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Published in:
2019 | OriginalPaper | Chapter
2. Vector and Tensor Analysis in Euclidean Space
Author : Mikhail Itskov
Published in: Tensor Algebra and Tensor Analysis for Engineers
Publisher: Springer International Publishing
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Abstract
In the following we consider a vector-valued function \(\varvec{x}\left( t\right) \) and a tensor-valued function \(\mathbf {A}\left( t\right) \) of a real variable t. Henceforth, we assume that these functions are continuous such that
for all \(t_0\) within the definition domain. The functions \(\varvec{x}\left( t\right) \) and \(\mathbf {A}\left( t\right) \) are called differentiable if the following limits exist and are finite. They are referred to as the derivatives of the vector- and tensor-valued functions \(\varvec{x}\left( t\right) \) and \(\mathbf {A}\left( t\right) \), respectively.
$$\begin{aligned} \lim \limits _{t \rightarrow t_0} \left[ \varvec{x}\left( t \right) - \varvec{x}\left( t_0 \right) \right] = \varvec{ 0 }, \quad \lim \limits _{t \rightarrow t_0} \left[ \mathbf {A}\left( t \right) - \mathbf {A}\left( {t_0 } \right) \right] = \mathbf {0} \end{aligned}$$
$$\begin{aligned} \frac{\mathrm{d}\varvec{x}}{\mathrm{d}t} = \lim \limits _{s \rightarrow 0} \frac{\varvec{x}\left( t + s\right) - \varvec{x}\left( t \right) }{s}, \quad \frac{\mathrm{d}\mathbf {A}}{\mathrm{d}t} = \lim \limits _{s \rightarrow 0} \frac{\mathbf {A}\left( t + s \right) - \mathbf {A}\left( t \right) }{s} \end{aligned}$$