Calculus I, II, and III: A Problem-Based Approach with Early Transcendentals:

“The only way of discovering the limits of the possible is to venture a little way past them into the impossible.” - Arthur C. Clarke

Welcome to calculus in three dimensions. The beauty of this material is how closely it parallels your first semester of calculus. After a brief introduction to the coordinate plane, you learned how to graph lines and parabolas. After a brief introduction to three-space, we will be graphing planes and paraboloids. Just as we defined continuity in terms of limits in Calculus I, we will define continuity of functions of several variables in terms of limits in this course. Lines were important as they allowed you to define tangent lines to functions and the derivative. Tangent planes to functions and surfaces will aid our definition of derivative. Once you understood the derivative, you used it to find maxima and minima of real valued functions and we will use the derivative of functions in three-space to find maxima and minima in our applications as well. Two of the most central ideas of your first calculus course were the chain rule and the fundamental theorem of calculus. We will extend your notion of the chain rule and the fundamental theorem in this course. Only near the end of the course will the work we do not have a parallel to your first course. At the end, we'll cover Green's and Gauss' Theorems, which are necessary tools in physics and engineering, but which had no parallel in Calculus I.

We will also use the notation, “\(\in\text{,}\)” to mean “is an element of.” Thus, “\(x \in \re\)” means “\(x\) is an element of \(\re,\)” or “x is a real number.” Similarly, \(x,y \in \re\) means “x and y are real numbers.” In Calculus I and II, you lived in two-space, or \(\re^2 = \{ (x,y) : x,y \in \re \}.\) Now you have graduated to 3-space!

Of course, there is no reason to stop with the number 3. More generally, \(n\)-space or \(\re^n\) is the set of all ordered sequences of \(n\) real numbers, but we will spend most of our time concerned with only \(\re, \re^2, \re^3,\) and occasionally, \(\re^4.\) Euclidean 4-space is handy since one might want to consider an object or shape in 3-space that is moving with respect to time, thus adding a \(4^{th}\) dimension.

We will write elements in \(\re^3\) just as letters; hence, by \(x \in \re^3\) we mean the element, \(x = (x_1, x_2, x_3)\) where \(x_1,x_2,x_3 \in \re.\) The origin is the element, \(o = (0,0,0).\)

Physicists and mathematicians often speak of a vector's magnitude and direction. Given a vector, \(\oa x,\) by magnitude (or norm) we mean the distance between the point, \(x,\) and the origin, \(o\text{.}\) By direction we mean the direction determined by the directed line segment \(\oa x\) that has base at \(o\) and tip at the point, \(x.\) When we say to “sketch the vector \(\oa x\)” we mean to draw the directed line segment from the origin to the point, \(x.\)

We are making a distinction between points and vectors. Points are the actual elements of 3-space and vectors are directed line segments. The word scalar will be used to refer to real numbers (and later in your mathematical career as complex numbers or elements of any field). The word, point may be used to mean a real number, an element of \(\re^2\text{,}\) an element of \(\re^3,\) etc.

Having carefully made clear the distinction between point and vector you will have to work hard to keep me honest; I tend to use the two more or less interchangeably.

Now, to be precise, we should also define vector addition and scalar multiplication for vectors. Since the definition is identical (except for placing arrows above the \(x\) and \(y\)) we omit this. As you can see, always making a distinction between a point and a vector can be cumbersome.

Consider the two vectors, \(\oa {xy}\) and \(\oa y - \oa x.\) Both vectors have the same direction and the same magnitude. They are different because the vector \(\oa y - \oa x\) has its base at the origin and its tip at the point \(y-x\) while \(\oa {xy}\) has its base at \(x\) and its tip at \(y.\)

We will be concerned primarily with vector valued functions where the range is \(\re^2\) or \(\re^3.\) If \(n=2\text{,}\) then such functions are also called planar curves and if \(n=3\) they are called space curves. We'll put a vector symbol, \(\oa{}\text{,}\) over such functions to remind us that the range is not a real number but a point in \({\re}^n.\)

In Calculus I and II you studied functions from \(\re\) to \(\re,\) for example \(f(x)=x^3,\) and functions from \(\re\) to \({\re}^2,\) for example \(\oa r(t)=(\cos(t),\sin(t))\text{.}\) Now we have added functions from \({\re}\) to \({\re}^3\) such as the examples in the previous few problems. We wish to add functions from \(\re^2\) to \(\re\) to our list next. Such functions are called real-valued functions of several variables.

1. real valued functions defined on \(\re\) \(f: \re \to \re\text{,}\) \(f(t) = t^2 + e^t\)

2. parametric curves (also called vector valued functions) defined on \(\re\)

   1. planar curves \(\oa f : \re \to \re^2\text{,}\) \(\oa f(t) = (e^{2t+1}, 3t+1)\)

   2. space curves \(\oa f : \re \to \re^3\text{,}\) \(\oa f(t) = (2t+1, (3t+1)^3, 4t+1)\)

3. real-valued functions of several variables, or multivariate functions

   1. real valued functions of two variables \(f: \re^2 \to \re\text{,}\) \(f(x,y) = x^2 + \sqrt{y}\)

   2. vector valued functions of three variables \(f: \re^3 \to \re^2\text{,}\) \(f(x,y,z) = (2xy, 3x + 4y^2)\)

   3. vector valued functions of several variables \(f: \re^n \to \re^m\) where \(m,n \ge 2\)

We defined the norm on vectors, but the same definition is valid for points in \(\re^3.\)

Again, we defined the dot product on vectors, but the same definition is valid for points in \(\re^3.\)

In the previous problem, the intersection of the graph with \(z=0, z=4, \mbox{ and } z=9\) are called level curves because each represents the path you would take if you walked around the graph always remaining at a certain height or level. Recall the Chain Rule from Calculus I.

Here is a reminder from Calculus II of the definition of arc length.

In the next problem the notation, \(| \cdot |,\) is used for both the absolute value (on the left side of the equation) and the norm (on the right side of the equation). Is this bad notation? Consider the definition for the norm, that

Suppose we take the norm of a vector in \(\re^1\text{,}\) such as \((x_1).\) Then,

Thus, the absolute value of \(x\) is the norm of \(x\) so you have been studying norms since high school (elementary school?) without knowing it!

The next result is known as the Triangle Inequality and it states essentially that the shortest distance between two points is the straight line. Look at a graph of \(\oa u, \oa v, \oa{u+v},\) and \(\oa{u(u+v)}.\) If you travel from the origin, along the vector \(\oa u\) and then along the vector \(\oa{u(u+v)}\text{,}\) then you have traveled further than if you traveled along the vector \(\oa{u+v}.\)