📂Functions

Identity Function

Definition¹

Given a set $X$, the following function $I_{X} : X \to X$ is called the identity function.

$$ I_{X}(x) = x,\quad \forall x \in X $$

Explanation

The following notations are commonly used.

$$ I,\quad \text{id},\quad \text{1} $$

Tangent vectors on a differentiable manifold are defined as follows in $\dfrac{d (f\circ \alpha)}{d t}$, where the function to be differentiated

$$ f \circ \alpha = f \circ I \circ \alpha = f \circ \mathbf{x} \circ \mathbf{x}^{-1} \circ \alpha $$

can be decomposed like this, allowing the tangent vector to be represented with respect to any coordinate system $\mathbf{x}$ while making it independent of the choice of the coordinate system.

Example

Identity Matrix

$$ I_{n\times n} = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix} $$