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Tangent/Velocity Vector, Acceleration Vector, Arc Length and Differential Arc Length of a Parametric Curve

1. Given a Curve in form of a Parametric Position Vector Function as
   
   \(\vec{R}=\vec{f(t)}=f_x(t)\hspace{1mm}\hat{\mathbf{i}} + f_y(t)\hspace{1mm}\hat{\mathbf{j}} + f_z(t)\hspace{1mm}\hat{\mathbf{k}}\)   ...(1)
   
   where \(f_x(t)\), \(f_y(t)\) and \(f_z(t)\) are Functions of a Single Variable Parameter \(t\), then the Tangent Vector of the Curve denoted by \(\vec{R\hspace{1mm}'}\) (also called Velocity Vector of the Curve denoted by \(\vec{V}\) ) is given by the Derivative of the Position Vector Function \(\vec{R}\) with respect to the Parameter Variable \(t\) as follows
   
   \(\vec{R\hspace{1mm}'}=\vec{V}={\Large\frac{d \vec{R}}{dt}}={\Large\frac{d (\vec{f(t)})}{dt}} = {\Large \frac{d (f_x(t))}{dt}}\hat{\mathbf{i}} + {\Large \frac{d (f_y(t))}{dt}}\hat{\mathbf{j}} + {\Large \frac{d (f_z(t))}{dt}}\hat{\mathbf{k}} =\vec{f\hspace{1mm}'(t)}= f_x\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{i}} + f_y\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{j}} + f_z\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{k}}\)   ...(2)
   
   The Tangent/Velocity Vector of a Curve at any Point on Curve gives the Instataneous Direction of Propagation of the Curve.
   
   The Derivative of the Tangent/Velocity Vector Function (\(\vec{R\hspace{1mm}'}\) or \(\vec{V}\) ) with respect to the Parameter Variable \(t\) (called the Acceleration Vector and denoted by \(\vec{R\hspace{1mm}''}\) or \(\vec{V\hspace{1mm}'}\) or \(\vec{A}\)) is given as
   
   \(\vec{R\hspace{1mm}''}=\vec{V\hspace{1mm}'}=\vec{A}={\Large\frac{d \vec{R\hspace{1mm}'}}{dt}}={\Large\frac{d \vec{V}}{dt}}={\Large\frac{d^2 \vec{R}}{dt^2}}={\Large\frac{d (\vec{f\hspace{1mm}'(t)})}{dt}} = {\Large \frac{d (f_x\hspace{1mm}'(t))}{dt}}\hat{\mathbf{i}} + {\Large \frac{d (f_y\hspace{1mm}'(t))}{dt}}\hat{\mathbf{j}} + {\Large \frac{d (f_z\hspace{1mm}'(t))}{dt}}\hat{\mathbf{k}} =\vec{f\hspace{1mm}''(t)}= f_x\hspace{1mm}''(t)\hspace{1mm}\hat{\mathbf{i}} + f_y\hspace{1mm}''(t)\hspace{1mm}\hat{\mathbf{j}} + f_z\hspace{1mm}''(t)\hspace{1mm}\hat{\mathbf{k}}\)   ...(3)
   
   If the Parameter Variable \(t\) represents Ellapsed Time, then the Position Vector Function \(\vec{R}\), Velocity Vector Function \(\vec{V}\) and Acceleration Vector Function \(\vec{A}\), give the Instantaneous Position, Velocity and Acceleration of an Object at time \(t\).
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
   
   \({\Large \frac{ds}{dt}} = | \vec{R\hspace{1mm}'}| = \sqrt{\vec{R\hspace{1mm}'} \cdot \vec{R\hspace{1mm}'}}= | \vec{f\hspace{1mm}'(t)}| = \sqrt{\vec{f\hspace{1mm}'(t)} \cdot \vec{f\hspace{1mm}'(t)}} = \sqrt{(f_x\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{i}} + f_y\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{j}} + f_z\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{k}}) \cdot (f_x\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{i}} + f_y\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{j}} + f_z\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{k}})}=\sqrt{f_x\hspace{1mm}'(t)\hspace{1mm}^2 + f_y\hspace{1mm}'(t)\hspace{1mm}^2 + f_z\hspace{1mm}'(t)\hspace{1mm}^2 }\)   ...(4)
4. Multiplying equation (2) 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
   
   \(d\vec{R}= \vec{R\hspace{1mm}'} dt= \vec{f\hspace{1mm}'(t)} dt= (f_x\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{i}} + f_y\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{j}} + f_z\hspace{1mm}'(t)\hspace{1mm}\hat{\mathbf{k}})\hspace{2mm}dt\)   ...(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), denoted by \(ds\) as follows
   
   \(ds= |\vec{R\hspace{1mm}'}| dt= |\vec{f\hspace{1mm}'(t)}| dt=\sqrt{f_x\hspace{1mm}'(t)\hspace{1mm}^2 + f_y\hspace{1mm}'(t)\hspace{1mm}^2 + f_z\hspace{1mm}'(t)\hspace{1mm}^2 }\hspace{2mm}dt\)   ...(6)
   
   Also, the Square of Differential of Arc Length of Curve \(ds\) is given by the Dot Product of Differential Change in Position Vector \(\vec{dR}\) with itself as follows
   
   \(ds^2= d\vec{R} \cdot d\vec{R} = \vec{R\hspace{1mm}'}dt\cdot \vec{R\hspace{1mm}'dt}\)   ...(7)
   
   \(\Rightarrow ds= \sqrt{(\vec{R\hspace{1mm}'}dt\cdot \vec{R\hspace{1mm}'}dt)}= \sqrt{(\vec{R\hspace{1mm}'}\cdot \vec{R\hspace{1mm}'})dt^2} = \sqrt{\vec{R\hspace{1mm}'}\cdot \vec{R\hspace{1mm}'}}dt = |\vec{R\hspace{1mm}'}| dt= |\vec{f\hspace{1mm}'(t)}| dt=\sqrt{f_x\hspace{1mm}'(t)\hspace{1mm}^2 + f_y\hspace{1mm}'(t)\hspace{1mm}^2 + f_z\hspace{1mm}'(t)\hspace{1mm}^2 }\hspace{2mm}dt\)   ...(8)
   
   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 \(s\) of a Curve is found by Integrating the Function given in equation(5) between any 2 Values \(t_1\) and \(t_2\) of Parameter Variable \(t\)
   
   \(s={\Large \int_{t1}^{t2}} ds= {\Large \int_{t1}^{t2}} \sqrt{f_x\hspace{1mm}'(t)\hspace{1mm}^2 + f_y\hspace{1mm}'(t)\hspace{1mm}^2 + f_z\hspace{1mm}'(t)\hspace{1mm}^2 }\hspace{2mm}dt\)   ...(9)