22. Parametric Surfaces and Surface Integrals

b. Tangent and Normal Vectors

2. Tangent Vectors

For a 2D curvilinear coordinate system \(\vec R(u,v)\), the coordinate tangent vectors are the tangent vectors to the coordinate curves. \[ \vec e_u=\dfrac{\partial\vec R}{\partial u} \qquad \text{and} \qquad \vec e_v=\dfrac{\partial\vec R}{\partial v} \] \(\vec e_u\) is tangent to the \(u\)-curves and points in the direction of increasing \(u\). \(\vec e_v\) is tangent to the \(v\)-curves and points in the direction of increasing \(v\). We do the same thing for parametric surfaces:

The vector \(\vec{e}_u\) (as well as \(\vec{e}_v\)) is analogous to the to tangent (or velocity) vector to a parametric curve, \(\vec r(t)=\left\langle x(t),y(t),z(t)\right\rangle\), given by: \[ \vec{v}=\dfrac{d\vec r}{dt} =\left\langle \dfrac{dx}{dt},\dfrac{dy}{dt},\dfrac{dz}{dt}\right\rangle \] with the following exceptions:

1. The parameter for \(\vec R(u,v_0)\) is \(u\) rather than \(t\); so we need to differentiate the with respect to \(u\).
2. Since the \(v\)-coordinate is held constant at \(v_0\), the derivative needs to be a partial derivative with respect to \(u\) instead of a total derivative with respect to \(t\).
3. We do not refer to the tangent vector as a velocity because the parameter, \(u\), represents a coordinate on the surface, not a time.
4. We use the symbols \(\vec e_u\) and \(\vec e_v\) rather than the letter \(\vec{v}\) for the tangent vectors, because they are not velocities and also because we have already used \(v\) for a parameter.

The tangent vectors to the \(\phi\) and \(\theta\) curves are \[\begin{aligned} \vec{e}_\phi&=\dfrac{\partial\vec R}{\partial\phi} =\left\langle 2\cos\phi\cos\theta,2\cos\phi\sin\theta,-2\sin\phi\right\rangle \\ \vec{e}_\theta&=\dfrac{\partial\vec R}{\partial\theta} =\left\langle -2\sin\phi\sin\theta,2\sin\phi\cos\theta,\quad 0\quad \right\rangle \end{aligned}\] Notice that \(\vec{e}_\theta\) is horizontal since it is tangent to a line of latitude and it points East in the direction of increasing \(\theta\). Similarly, \(\vec{e}_\phi\) points South, tangent to a line of longitude, in the direction of increasing \(\phi\).