Tensor Analysis in Euclidean Space

Abstract

After briefly reviewing differentiability in $\mathbb {R}$, a generalization of differentiability based on the directional derivative in $\mathbb {R}^N$ is established. Gradients of scalar and vector fields are first discussed in $\mathbb {R}^N$ before adapting these concepts for general euclidean spaces in combination with global charts. Afterwards, nonlinear chart relations, also known as curvilinear coordinates, are examined, and the concept of tangent space at a point is introduced. In this context, the covariant derivative is derived, insinuating its character as special case of the covariant derivative for smooth manifolds. Aspects of integration based on differential forms are discussed together with the exterior derivative and Stoke’s theorem in $\mathbb {R}^N$.