Previously, the properties of the Lie group G, which is an equivalence group of the eikonal equation, wave equation, and other differential equations (DEs), have been studied by the author in the two-dimensional case; various applications to mathematical physics and differential geometry have been obtained. This paper presents a study of the three-dimensional analogue of the G group, the ten-parameter G ₁₀ group, which is a subgroup of the main equivalence group of the three-dimensional eikonal equation, acoustics equation, and other DEs. Its differential invariants (DIs) up to the third order and invariant differentiation operators (IDOs) were calculated. The geometric meaning of some DIs of the group G ₁₀ (the scalar curvature R of Riemannian space with the metric dl ² = n ²(x, y, z)(dx ² + dy ² + dz ²), its first and second Beltrami differential parameters Δ₁ u and Δ₂ u, and other quantities) and IDOs was found. An expression for R was derived in terms of other DIs of the group G ₁₀. To obtain this expression, and DIs and IDOs of the group G ₁₀, we use the geometric analogy with the two-dimensional case and differential and Riemannian geometry.

以前，作者已经在二维情况下研究了Lie群G的性质，该群是eikonal方程，波动方程和其他微分方程（DEs）的等价群。已经获得了数学物理和微分几何的各种应用。本文介绍了G组的三维类似物，十参数G ₁₀组的研究，G ₁₀组是三维电子方程，声学方程和其他DE的主要等价类的子组。计算了其三阶微分不变量（DI）和不变微分算子（IDO）。G ₁₀组中某些DI的几何含义（标量曲率ř的黎曼空间与所述度量DL ² = Ñ ²（X，Y，Z）（DX ² + DY ² + DZ ²），其第一和第二贝尔特拉米差分参数Δ ₁ ù和Δ ₂ ü，和其他数量）和IDO。根据组G ₁₀的其他DI得出R的表达式。为了获得该表达式以及组G _10的DI和IDO ，我们将几何比喻与二维情况以及微分和黎曼几何一起使用。