Surfaces¶
Regular surfaces¶
defintion of regular surfaces
A subset \(S\subset \mathbb{E}^3\) is a regular surface if for each \(p\in S\), there exsits a neighborhood \(V\) in \(\mathbb{E}^3\) and a map \(\pmb{x}: U\rightarrow V\cap S\) of an open set \(U\in \mathbb{E}^2\) onto \(V\cap S\subset \mathbb{E}^3\) such that
(i) \(\pmb{x}\) is differentiable, i.e.
whose component functions have continuous partial derivatives of all orders in \(U\).
(ii) \(\pmb{x}\) is a homeomorphism.
(iii) regularity condition. For each \(q\in U\), the differential \(d\pmb{x}_q: \mathbb{E}^2\rightarrow \mathbb{E}^3\) is one-to-one.
Condition (i) is necessary if we want to do some geometric ananlysis on \(S\). The homeomorphism in Condition (ii) prevents the self-intersections in regular surfaces, otherwise it would induce ambiguous tangent plane at the intersection point. Consition (iii) guarantee the existence of a tangent plane at all points of \(S\). A more familiar form of condition (iii) is given as follows.
Interpretation of condition (iii)
Let us compute the matrix of the linear map \(d\pmb{x}_q\) in the canonical bases \(e_1, e_2\) of \(\mathbb{R}^2\) with coordinate \((u,v)\) and \(f_1, f_2, f_3\) of \(\mathbb{R}^3\) with coordinate \((x,y,z)\).
Let \(q=(u_0,v_0)\), then \(e_1=(1,0)\) is tangent to the curve \(u\mapsto (u, v_0)\) on \(\mathbb{R}^2\) whose image is \(u\mapsto (x(u,v_0), y(u, v_0), z(u,v_0))\) (This image curve is called the coordinate curve \(v=v_0\), or with ODE \(dv=0\)), which lies on \(S\) and has a tangent vector at \(\pmb{x}_q\)
Similarly, we have
So we could write the matrix of \(d\pmb{x}_q\)
condition (iii) requires the matrix to be full rank. Equivalently speaking, we need \(\frac{\partial \pmb{x}}{\partial u}\times \frac{\partial \pmb{x}}{\partial v}\neq 0\); or one of the minors of order \(2\) of the matrix of \(d\pmb{x}_q\), that is, one of the Jacobian determinants
does not vanish at \(q\).
Condition (iii) is also of great importance for \(\pmb{x}^{-1}\) to be a so-called differentiable function, that is, if we lift its range to three dimension, then the map is differentiable. Details could be found in Change of parameters.
Actually, it would be tiresome if we test all the three conditions one by one. The following theorems gives a cheaper method to the testing by utilizing the image of a multi-variable function.
Images¶
images implies regularity
If \(f:U\rightarrow \mathbb{E}\in C^1(U)\) where \(U\subset \mathbb{E}^2\) is an open set, then the graph of \(f\), viewed in \(\mathbb{E}^3\), i.e. the subset of \(\mathbb{E}^3\) given by \((x,y,f(x,y))\) for \((x,y)\in U\) is a regular surface.
Now we give some definitions about the following application.
regular point, critical point
Given a differentiable map \(F:U\subset \mathbb{E}^n\rightarrow \mathbb{E}^m\) where \(U\) is open, a point \(p\in U\) is called a critical point of \(F\) if the differential \(dF_p:\mathbb{E}^n\rightarrow \mathbb{E}^m\) is not a surjective mapping.
The image \(F(p)\in\mathbb{E}^m\) of a critical point is called the critical value of \(F\). A non-critical value of \(\mathbb{E}^m\) is called the regular value of \(F\).
The above terminology is inspired by a real-valued function of a real variable.
