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 \(r\).

22. Parametric Surfaces and Surface Integrals

b. Tangent and Normal Vectors

1. Coordinate Curves

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