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Tangent Plane and Normal to the Tangent Plane of a Parametric Surface

1. Given a Surface in form of a Parametric Position Vector Function as
   
   \(\vec{R}=\vec{f(u,v)}=f_x(u,v)\hspace{1mm}\hat{\mathbf{i}} + f_y(u,v)\hspace{1mm}\hat{\mathbf{j}} + f_z(u,v)\hspace{1mm}\hat{\mathbf{k}}\)   ...(1)
   
   where \(f_x(u,v)\), \(f_y(u,v)\) and \(f_z(u,v)\) are Functions of 2 Variable Parameters \(u\) and \(v\), then the Tangent Plane of the Surface at any Point on the Surface is a Plane Containing 2 Non Parallel Vectors \(\vec{R_u}\) and \(\vec{R_v}\) which are Partial Derivatives of the Position Vector Function \(\vec{R}\) with respect to the Parameter Variables \(u\) and \(v\) respectively as follows
   
   \(\vec{R_u}={\Large\frac{\partial \vec{R}}{\partial u}}={\Large\frac{\partial (\vec{f(u,v)})}{\partial u}} = {\Large \frac{\partial (f_x(u,v))}{\partial u}}\hat{\mathbf{i}} + {\Large \frac{\partial (f_y(u,v))}{\partial u}}\hat{\mathbf{j}} + {\Large \frac{\partial (f_z(u,v))}{\partial u}}\hat{\mathbf{k}} =\vec{f_u(u,v)}= f_{xu}(u,v)\hspace{1mm}\hat{\mathbf{i}} + f_{yu}(u,v)\hspace{1mm}\hat{\mathbf{j}} + f_{zu}(u,v)\hspace{1mm}\hat{\mathbf{k}}\)   ...(2)
   
   \(\vec{R_v}={\Large\frac{\partial \vec{R}}{\partial v}}={\Large\frac{\partial (\vec{f(u,v)})}{\partial v}} = {\Large \frac{\partial (f_x(u,v))}{\partial v}}\hat{\mathbf{i}} + {\Large \frac{\partial (f_y(u,v))}{\partial v}}\hat{\mathbf{j}} + {\Large \frac{\partial (f_z(u,v))}{\partial v}}\hat{\mathbf{k}} =\vec{f_v(u,v)}= f_{xv}(u,v)\hspace{1mm}\hat{\mathbf{i}} + f_{yv}(u,v)\hspace{1mm}\hat{\mathbf{j}} + f_{zv}(u,v)\hspace{1mm}\hat{\mathbf{k}}\)   ...(3)
   
   Vectors \(\vec{R_u}\) and \(\vec{R_v}\) are called Tangent Vectors of Surface \(\vec{R}\).
2. Multiplying equation (2) and equation (3) with Differential of Parameter Variable \(u\) (denoted by \(du\)) and Differential of Parameter Variable \(v\) (denoted by \(dv\)) respectively, we get the Differential Change in Tangent Vectors \(\vec{R_u}\) (denoted by \(d\vec{R_u}\)) and \(\vec{R_v}\) (denoted by \(d\vec{R_v}\)) and as
   
   \(d\vec{R_u}={\Large\frac{\partial \vec{R}}{\partial u}}du=\vec{R_u}du\)   ...(4)
   
   \(d\vec{R_v}={\Large\frac{\partial \vec{R}}{\partial v}}dv=\vec{R_v}dv\)   ...(5)
   
   As per Total Differential Rule of Function of Multiple Variables, Sum of the Differential Change in Tangent Vectors \(\vec{R_u}\) and \(\vec{R_v}\) gives the Differential Change in Position Vector \(\vec{R}\) on the Surface denoted by \(d\vec{R}\) as follows
   
   \(d\vec{R} = d\vec{R_u} + d\vec{R_v}=\vec{R_u}du + \vec{R_v}dv\)   ...(6)
3. The Unit Vector \(\hat{R_n}\) Normal to the Tangent Plane at any Point of the Surface \(\vec{R}\) is given by the Unit Vector Corresponding to the Cross Product of Tangent Vectors \(\vec{R_u}\) and \(\vec{R_v}\) at that Point as follows
   
   \(\hat{R_n}= {\Large \frac{\vec{R_u} \times \vec{R_v}}{|\vec{R_u} \times \vec{R_v}|}}\)   ...(7)
4. The Rate of Change of Unit Vector Normal to the Tangent Plane at any Point of the Surface \(\vec{R}\) with respect to the Parameter Variables \(u\) and \(v\) is given by the Partial Derivatives of the Unit Normal Vector \(\hat{R_n}\) with respect to the Parameter Variables \(u\) and \(v\) respectively as follows
   
   \(\vec{R_{n_{u}}}={\Large\frac{\partial \hat{R_n}}{\partial u}}\)   ...(8)
   
   \(\vec{R_{n_{v}}}={\Large\frac{\partial \hat{R_n}}{\partial v}}\)   ...(9)
5. Multiplying equation (8) and equation (9) with Differential of Parameter Variable \(u\) (denoted by \(du\)) and Differential of Parameter Variable \(v\) (denoted by \(dv\)) respectively, we get the Differential Change in Vectors \(\vec{R_{n_{u}}}\) (denoted by \(d\vec{R_{n_{u}}}\)) and \(\vec{R_{n_{v}}}\) (denoted by \(d\vec{R_{n_{v}}}\)) and as
   
   \(d\vec{R_{n_{u}}}={\Large\frac{\partial \hat{R_n}}{\partial u}}du=\vec{R_{n_{u}}}du\)   ...(10)
   
   \(d\vec{R_{n_{v}}}={\Large\frac{\partial \hat{R_n}}{\partial v}}dv=\vec{R_{n_{v}}}dv\)   ...(11)
   
   As per Total Differential Rule of Function of Multiple Variables, Sum of the Differential Change in Vectors \(\vec{R_{n_{u}}}\) and \(\vec{R_{n_{v}}}\) gives the Differential Change in Unit Normal Vector \(\hat{R_n}\) on the Surface denoted by \(d\hat{R_n}\) as follows
   
   \(d\hat{R_n} = d\vec{R_{n_{u}}} + d\vec{R_{n_{v}}}=\vec{R_{n_{u}}}du + \vec{R_{n_{v}}}dv\)   ...(12)