Tangent Vector to a Curve and Arc Length / Differential Arc Length of a Curve

1. The Tangent Vector to a Curve at any Point on Curve gives the Instataneous Direction of Propagation of the Curve. Given a Curve in form of a Parametric Position Vector Function as
   
   \(\vec{R}=x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}\)   ...(1)
   
   where \(x\), \(y\) and \(z\) are Functions of a Single Variable Parameter \(t\) given as
   
   \(x=f_x(t)\),   \(y=f_y(t)\),   \(z=f_z(t)\)   ...(2)
   
   then the Tangent Vector to the Curve denoted by \(\vec{R\hspace{1mm}'}\), is given by the Derivative of the Position Vector Function \(\vec{R}\) with respect to the Parameter Variable \(t\) as follows
   
   \(\vec{R\hspace{.1cm}'}={\Large\frac{d \vec{R}}{dt}} = {\Large \frac{dx}{dt}}\hat{\mathbf{i}} + {\Large \frac{dy}{dt}}\hat{\mathbf{j}} + {\Large \frac{dz}{dt}}\hat{\mathbf{k}} = {\Large \frac{d (f_x(t))}{dt}}\hat{\mathbf{i}} + {\Large \frac{d (f_y(t))}{dt}}\hat{\mathbf{j}} + {\Large \frac{d (f_z(t))}{dt}}\hat{\mathbf{k}} \)   ...(3)
   
   If the Parameter Variable \(t\) represents Ellapsed Time, then the Tangent Vector to the Curve denoted by \(\vec{R\hspace{1mm}'}\) is also called the Instantaneous Velocity Vector.
2. The Arc Length of a Curve is given by the Length of the Curve between any 2 Points on the Curve. It is denoted by \(s\).
3. The Magnitude of the Tangent Vector \(\vec{R\hspace{1mm}'}\), gives the Derivative of Arc Length of Curve with repect to Parameter Variable \(t\) as follows
   
   \({\Large \frac{ds}{dt}} ={|\Large\frac{d \vec{R}}{dt}|} = | \vec{R\hspace{.1cm}'}|={\Large\sqrt{\frac{d \vec{R}}{dt} \cdot \frac{d \vec{R}}{dt}}} = \sqrt{\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}'}} = \sqrt{({\Large \frac{dx}{dt}}\hat{\mathbf{i}} + {\Large \frac{dy}{dt}}\hat{\mathbf{j}} + {\Large \frac{dz}{dt}}\hat{\mathbf{k}}) \cdot ({\Large \frac{dx}{dt}}\hat{\mathbf{i}} + {\Large \frac{dy}{dt}}\hat{\mathbf{j}} + {\Large \frac{dz}{dt}}\hat{\mathbf{k}})} =\sqrt{{\Large (\frac{dx}{dt})}^2 + {\Large (\frac{dy}{dt})}^2 + {\Large (\frac{dz}{dt})}^2}\)   ...(4)
4. Multiplying equation (3) with Differential of Parameter Variable \(t\) (denoted by \(dt\)), we get the Differential Change in Position Vector \(\vec{R}\) denoted by \(d\vec{R}\) as
   
   \(d\vec{R}= \vec{R\hspace{.1cm}'} dt= ({\Large \frac{dx}{dt}}\hat{\mathbf{i}} + {\Large \frac{dy}{dt}}\hat{\mathbf{j}} + {\Large \frac{dz}{dt}}\hat{\mathbf{k}})\hspace{2mm}dt = ({\Large\frac{d (f_x(t))}{dt}}\hat{\mathbf{i}} + {\Large \frac{d (f_y(t))}{dt}}\hat{\mathbf{j}} + {\Large \frac{d (f_z(t))}{dt}}\hat{\mathbf{k}})\hspace{2mm}dt\)   ...(5)
5. Multiplying equation (4) with Differential of Parameter Variable \(t\) (denoted by \(dt\)) gives the Differential of Arc Length of Curve (also called Line Element or Length Element or the Magnitude of the Differential Change in Position Vector \(\vec{R}\), \(d\vec{R}\)), denoted by \(ds\) as follows
   
   \(ds=|d\vec{R}|=\sqrt{d\vec{R} \cdot d\vec{R}}= |\vec{R\hspace{.1cm}'}| dt= \sqrt{({\Large \frac{dx}{dt}}\hat{\mathbf{i}} + {\Large \frac{dy}{dt}}\hat{\mathbf{j}} + {\Large \frac{dz}{dt}}\hat{\mathbf{k}}) \cdot ({\Large \frac{dx}{dt}}\hat{\mathbf{i}} + {\Large \frac{dy}{dt}}\hat{\mathbf{j}} + {\Large \frac{dz}{dt}}\hat{\mathbf{k}})}\hspace{2mm}dt =\sqrt{{\Large (\frac{dx}{dt})}^2 + {\Large (\frac{dy}{dt})}^2 + {\Large (\frac{dz}{dt})}^2}\hspace{2mm}dt\)   ...(6)
   
   Hence, the Differential of Arc Length of Curve is given by the Product of Magnitude of Tangent Vector Function and the Differential of Parameter Variable \(t\), \(dt\).
6. The actual Arc Length of a Curve \(s\) is found by Integrating the Function given in equation(6) between any 2 Values of Parameter Variable \(t\)
   
   \(s={\Large \int_{t1}^{t2}} ds= {\Large \int_{t1}^{t2}} \sqrt{{\Large (\frac{dx}{dt})}^2 + {\Large (\frac{dy}{dt})}^2 + {\Large (\frac{dz}{dt})}^2}\hspace{2mm}dt\)   ...(7)