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MATHEMATICS
S. V. BAKHVALOV
SOME GEOMETRIC PROPERTIES OF NOMOGRAPHABLE EQUATIONS
(Presented by Academician P. S. Aleksandrov, 15 III 1960)
Consider a function
\[ w=f(u,v), \tag{1} \]
satisfying the following conditions:
1) the function \(f(u,v)\) is defined for all values \(u,v\) in some neighborhood \(g\) of the point \(u_0,v_0\);
2) in \(g\) the function \(f(u,v)\) and its derivatives up to and including the second order are continuous;
3)
\[ \frac{\partial f(u,v)}{\partial v}\ne 0 \quad \text{for } u=u_0,\; v=v_0. \tag{2} \]
If, for the given function \(w=f(u,v)\), one can find a determinant
\[ \left| \begin{array}{ccc} A_1(u) & A_2(u) & A_3(u)\\ B_1(v) & B_2(v) & B_3(v)\\ C_1(w) & C_2(w) & C_3(w) \end{array} \right|\ne 0, \tag{3} \]
satisfying the condition
\[ \left| \begin{array}{ccc} A_1(u) & A_2(u) & A_3(u)\\ B_1(v) & B_2(v) & B_3(v)\\ C_1(f(u,v)) & C_2(f(u,v)) & C_3(f(u,v)) \end{array} \right|\equiv 0, \tag{4} \]
then the function \(w=f(u,v)\) is called nomographable. In this case, for equation (1) one can construct a nomogram of aligned points.
From (4) it follows that
\[ C_1(f)\,p_{23}(u,v)+C_2(f)\,p_{31}(u,v)+C_3(f)\,p_{12}(u,v)\equiv 0, \tag{5} \]
where
\[ p_{ik}= \left| \begin{array}{cc} A_i(u) & A_k(u)\\ B_i(v) & B_k(v) \end{array} \right| \]
and they satisfy the conditions
\[ \left| \begin{array}{ccc} p_{23} & p_{31} & p_{12}\\ \dfrac{\partial p_{23}}{\partial u} & \dfrac{\partial p_{31}}{\partial u} & \dfrac{\partial p_{12}}{\partial u}\\ \dfrac{\partial^2 p_{23}}{\partial u^2} & \dfrac{\partial^2 p_{31}}{\partial u^2} & \dfrac{\partial^2 p_{12}}{\partial u^2} \end{array} \right|\equiv 0; \tag{6} \]
\[ \left| \begin{array}{ccc} p_{23} & p_{31} & p_{12}\\ \dfrac{\partial p_{23}}{\partial v} & \dfrac{\partial p_{31}}{\partial v} & \dfrac{\partial p_{12}}{\partial v}\\ \dfrac{\partial^2 p_{23}}{\partial v^2} & \dfrac{\partial^2 p_{31}}{\partial v^2} & \dfrac{\partial^2 p_{12}}{\partial v^2} \end{array} \right|\equiv 0. \tag{7} \]
With the aid of the functions \(p_{ik}\), the functions \(A_i(u), B_k(v)\) are determined \((^{1,2})\).