Now particularly we consider \(f:U\subset \mathbb{E}^3\rightarrow \mathbb{E}\), which takes \(m=1,n=3\). With a similar logic as we have in Interpretation of condition (iii), for canonical bases \(f_1,f_2,f_3\), we have the matrix form
In this case, \(df_p\) is not surjective at \(p\), iff \(f_x=f_y=f_z=0\) at \(p\).
regular surfaces by images
If \(f:U\subset \mathbb{E}^3\rightarrow \mathbb{E}\) is a differentiable function and \(a\in f(U)\) is a regular value of \(f\), then \(f^{-1}(a)\) is a regular surface in \(\mathbb{E}^3\).
Let \(p=(x_0,y_0,z_0)\) be a point of \(f^{-1}(a)\). Since \(a\) is a regular value of \(f\), we may assume without loss of generality that \(f_z\neq 0\) at \(p\). Then define a map like an image
and we indicate by \((u,v,t)\) the coordinates of a point in \(\mathbb{E}^3\) where \(F\) takes its values. The matrix of the differential map \(dF_p\) is given by
as we illustrated in special case for function on \(\mathbb{E}^3\). Whence \(det(dF_P)=f_z\neq 0\). We may apply the inverse function theorem, which guarantees the existence of neighborhood \(V\) of \(p\) and \(W\) of \(F(p)\) such that \(F:V\rightarrow W\) is invertible, and the inverse \(F^{-1}:W\rightarrow V\) is differentiable. It follows that
are differentiable. In particular, \(z=g(u,v,t=a)=h(x,y)\) is differentiable defined in the projection of \(V\) onto \(xy\) plane. Since
we conclude that the graph of \(h\) (i.e. \(z\)) is \(f^{-1}(a)\cap V\). By images implies regularity, we have \(f^{-1}(a)\cap V\) is a coordinate neighborhood of \(p\). Therefore, every point \(p\in f^{-1}(a)\) can be covered by a coordinate neighborhood and \(f^{-1}(a)\) is a regular surface.
\(\square\)
It would be good if readers could recall the implicit function theorem and its application -- inverse function theorem.
The following proposition shows that any regular surface is locally the graph of a differeniable function.
Find differentiable function using projection
Let \(S\subset \mathbb{E}^3\) is a regular surface, and \(p\in S\). Then there exsits a neighborhood \(V\) of \(p\) in \(S\) such that \(V\) is the graph of a differentiable function, which belongs to one of the three forms \(z=f(x,y), y=g(x,z), x=h(y,z)\).
Using projection and by inverse function theorem.
Using the above proposition, we claim that, for a regular surface, and any other parametrization \(\pmb{x}\), we do not need to test continuity of \(\pmb{x}^{-1}\), provided that the other conditions hold.
omit the test of continuity of inverse map
Let \(p\in S\) be a point of a regular surface \(S\) and \(\pmb{x}:U\subset \mathbb{E}^2\rightarrow \mathbb{E}^3\) is a parametrization with \(p\in \pmb{x}(U)\) such that condition 1 and 3 of defintion of regular surfaces hold. Assume \(\pmb{x}\) is one-to-one, then \(\pmb{x}^{-1}\) is continuous.
\(\square\)
Change of parameters¶
Differentiability of change of parameters
Let \(p\) be a point in a regular surface \(S\subset \mathbb{E}^3\), and two parametrizations \(\pmb{x}:U\subset \mathbb{E}^2\rightarrow \mathbb{E}^3\) and \(\pmb{y}:V\subset \mathbb{E}^2\rightarrow \mathbb{E}^3\), parametrized by \((u,v)\) and \((\xi, \eta)\), respectively. Suppose \(p\in \pmb{x}(U)\cap \pmb{y}(V)=W\). Then the change of parameters \(h=\pmb{x}^{-1}\circ \pmb{y}: \pmb{y}^{-1}(W)\rightarrow \pmb{x}^{-1}(W)\) is a diffeomorphism.
Utilizing the map
is a diffeomorphism by condition (iii). Restrict the map on a slice \(U\times {0}\) and \(F^{-1}\) is differentiable.
\(\square\)