Questions and discussion for this lecture live here. Fire away by hitting Reply below 
Can you please elaborate on the calculation of slopes here in case of shear strains?
Sure, the slopes beta_1 and beta_2 are the slopes of the inclined sides of the stress element (dashed red line), with respect to their respective axes (or the undeformed position of the side - dashed black line).
In each case, we’re using the standard definition of the slope of a line to define beta…i.e. the slope is the ‘rise over the run’, or in other words, the rate of change in height of the line (above the axis base line) with respect to distance along the line.
Does this help?
S