I'm unsure whether this fits in topology or differential geometry.
I have a basic intuitive idea, and I'm trying to find either the appropriate definition or theorem relating to this idea within mathematics. It seems almost tautological.
Suppose we have some filled up shape S in the euclidean plane. It can be a square, circle or a blob of some kind.
Let P be some point on the boundary of S. The boundary of S is smooth at P.
The idea is this: If I take a curve Y located inside S and goes through P and is smooth at P, then the tangent to Y at P is coincident with the tangent to the boundary of S at P.
Firstly, is this idea true? And if so, what is the appropriate formal mathematical definition or theorem relating to this idea?
Thanks.
I'd say this fits in differential geometry/calculus. I think the assertion is true.
Assume that the boundary $\partial S$ of $S$ is smooth. Consider a point $P$ on $\partial S$. Rotate and translate $S$ such that $P=(0,0)$ and the tangent line is the $x$-axis. Since $\partial S$ is smooth, by the Implicit Function Theorem there exists an interval $U$ around $x=0$ and a smooth function $f\colon U\to \mathbb{R}$ such that the graph of $f$ is (locally) $\partial S$. We can take $U$ small enough such that points $(x,y)$ inside $S$ satisfy $x \in U$ and $y > f(x)$ (that is, the inside of $S$ lies above the graph of $f$).
Now consider a curve in $S$ that passes through $(0,0)$. We assume, by contraposition, that the tangent of the curve is not equal to the $x$-axis. Since the curve is smooth, by the Implicit Function Theorem again, there is a function $g$ such that $g(0)=0$ and $g'(0)\neq 0$. (Note: if the tangent line of the curve is the $y$-axis, IFT doesn't work, but one can proceed with a similar argument, I guess.)
Finally, by Taylor's theorem we have that
$$f(x)-g(x) = f(0)-g(0) + (f'(0)-g'(0))x + R(x) = -g'(0)x + R(x),$$
with $R(x) = O(x^2)$ for $x\to 0$. This means that $f(x)-g(x)$ is positive on one side of $x=0$ and negative on the other side (for $x$ small enough). Geometrically, this means that part of the curve lies outside $S$.