Vector Laplacian in Spherical Coordinates: An Unnoticed Typo in Landau and Lifshitz's Fluid Mechanics Course

A previously unaccounted fundamental typo has been discovered in Course of Theoretical Physics, vol. 6, by Landau and Lifshitz Fluid Mechanics, 1987, Pergamon, which corresponds to the same typo in the Russian original of this book. This concerns the first of the Navier-Stokes equations in spherical coordinates (15.21), which includes the radial component of the vector Laplacian, namely, an extra square in the denominator of the sine. This error migrates to secondary publications and can complicate theoretical studies related to the application of the Navier-Stokes equations, which the author of this note has encountered first-hand. This typo is not present in An Introduction to Fluid Dynamics by Batchelor. The present short article makes a detailed derivation of the radial component of the vector Laplacian from general principles to show what the corresponding Navier-Stokes equation should look like, and that the typo is indeed present. In addition, more compact equivalent forms of the Navier-Stokes equations in spherical coordinates are proposed.