In this paper the notions of vector field and differential form are extended to locally compact groups which are the inverse limit of Lie groups. This is done using Bruhat's definition of [unk]c∞ functions on such a group. Vector fields are defined as derivations on the [unk]c∞ functions. Then tangent vectors at a point are defined as elements of the inverse limit of the tangent spaces of the Lie groups. Tangent vectors then are put together to form vector fields, corresponding to a bundle definition, and the two notions are shown to be equivalent. Differential forms are defined using a bundle type definition from continuous linear functional on the tangent space. An existence and uniqueness theorem is proven for the exterior differential. Then an analog of the Poincaré lemma leads to the de Rham theorem relating the Cech cohomology with real coefficients to the cohomology of the differential forms.