Unit normal vector
In geometry, a normal is an object that is perpendicular to a given object. Given an arbitrary surface \(S\), we can construct two non-collinear vectors \(\mathbf{u}\) and \(\mathbf{v}\) tangent to \(S\) at some point \(P\) (Schey 2005). Their cross product \(\mathbf{u}\times\mathbf{v}\) is by definition normal to both vectors and thus normal to \(S\) at \(P\). To make this a unit vector, we simply divide the normal vector by its magnitude: \[\begin{equation} \mathbf{\hat{n}} % = \frac{ % \mathbf{u}\times\mathbf{v} % }{|\mathbf{u}\times\mathbf{v}|} % \thinspace , \end{equation}\] \(\mathbf{\hat{n}}\) is a unit vector normal to \(S\) at \(P\). The special case where the unit vector is normal to the surface \(z=f(x,y)\), where \[\begin{align} \require{physics} \mathbf{u} % &= \qty( u_x, 0, \pdv{f}{x} % ) % \\ \mathbf{v} % &= \qty( % 0, v_y, \pdv{f}{y} % ) % \thinspace , \end{align}\] is written as \[\begin{equation} \mathbf{\hat{n}} % = \frac{ \mathbf{u}\times\mathbf{v} }{ \norm{ \mathbf{u}\times\mathbf{v} } } % = \frac{ -\mathbf{i}\pdv{f}{x} - \mathbf{j}\pdv{f}{y} + \mathbf{k} }{\sqrt{ 1 + \qty( \pdv{f}{x} )^2 + \qty( \pdv{f}{y} )^2 }} % \thinspace . \end{equation}\] This form proves to be quite useful when projecting a 3D surface to a 2D plane. Note that this result is independent of the two arbitrary quantities \(u_x\) and \(v_y\).