  We introduce new techniques for calculations in Gauge theories with extended
supersymmetry. We are working in Projective Superspace where the $SU(2)$
R-symmetry is realized geometrically by including an auxilliary $\mathbb{CP}^1$
component in the superspace. Different gauge representations are associated
with different dependence on the $\mathbb{CP}^1$ coordinate $\zeta$ and using
contour integrals on $\mathbb{CP}^1$ we define natural projection operators on
these different representations which leads to elegant formulas for all
relevant objects. The new techniques lead to compact expressions for
lagrangians and field strengths in terms of the gauge prepotential but also to
effective ways of reducing superspace expressions to components, i.e. to write
them in terms of fields transforming covariantly only under a subgroup of the
supersymmetry group. We illustrate our findings in several examples in three
and four dimensions.
