# 22. Parametric Surfaces and Surface Integrals

## b. Tangent and Normal Vectors

## 1. Coordinate Curves

For a 2D curvilinear coordinate system \(\vec R(u,v)\), the coordinate curves (the \(u\)-curves and \(v\)-curves) are obtained by holding one coordinates fixed while the other coordinate changes. \(u\) changes on a \(u\)-curve. \(v\) changes on a \(v\)-curve. In the figure the blue curves are the \(u\)-curves and the red curves are the \(v\)-curves.

For a parametric surface, \(\vec R(u,v)\), the
coordinate curves (the \(u\)-curves and
\(v\)-curves) are obtained by holding one coordinate fixed while the
other coordinate changes. \(u\) changes on a \(u\)-curve. \(v\) changes
on a \(v\)-curve.

The \(u\)-curve (in blue) with \(v=v_0\)
and parameter \(u\) is:
\[
\vec R(u,v_0)=\left\langle x(u,v_0),y(u,v_0),z(u,v_0)\right\rangle
\]
The \(v\)-curve (in red) with \(u=u_0\)
and parameter \(v\) is:
\[
\vec R(u_0,v)=\left\langle x(u_0,v),y(u_0,v),z(u_0,v)\right\rangle
\]

Identify the coordinate curves for the sphere of radius \(\rho=2\), parametrized by \[ \vec R(\phi,\theta) =\left\langle 2\sin\phi\cos\theta,2\sin\phi\sin\theta,2\cos\phi\right\rangle \]

A \(\phi\)-curve (in red) is a line of
longitude running southward, from the North pole to the South pole
with a constant value of \(\theta\).
A \(\theta\)-curve (in yellow) is a
line of latitude running counterclockwise from West to East
with a constant value of \(\phi\).

Redo plot to switch to red and blue.

Identify the coordinate curves for the elliptic paraboloid \(z=x^2+y^2\)
parametrized by
\(\vec R(r,\theta)
=\left\langle r\cos\theta,r\sin\theta,r^2\right\rangle\).

PY: Change the grid on the parabola to polar.

An \(r\)-curve (in red) is a parabola starting at the vertex of the paraboloid and running up a side with a constant value of \(\theta\). A \(\theta\)-curve (in yellow) is a circle running around the paraboloid with a constant value of \(r\).

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