91Ó°ÊÓ

When we deal with vector functions, each component of the vector function can depend on a single parameter, usually denoted as \( t \). To find the derivative of a vector function, we simply take the derivative of each component with respect to \( t \).
Let's consider the function \( \alpha(t) = (\cosh t, \sinh t, t) \). The derivative, denoted by \( \alpha'(t) \), is formed by differentiating each component:

• The derivative of \( \cosh t \) is \( \sinh t \).
• The derivative of \( \sinh t \) is \( \cosh t \).
• The derivative of \( t \) is \( 1 \).

Thus, \( \alpha'(t) = (\sinh t, \cosh t, 1) \). For curves such as \( \boldsymbol{\beta}(t) = (\cos t, \sin t, t) \), simply apply the same component-wise differentiation, resulting in \( \boldsymbol{\beta}'(t) = (-\sin t, \cos t, 1) \).
This process allows us to analyze the velocity of a particle moving along a path described by the vector.

The norm, or magnitude, of a vector is a measure of its length or size. For a vector \( \mathbf{v} = (v_1, v_2, v_3) \), the norm is calculated as \( \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + v_3^2} \). This formula applies to any vector in 3D space, regardless of direction.
In the process of finding arc length, particularly for the derivative vectors \( \alpha'(t) = (\sinh t, \cosh t, 1) \) and \( \boldsymbol{\beta}'(t) = (-\sin t, \cos t, 1) \), we compute their norms:

• For \( \alpha \), use \( \sqrt{(\sinh t)^2 + (\cosh t)^2 + 1} \) and apply hyperbolic identities to simplify to \( \sqrt{2} \cosh t \).
• For \( \beta \), compute \( \sqrt{(-\sin t)^2 + (\cos t)^2 + 1} \), simplifying to \( \sqrt{2} \).

This norm calculation gives the instantaneous 'stretch' or 'rate of change' of position along the path, necessary for further calculations like finding the arc-length.

Integration of a vector function often involves calculating the area under its curve, analogous to accumulating change over time. This process gives us crucial insights, like arc-length, which represent the total distance traveled along the curve.
To find the arc-length \( s(t) \), we integrate the norm of the derivative vector from the starting point \( a \) to a particular time \( t \). For the curve \( \alpha(t) \), the integral becomes:\[ s(t) = \int_0^t \sqrt{2} \cosh \tau \, d\tau \]This calculates to \( s(t) = \sqrt{2} \sinh t \), representing the distance traveled from time 0 to \( t \) along \( \alpha \).
Similarly, for \( \beta(t) \), the integration is straightforward due to the constant norm value. Despite the rotational and linear components, the integral:\[ s(t) = \int_0^t \sqrt{2} \, d\tau = \sqrt{2}t \]gives the path distance, emphasizing that even uniform circular paths yield simple linear dependency when calculated for arc-length.