text: The scalar approach defines flux density as a scalar-valued function of a direction and sense in space prescribed by the investigator at a point prescribed by the investigator. Sometimes this approach is indicated by the use of the term hemispheric flux. For example, an investigator of thermal radiation, emitted from the material substance of the atmosphere, received at the surface of the earth, is interested in the vertical direction, and the downward sense in that direction. This investigator thinks of a unit area in a horizontal plane, surrounding the prescribed point. The investigator  wants to know the total power of all the radiation from the atmosphere above in every direction, propagating with a downward sense, received by that unit area. For the flux density scalar for the prescribed direction and sense, we may write
  where with the notation above,  indicates that the integration extends only over the solid angles of the relevant hemisphere, and  denotes the angle between  and the prescribed direction. The term  is needed on account of Lambert's law. Mathematically, the quantity  is not a vector because it is a positive scalar-valued function of the prescribed direction and sense, in this example, of the downward vertical. In this example, when the collected radiation is propagating in the downward sense, the detector is said to be "looking upwards". The measurement can be made directly with an instrument (such as a pyrgeometer) that collects the measured radiation all at once from all the directions of the imaginary hemisphere; in this case, Lambert-cosine-weighted integration of the spectral radiance (or specific intensity) is not performed mathematically after the measurement; the Lambert-cosine-weighted integration has been performed by the physical process of measurement itself.