Final answer:
To find the arc length of a vector-valued function, one must integrate the magnitude of the function's derivative over the given interval. The magnitude is determined by the square root of the sum of the squares of the derivatives of the components of the function. This calculation provides us with the total arc length of the curve.
Step-by-step explanation:
The given information seems to mix concepts from different areas of mathematics and physics, which might not apply directly to the calculation of the arc length of a vector-valued function. However, if we focus on the mathematically accurate process for finding the arc length of the curve represented by ➻r(t)=⟩f(t),g(t),h(t)⟫, we need to use calculus to derive the arc length formula for a parametric curve.
The formula for the arc length s of a vector-valued function ➻r(t) from t = a to t = b is given by:
s = ∫_a^b |➻r'(t)| dt,
where ➻r'(t) is the derivative of ➻r(t) with respect to t. By finding the magnitude of ➻r'(t), we are calculating the speed at which the point moves along the path defined by ➻r(t), and integrating this speed over the interval from a to b gives us the total arc length of the curve.
The magnitude of ➻r'(t) is calculated as the square root of the sum of the squares of its components' derivatives: |➻r'(t)| = √[(df/dt)² + (dg/dt)² + (dh/dt)²]. The integral of this quantity from a to b provides the arc length s.