In this problem, I know that the hypothesis of Green's theorem must ensure that the simple closed curve is smooth, but what is smooth? Could you give a definition and an intuitive explanation?
What does smooth curve mean?
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0Relevant:https://en.wikipedia.org/wiki/Smoothness – 2016-01-20
3 Answers
There are many ways you can characterize the smoothness of a curve.
Typically, people use the notation $C^{(n)}(\Omega)$ where $n \in \mathbb{N}$.
So when we say $f(x) \in C^{(n)}(\Omega)$, we mean that $f(x)$ has $n$ derivatives in the entire domain ($\Omega$ denotes the domain of the function) and the $n^{th}$ derivative of $f(x)$ is continuous i.e. $f^{n}(x)$ is continuous.
Also by convention, if $f(x)$ is just continuous, then we say $f(x) \in C^{(0)}(\Omega)$.
Also, $f(x) \in C^{(\infty)}$ if the function is differentiable any number of times. For instance, $e^{x} \in C^{(\infty)}$
An example to illustrate is to consider the following function $f: \mathbb{R} \rightarrow \mathbb{R}$. $f(x) = \begin{cases}0, &\mbox{if }x \leq 0 \\ x^2, &\mbox{if }x>0\end{cases}$
This function is in $C^{(1)}(\mathbb{R})$ but not in $C^{(2)}(\mathbb{R})$.
When the domain of the function is the largest set over which the function definition makes sense, we omit $\Omega$ and write that $f \in C^{(n)}$, the domain being understood as the largest set over which the function definition makes sense.
Note that $C^{(n)} \subseteq C^{(m)}$ whenever $n>m$.
EDIT:
In case of Green's theorem, when we apply the formula $\oint_c (L\,dx + M\,dy) = \iint_D \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right)\,dx\,dy$ we need $L,M \in C^1{(\Omega)}$, where $\Omega$ is a domain containing the curve and the interior of the curve viz $D$.
The simple closed curve $C$ should be piecewise smooth or more generally the curve $C$ should be in $C^{(0)}$.
We say that the curve $C$ is piecewise smooth curve when the two conditions below are satisfied:
(i) $C \in C^{(0)}$
(ii) The domain over which the curve is defined can be partitioned into disjoint subsets such that the curve is in $C^{(\infty)}$ (or sufficiently smooth i.e. the curve is in $C^{(n)}$ for some $n$ till which we are interested) over each of these subsets.
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0@Jichao: Yes... – 2010-11-27
Consider the following curve in the plane, $(x(t),y(t))$, this curve is called smooth if the functions $x(t)$ and $y(t)$ are smooth, which simply means that for all $N$, the derivatives $\frac{d^Nx}{dt^N}$ and $\frac{d^Ny}{dt^N}$ exist.
Smooth means differentiable (at least once). In other words, smooth means continuous and without "corners".
Edit: changed "edges" to "corners" after J.M.
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3@Matt: It depends on the author. Some authors define "smooth" to mean $C^\infty$, others define it to mean $C^1$, and still others define it to mean $C^1$ and has non-vanishing derivative. – 2011-01-31