Section 1.1 Functions
This is a course about function, of course, since the solution of a differential equation is a function. This preliminary section is here to remind you of some definition and notations that will be useful in the course. First, there is some useful notation for sets of function on an interval.
Definition 1.1.1.
Let \(A\) be the \(\RR\) or an interval subset of \(\RR\text{.}\)
\(C(A)\) is the set of continuous real-valued function on \(A\text{.}\)
\(C^1(A)\) is the set of continuously differentiable functions on \(A\text{.}\)
\(C^n(A)\) is the set of functions on \(A\) with \(n\) continuous derivatives.
\(C^\infty(A)\) is the set of infinitly differentiable functions on \(A\text{.}\)
I cannot always expect continuous solutions to differential equations. However, almost all of the solutions in this course will be at least piecewise-continuous. That definition is worth reviewing.
Definition 1.1.2.
Let \(A\) be a subset of \(\RR\text{.}\) A function \(f:A
\rightarrow \RR\) is piecewise-continuous on \(A\) if it continous everywhere on \(A\) except for a set of isolated point. Likewise, \(f\) is piecewise-differentiable on \(A\) if it differentiable everywhere on \(A\) except for a set of isolated points. Note that \(A\) is the domain of the function: these piecewise functions can have jumps or break, but they are defined on all of \(A\text{.}\) They cannot have, for example, asymptotes at these isolated points.
There are many exponential functions in this course, since exponential functions solve many important differential equations. As in other calculus courses, I almost exclusively use the base \(e\) in this course, since it is the easiest base for derivatives and integrals. Since any other base \(a^t\) can be written \(a^t = e^{\ln a t}\text{,}\) I will very frequently work with functions of the form \(e^{\alpha t}\) for some \(\alpha \in
\RR\text{.}\) You can expect that typical exponential function to have this form.
In
Section 2.5, it will be necessary to talk about functions of two variables. The study of those functions is covered in Calculus III, but since that course is not prerequisite to this material, I should say a little bit about such functions. A function
\(F(x,y)\) of two variables is an expression in the two variables; alternatively, it is a model which depends on two different independent quantities.
Such function inherent many properties of ordinary function by treating each variable individually. That is, if I momentarily pretent that
\(y\) is just some constant, I can think of
\(F(x,y)\) as a single variable function of
\(x\text{;}\) and vice-versa when I think of
\(x\) momentarily as a constant. This perspective gives the two properties that I will need in this course.
Definition 1.1.3.
A function \(F(x,y)\) of two variables is continuous when it is continuous in each variable independently. Likewise, it is differentiable if is differentiable in each variable independently. Moreover, I can actually take derivatives this way. I can pretend that \(y\) is momentarily constant and differentiate in \(x\text{;}\) or vice versa when I pretend that \(x\) is momentarily constant and I differentiate in \(y\text{.}\) These derivatives are called partial derivatives. The notation for partial derivatives is an adjustment of the usual Leibniz notation.
\begin{align*}
\amp \frac{\del f}{\del x} \amp \amp \frac{\del f}{\del y}
\end{align*}
