Herein, we demonstrate that first-principles calculations can be used for mapping electronic properties of two-dimensional (2d) materials with respect to non-uniform strain. By investigating four representative single-layer 2d compounds with different symmetries and bonding characters, namely 2d-${MoS_2}$, phosphorene, ${alpha}$-Te, and ${beta}$-Te, we reveal that such a mapping can be an effective guidance for advanced strain engineering and development of strain-tunable nanoelectronics devices, including transistors, sensors, and photodetectors. Thus, we show that ${alpha}$-Te and ${beta}$-Te are considerably more elastic compared to the 2d compounds with strong chemical bonding. In case of ${beta}$-Te, the mapping uncovers an existence of curious regimes where non-uniform deformations allow to achieve unique localization of band edges in momentum space that cannot be realized under either uniform or uniaxial deformations. For all other systems, the strain mapping is shown to provide deeper insight into the known trends of band gap modulation and direct-indirect transitions under strain. Hence, we prove that the standard way of analyzing selected strain directions is insufficient for some 2d systems, and a more general mapping strategy should be employed instead.
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