more closely to the notation and language commonly employed in physics curriculum, touching only briefly on the more abstract connections. Motivation of the covariant derivative through gauge covariance requirement Consider a generic (possibly non-Abelian) gauge transformation acting on a component field . The main examples in field theory have a compact gauge group and we write the symmetry operator as where is an element of the Lie algebra associated with the Lie group of symmetry transformations, and can be expressed in terms of the hermitian generators of the Lie algebra (i.e. upto a factor , the infinitesimal generators of the gauge group), , as . It acts on the field as Now the partial derivative transforms, accordingly, as . Therefore, a kinetic term of the form in a Lagrangian is not invariant under gauge transformations. Definition of the gauge covariant derivative The root cause of the non gauge invariance is that in writing the field as a row vector or in index notation , we have implicitly made a choice of basis frame field i.e. a set of fields such that every field can be uniquely expressed as for functions (using Einstein summation), and assumed the frame fields are constant. Local (i.e. dependent) gauge invariance can be considered as invariance under the choice of frame. However, if one basis frame is as good as any gauge equivalent other one, we can not assume a frame fields to be constant without breaking local gauge symmetry. We can introduce the gauge covariant derivative as a a generalisation of the partial derivative that acts directly on the field rather than its components with respect to a choice of frame. A gauge covariant derivative is defined as an operator satisfying a product rule for every smooth function (this is the defining property of a connection). To go back to index notation we use the product rule . For a fixed , is a field, so can be expanded w.r.t. the frame field. Hence a gauge covariant derivative and frame field defines a (possibly non Abelian) gauge potential (the factor is conventional for compact gauge groups and is interpreted as a coupling constant). Conversely given the frame and a gauge potential , this uniquely defines the gauge covariant derivative. We then get . and with suppressed frame fields this gives in index notation which by abuse of notation is often written as . This is