Differentiating identities (5) with respect to \(u, v\), from the relation \(f'_u/f'_v\) we obtain
\[ C_1(f)\left[\frac{\partial p_{23}}{\partial u}f'_v-\frac{\partial p_{23}}{\partial v}f'_u\right] + C_2(f)\left[\frac{\partial p_{31}}{\partial u}f'_v-\frac{\partial p_{31}}{\partial v}f'_u\right] + C_3(f)\left[\frac{\partial p_{12}}{\partial u}f'_v-\frac{\partial p_{12}}{\partial v}f'_u\right]\equiv 0. \tag{8} \]
From (5) and (8) it follows that
\[ \frac{C_1(f)} { \left|\begin{array}{cc} \dfrac{\partial p_{31}}{\partial u} & \dfrac{\partial p_{12}}{\partial u}\\ p_{31} & p_{12} \end{array}\right| f'_v - \left|\begin{array}{cc} \dfrac{\partial p_{31}}{\partial v} & \dfrac{\partial p_{12}}{\partial v}\\ p_{31} & p_{12} \end{array}\right| f'_u } = \frac{C_2(f)} { \left|\begin{array}{cc} \dfrac{\partial p_{12}}{\partial u} & \dfrac{\partial p_{23}}{\partial u}\\ p_{12} & p_{23} \end{array}\right| f'_v - \left|\begin{array}{cc} \dfrac{\partial p_{12}}{\partial v} & \dfrac{\partial p_{23}}{\partial v}\\ p_{12} & p_{23} \end{array}\right| f'_u } = \]
\[ = \frac{C_3(f)} { \left|\begin{array}{cc} \dfrac{\partial p_{23}}{\partial u} & \dfrac{\partial p_{31}}{\partial u}\\ p_{23} & p_{31} \end{array}\right| f'_v - \left|\begin{array}{cc} \dfrac{\partial p_{23}}{\partial v} & \dfrac{\partial p_{31}}{\partial v}\\ p_{23} & p_{31} \end{array}\right| f'_u }. \tag{9} \]
It is easy to show that the functions \(P_{ik}=p_{ik}\lambda(u,v)\) satisfy conditions (6), (7). Define \(\lambda(u,v)\) so that
\[ P_{23}^{2}+P_{31}^{2}+P_{12}^{2}=1. \]
Then \(P_{ik}\) may be regarded as the coordinates of a point on a sphere whose radius is equal to 1 and whose center is at the origin of the coordinate system.
The vector \(\mathbf N=\{P_{23},P_{31},P_{12}\}\) is the unit normal vector to the sphere, and the coefficients \(D,D',D''\) of the second quadratic form of the sphere have the values
\[ D=-E,\qquad D'=-F,\qquad D''=-G, \]
where \(E,F,G\) are the coefficients of the first quadratic form of the sphere.
From (6), (7) it follows that
\[ \frac{\partial^{2}P_{ik}}{\partial u^{2}} = \lambda_1\frac{\partial P_{ik}}{\partial u} +\mu_1P_{ik} = \left\{\begin{array}{c} 11\\ 1 \end{array}\right\} \frac{\partial P_{ik}}{\partial u} - EP_{ik}; \]
\[ \frac{\partial^{2}P_{ik}}{\partial v^{2}} = \lambda_2\frac{\partial P_{ik}}{\partial v} +\mu_2P_{ik} = \left\{\begin{array}{c} 22\\ 2 \end{array}\right\} \frac{\partial P_{ik}}{\partial v} - GP_{ik}. \]
Thus,
\[ \left\{\begin{array}{c} 11\\ 2 \end{array}\right\} = \left\{\begin{array}{c} 22\\ 1 \end{array}\right\} =0. \tag{10} \]
Hence it follows that the lines \(u=\mathrm{const},\ v=\mathrm{const}\) are geodesic lines on the sphere.
Since
\[ \frac{C_1(f)}{C_3(f)} = \frac{ \left|\begin{array}{cc} \dfrac{\partial P_{31}}{\partial u} & \dfrac{\partial P_{12}}{\partial u}\\ P_{31} & P_{12} \end{array}\right| f'_v - \left|\begin{array}{cc} \dfrac{\partial P_{31}}{\partial v} & \dfrac{\partial P_{12}}{\partial v}\\ P_{31} & P_{12} \end{array}\right| f'_u } { \left|\begin{array}{cc} \dfrac{\partial P_{23}}{\partial u} & \dfrac{\partial P_{31}}{\partial u}\\ P_{23} & P_{31} \end{array}\right| f'_v - \left|\begin{array}{cc} \dfrac{\partial P_{23}}{\partial v} & \dfrac{\partial P_{31}}{\partial v}\\ P_{23} & P_{31} \end{array}\right| f'_u } =\delta_1, \]
