1st Order Nonlinear PDE: Understanding Envelopes and Monge Cones

Question

1. I have a question about envelopes of surfaces. In a book I am reading the following:

   Suppose $S_a$ is a one parameter family of surfaces in $R^3$ given by $z=w(x,y;a)$ where $w$ depends smoothly on $x,y$ and the real parameter $a$. Consider also the equation $\partial_a w(x,y;a)=0$. For a fixed values of $a$, these two equations determine a curve $\gamma_a$. The envelope $E$ of the family of surfaces $S_a$ is just the union of these curves $\gamma_a$. The equation for $E$ is found simply by solving $\partial_a w(x,y,a)=0$ for $a$ as a function of $x$ and $y$, $a=f(x,y)$, and then substituting into $z=w(x,y,f(x,y))$. Moreover, along $\gamma_a$, $a$ is constant and we have $$dz = w_xdx + w_ydy \\0 = w_{ax}dx + w_{ay}dy$$ For instance, if $S_a$ is a one-parameter family of 2-spheres: $(x-a)^2+y^2+z^2 = 1$, then the envelope is a cylinder of radius 1.

Can anyone provide an "intuitive" geometric reason (read: has a picture in their head) for why taking the derivative with respect to the parameter, setting it equal to zero, and plugging it back into $F$ gives the envelope? I see that it works in the example of the sphere, I obtain a cylinder $y^2 + z^2 = 1$.

2. In the procedure descirbed above they use the notation $$dz = w_xdx + w_ydy \\0 = w_{ax}dx + w_{ay}dy$$ Is this a formal expression? When I read it as $$\frac{dz}{dt} = w_x\frac{dx}{dt} + w_y\frac{dy}{dt} \\0 = w_{ax}\frac{dx}{dt} + w_{ay}\frac{dy}{dt}$$ It makes sense to me (Namely they are ODEs valid on a characteristic curve parameterized by $t$). I looked this up and saw some stuff on cotangent spaces, but couldn't understand how it was related to the discussion above.

3. They introduce a notion of Monge Cone in the following way:

   Consider the 1st order PDE: $F(x,y,z,p,q)=0$. At any point $(x_0,y_0,z_0)$, $F$ establishes a functional relation between $p$ and $q$. Assuming $F_q(x_0,y_0,z_0,p,q)\neq 0$, implicit function theorem gives us: $F(x_0,y_0,z_0,p,q(p))=0$ for all $p$. The possible tangent planes to the graph $z=u(x,y)$ are given by: $$(z-z_0) = p(x-x_0)+q(p)(y-y_0)$$ which, as $p$ varies, describe a one-parameter family of planes through the point $(x_0,y_0,z_0)$.

Using the equations in #2, they solve for the envelope of planes at $(x_0,y_0,z_0)$ (parameter is $p$) and find: $$dz = pdx + qdy \\ 0 = dx + \frac{dq}{dp}dy$$

How do I see this is a cone at $(x_0,y_0,z_0)$?

Note: This is cross posted to Math Stackexchange - so go and collect bounty if you answer this!

Answer 1

1. A geometric interpretation of the envelope of a one-parameter family of surfaces is the following (slightly adapted from Wikipedia): a point on the envelope is a point in the intersection of two "adjacent" surfaces. If you don't mind infinitesimals this could be rephrased as: $(x_0,y_0,z_0)$ is a point on the envelope, if $(x_0,y_0,z_0)\in S_{a_0}\cap S_{a_0+da}$ (where $S_a$ denotes the surface and 0-subscripts denote values of the variables $x,y,z,a$.). But this condition is equivalent to the following: The values $(x_0,y_0,z_0)$ satisfy the equation $z=w(x,y;a)$ for a certain value $a_0$, and moreover, as we change $a$ slightly, keeping $x=x_0, y=y_0$ fixed, the value of $z$ does not change to first order. This is the condition $\frac{\partial w(x,y;a)}{\partial a}=0$.

2. I'm not sure what it means in mathematics for something to be formal or not. So I assume you are just expressing your discomfort with differentials $dz$, $dx$ etc. Unfortunately calculus books tend to discard them as "notation without meaning". The current mainstream approach to give them meaning, is to interpret them as differential forms in the sense of differential geometry. I suggest you look at any book on the topic. You would then interpret the variables $x,y,z$ as maps $x:\mathbb{R}^3\to \mathbb{R}$ etc. (And you will also be confronted with the cotangent space.) From this point of view the equations $dz=w_xdx+w_ydy$ and $0=w_{ax}dx+w_{ay}dy$ can be seen as the result of applying the de Rham differential to both sides of the equations $z=w(x,y;a)$ and $0=w_a(x,y;a)$, which (for fixed value of $a$) are just the defining equations of $\gamma_a$. Interpreting these differential forms as fields of covectors on $\mathbb{R}^3$, the system of these two equations can be seen as determining a field of one dimensional subspaces of the tangent space (the kernel of the covectors.). This then amounts to the same as your point of view, of parametrizing the curve $\gamma_a$ with a parameter $t$, and writing down the equations you wrote.

3. Since they are talking about a one parameter family of flat planes through a point, it should be geometrically clear, that the intersection of two adjacent planes is a line through $(x_0,y_0,z_0)$, hence the envelope is a union of such lines and is hence a cone.