How to write clear proofs involving curves and surfaces, when switching between different frames?

Question

I'm having trouble wtih the exposition to problems such as the one below, and I'd therefore like to request suggestions on how to improve it. The problem, my solution, and my struggles are below.

Problem

The surface of a football is the ellipsoid $4x^2 + y^2 + 4z^2 = 9$. A drop of water starts at $(1,1,1)$ and rolls down the path of steepest descent. Describe its path.

(Source: Shifrin's Multivariable Mathematics)

Solution

The drop takes the path $$p(t) = \begin{bmatrix}t^4 \\ t \\ \frac 1 2 {\sqrt{9 - 4t^8 - t^2}} \end{bmatrix}$$ with $t_0 =1$, for $t \geq 1, {\sqrt{9 - 4t^8 - t^2}} \geq 0$. I am assuming that when the drop reaches the widest part of the football, it stops or falls off, but doesn't continue rolling underneath.

Proof: Given surface $S$ defined by $z(x,y)$, any curve $\gamma: [a,b] \to \mathbb R ^3$ is a path of steepest descent of $S$ iff two conditions are met:

1. For all $t \in [a,b], \gamma(t) \in S$
2. For all $t \in [a,b]$, there exists an $m > 0$ such that $$(D \gamma(t))_x = m (\nabla z)_x \\ (D \gamma(t))_y = m (\nabla z)_y.$$

Since $p$ clearly meets condition 1, it suffices to show $p$ meets condition 2 as well.

For all $(x,y,z), z \in [0,1]$ on the ellipsoid's surface, we have $$z = \frac 1 2 {\sqrt{9 - 4x^2 - y^2}}$$ and there exists a function $g: \mathbb R^2 \to \mathbb R$ such that $$\nabla z = |g(x,y)| \begin{bmatrix}4x \\ y\end{bmatrix}.$$ Thus, curve $p$ will be a path of steepest descent iff $$Dp(t) = |h(t)|\begin{bmatrix}4(p(t))_x \\ (p(t))_y \\ \zeta(t)\end{bmatrix}$$ for some functions $h, \zeta: \mathbb R \to \mathbb R$.

To show $p$ meets this, reparameterize $p$ as $q(u) = p(e^u)$, so that $$q(u) = \begin{bmatrix}e^{4u} \\ e^u \\ \xi(u) \end{bmatrix}$$ for some $\xi: \mathbb R \to \mathbb R$. Therefore, $$Dq(u) = \begin{bmatrix} 4e^{4u} \\ e^u \\ \xi(u) \end{bmatrix} = \begin{bmatrix} 4(q(u))_x \\ (q(u))_y \\ \xi(u) \end{bmatrix}$$ and the proof is complete.

Struggles

I believe my solution correct and my proof rigorous. However, the exposition is difficult, largely because it involves looking at the curve and surface from shifting perspectives:

• At some points I want all three dimensions, but at some points only the projection to $(x,y)$. For example, I would have liked to simply write in condition 2 "$D \gamma(t) = m \nabla z$", but this would be incorrect, as $\gamma$ is three dimensional. I handled this by defining dummy functions $\zeta, \xi$ for $z$, and using subscripts $\cdot _x \cdot _y$, but this obfuscates the main point.
• At some points I need to use implicit definitions of the surface whereas at others I need a function $z(x,y)$.
• At some points I need the gradient whereas at others the derivative (Jacobian).

I tried many variations of the exposition, such as taking the projection of $p$ to the $xy$ plane, or using Leibnitz partial derivative notation; the best result is what I posted above. Still, the result seems to be hard to read, and so I request your help.

Please note that there are similar questions on math.SE asking for the direction of the drop at one particular point; this question, however, asks for the entire path. Moreover, my question is how to improve exposition and writing of this proof (not how to find a different proof).

Answer 1

This is a cool question!

I feel like there are several ways to look at it:

1.) Physics,

2.) Math,

3.) A combination

In terms of physics and point 1.) I believe you are correct on the drop falling off, but there may be other forces keeping it on past the "edge" of the surface. It depends on how you approximate the physics of water. For example when we say the drop will fall off the shape, we might be assuming that the ellipsoid is in the presence of a gravitational field (on Earth, Jupiter, etc.) for example.

In terms of Math 2.), is the "drop of water" completely constrained to the surface? I imagine the path of steepest descent takes on a different meaning depending on the outside forces. In the lack of these, and being constrained to the surface, the drop would never fall off, but this isn't exactly physical.

In terms of the combination of both 3.) I think the question could be refined and clarified better, but at the end of the day I also think it is fruitful to take both approaches. At least for this particular question, constraining the water to be on the sphere and calculating the path of steepest descent should be fine. In terms of this solution, the unboundedness of the y coordinate as a function of t seems particularly problematic. However, with the proper physical assumptions, it isn't necessarily bad. Several physics texts give good examples of particles constrained to surfaces. I suggest that you consult the problem of a particle constrained to a surface. Often these porblems also give you the force necessary to constrain the particle to the surface as well. Then when the gravitational force acting on the particle overcomes the force of constraint, that is when the particle will fall off the surface.

All in all, this should help you discuss the the entire path as you wanted. On the part you asked about on how to improve exposition and writing of this proof, stating the definition is always excellent. However, this seems more like a derivation than a proof. Explaining the derivation seems to make more sense than a proof. But, explaining how and why you answered the problem the way you did is good. The subscripts look a lot like partial derivatives. Elaborating on what your notation means is very good, unless it is already a well known notation. The column vector you presented for the curve is a widely used notation in differential geometry for curves. Finally, at one point in your proof it seems you are using what you want to prove, i.e. the "there exists a function g" all you need is to find one, and present it that matches it. Using there exists in this contexts seems to be overkill.

Hope this helps, and best of luck as you continue to learn more!