\[ \frac{C_2(f)}{C_3(f)} = \frac{ \left|\begin{array}{cc} \dfrac{\partial P_{12}}{\partial u} & \dfrac{\partial P_{23}}{\partial u}\\ P_{12} & P_{23} \end{array}\right| f'_v - \left|\begin{array}{cc} \dfrac{\partial P_{12}}{\partial v} & \dfrac{\partial P_{23}}{\partial v}\\ P_{12} & P_{23} \end{array}\right| f'_u } { \left|\begin{array}{cc} \dfrac{\partial P_{23}}{\partial u} & \dfrac{\partial P_{31}}{\partial u}\\ P_{23} & P_{31} \end{array}\right| f'_v - \left|\begin{array}{cc} \dfrac{\partial P_{23}}{\partial v} & \dfrac{\partial P_{31}}{\partial v}\\ P_{23} & P_{31} \end{array}\right| f'_u } =\delta_2 \]
depends only on \(f(u,v)\), then from the conditions
\[ f'_v\frac{\partial}{\partial u}\delta_k - f'_u\frac{\partial \delta_k}{\partial v} =0 \qquad (k=1,2) \]
we obtain Gronwall’s condition
\[ M=\left(\left\{\begin{matrix}22\\ 2\end{matrix}\right\}-2\left\{\begin{matrix}12\\ 1\end{matrix}\right\}\right)f'_u+ \left(\left\{\begin{matrix}11\\ 1\end{matrix}\right\}-2\left\{\begin{matrix}12\\ 2\end{matrix}\right\}\right)f'_v, \tag{11} \]
where
\[ M=\frac{f''_{uu}f'^2_v-2f''_{uv}f'_u f'_v+f''_{vv}f'^2_u}{f'_u f'_v}. \tag{12} \]
From condition (5) we find that the vector
\[ \mathbf{C}(f)=\{C_1(f),\ C_2(f),\ C_3(f)\} \]
is situated in the tangent plane to the sphere.
At the points of the line \(f(u,v)=c\) the vector \(\mathbf{C}\) has a constant direction and is situated at each point of this line in the tangent plane to the sphere; consequently, the lines \(f(u,v)=c\) are arcs of great circles or geodesic lines on the sphere.
On the basis of conditions (2), from the equation \(f(u,v)=c\) we determine
\[ v=\sigma(u,c),\qquad \frac{dv}{du}=-\frac{f'_u}{f'_v} \]
along the geodesics. Substituting the values \(v=\sigma(u,c)\), \(dv/du=-f'_u/f'_v\) into the differential equation of the geodesics, we obtain
\[ M=\left(\left\{\begin{matrix}22\\ 2\end{matrix}\right\}-2\left\{\begin{matrix}12\\ 1\end{matrix}\right\}\right)f'_u+ \left(\left\{\begin{matrix}11\\ 1\end{matrix}\right\}-2\left\{\begin{matrix}12\\ 2\end{matrix}\right\}\right)f'_v; \tag{13} \]
\[ M=\frac{f''_{uu}f'^2_v-2f''_{uv}f'_u f'_v+f''_{vv}f'^2_u}{f'_u f'_v} \tag{14} \]
for \(v=\sigma(u,c)\).
Conditions (13), (14) are obtained from Gronwall’s conditions (11), (12) for \(v=\sigma(u,c)\). But since for any values \((u,v)\) one can determine \(c\) so that \(v=\sigma(u,c)\), conditions (13), (14) will hold for any values \((u,v)\).
Thus, Gronwall’s condition is the condition that any values \((u,v)\subset g\) and the derivative \(dv/du=-f'_u/f'_v\), determined from the equation \(f(u,v)=c\), satisfy the differential equation of the geodesics.
On the basis of the results obtained, one can construct a differential-geometric theory of nomographable equations.
The coefficients of the first quadratic form of the sphere satisfy Gronwall’s conditions (11), (12), the Gauss equation, and equations (10). The Peterson–Codazzi conditions are satisfied identically by virtue of conditions (10).
Moscow State University
named after M. V. Lomonosov
Received
11 III 1960
REFERENCES CITED
\(^{1}\) A. I. Mollaver, Nomographic Collection, Moscow–Leningrad, 1935, p. 47. \(^{2}\) S. V. Bakhvalov, Scientific Notes of the Moscow Regional Pedagogical Institute named after N. K. Krupskaya, 177 (1950). \(^{3}\) T. H. Gronwall, J. de Math. pures et appl., ser. 6, 8 (1912).