrow x_0} A(x) \times \lim_{x \rightarrow x_0} B(x) \$.

4. Using the definition of "limit" given above, find $\lim_{x \rightarrow 0} \frac{\sin(x)}{x} \$.

5. Using the limit rules given in question 3 and the definitions of derivative, prove

$\frac{d}{dx} (f(x)+g(x)) = f'(x) + g'(x) \$

and $\frac{d}{dx} x^n = nx^{n-1} \$

6. Find the derivative of $5+e^x-2ax^3 + \sin(x^2) \$, where $a \$ is a constant.

7. Give an antiderivative of $\log(x) + 3x^2 \$

8. Find $\int_0^1 \sqrt{1-x^2}dx \$. You may consult a table of integrals.

9. Find $\int_1^{e^{\pi-2}} \sin(2+\log(x)) dx \$

## Main Problems

1. What sort of objects are described by these equations:

$z=3x+\pi y - \frac{1}{7} \$

$y=z^2+2y^2 \$

$x=t, y=3t-1, z=t+2 \$

2. Given what you know about plane geometry, describe the shapes described by

$\frac{x^2}{4} + y^2 + \frac{z^2}{9} = 1 \$

$x^2 + {y^2 \over 16} - {z^2 \over 4} = 1 \$

$- {x^2 \over 9} - y^2 + z^2 = 1 \$

3. Consider the shape described by $z=-2x+\frac{y}{2} \$ and $x=t, y=\frac{t}{3}, z=-t \$. What are these two shapes? Do they intersect? Where?

4. Sketch some level curves in the plane of $f(x,y) = \frac{\cos(x^2 + y^2)}{2} \$. Sketch the surface $z=f(x,y) \$.

## Challenging Problems

Not only lines can be described with equations like $x=x(t), y=y(t), z=z(t) \$. When the coordinates are not linear functions of $t \$, these equations describe a curve. These kinds of equations are called parametric, because instead of describing shapes with only the coordinates $x,y,z \$, they use a parameter, in this case, $t \$. We'll discuss this more in later lectures.

1. Describe the curve $x=\cos(t), y=\sin(t), z=t \$. What shape does this curve have?

2. Curves are frequently encountered when two surfaces intersect. Do the surfaces $z=x^2-y^2 \$ and $x^2 + y^2 - z^2 = 1 \$ intersect? Can you give a parametric description of the curve of intersection??