The local kinetic energy and the closely related local electronic stress tensor are commonly used to elucidate chemical bonding patterns, especially for covalent bonds. We use three different approaches transformation properties of the stress tensor, quasiprobability distributions, and the virial theorem from density-functional theory to clarify the inherent ambiguity in these quantities, discussing the implications for analyses based on the local kinetic energy and stress tensor An expansive but not universal family of local kinetic energy forms that includes the most common choices and is suitable for both chemical-bonding and atoms-in-molecule analysis is derived. A family of local electronic stress tensors is also derived Several local kinetic energy functions that are mathematically justified, but unlikely to be conceptually useful, are derived The implications of these forms for atoms-in-molecule analysis are discussed