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3.3 Curves on Surfaces

3.3.1 Parametric curves in the x–y–plane

Definition: A curve in the x–y–plane is a map

\begin{eqnarray} R → {R}^{2},\quad t → x(t) := (x(t),y(t))& & %&(3.8) \\ \end{eqnarray}

which associates with each values of the parameter t (‘time’ t) a point (x(t),y(t)) in the x–y–plane.

Examples

1. The circle around the origin,

\begin{eqnarray} (x(t),y(t)) = (r\mathop{cos}\nolimits (t),r\mathop{sin}\nolimits (t)).& & %&(3.9) \\ \end{eqnarray}

Check that x{(t)}^{2} + y{(t)}^{2} = {r}^{2} for all t.

2. The curve line

\begin{eqnarray} (x(t),y(t)) = ({t}^{2},t).& & %&(3.10) \\ \end{eqnarray}

Sketch this!

3.3.2 Parametric urves on Surfaces

Consider a function f(x,y), i.e. a surface z = f(x,y) above the x–y–plane. Consider a curve (x(t),y(t)) in the x–y–plane. This curve defines a corresponding curve

\begin{eqnarray} z(t) = f(x(t),y(t))& & %&(3.11) \\ \end{eqnarray}

on the surface. Example: for f(x,y) = {x}^{2} + {y}^{2} and (x(t),y(t)) = (r\mathop{cos}\nolimits (t),r\mathop{sin}\nolimits (t)), z(t) = {r}^{2}. The circle in the x–y–plane corresponds to a ring hovering at a distance {r}^{2} above the plane, being part of the surface of the paraboloid {x}^{2} + {y}^{2}. Sketch the corresponding picture (lecture)!

3.3.3 Change of height along a Curve

Reminder: Total change of height (total differential)

\begin{eqnarray*} df(x,y) := {∂\over ∂x}f(x,y)dx + {∂\over ∂y}f(x,y)dy& & %& \\ \end{eqnarray*}

is called the total differential of f(x,y) at the point (x,y) and gives the total change of the height of the surface f(x,y) (measured from the x–y–plane) at the point (x,y), if we move a tiny step dx along the x–direction and a tiny step dy along the y–direction.

From this, we can calculate the change of the height of the curve z(t) = f(x(t),y(t)):

\begin{eqnarray} {dz(t)\over dt} = {df(x(t),y(t))\over dt} = {∂\over ∂x}f(x,y){dx(t)\over dt} + {∂\over ∂y}f(x,y){dy(t)\over dt} .& & %&(3.12) \\ \end{eqnarray}

This is a chain rule

\begin{eqnarray} {df(x(t),y(t))\over dt} = {∂f\over ∂x} {dx\over dt} + {∂f\over ∂y} {dy\over dt} .& & %&(3.13) \\ \end{eqnarray}

Example: f(x,y) = {x}^{2} + {y}^{2} and (x(t),y(t)) = ({t}^{2},t)

We have

\begin{eqnarray*} {dz(t)\over dt} & =& = {∂\over ∂x}f(x,y){dx(t)\over dt} + {∂\over ∂y}f(x,y){dy(t)\over dt} = 2x(t) ⋅ 2t + 2y(t) ⋅ 1 = 4{t}^{3} + 2t.%& \\ \end{eqnarray*}

We can check this by direct calculation, z(t) = {t}^{4} + {t}^{2} ⇝ dz(t)∕dt = 4{t}^{3} + 2t. The general formula, however, makes it clear that there a two contributions to the change of the curve z(t) on the surface: 1. the ‘geometric change’ (partial derivatives {f}_{x}, {f}_{y}) of the surface. 2. the ‘kinematic change’, i.e. the time derivatives dx(t)∕dt, dy(t)∕dt that determine the speed by which we sweep along the curve z(t).

Example: f(x,y) = {x}^{2} − {y}^{2} and (x(t),y(t)) = (\mathop{cos}\nolimits t,\mathop{sin}\nolimits t)

\begin{eqnarray*} {dz(t)\over dt} & =& = {∂\over ∂x}f(x,y){dx(t)\over dt} + {∂\over ∂y}f(x,y){dy(t)\over dt} = 2x(t) ⋅ (−\mathop{sin}\nolimits (t)) − 2y(t) ⋅\mathop{ cos}\nolimits (t)%& \\ & =& −2\mathop{cos}\nolimits (t)\mathop{sin}\nolimits (t) − 2\mathop{sin}\nolimits (t)\mathop{cos}\nolimits (t) = −2\mathop{sin}\nolimits (2t). %& \\ \end{eqnarray*}

Direct check with z(t) ={\mathop{ cos}\nolimits }^{2}(t) −{\mathop{ sin}\nolimits }^{2}(t) =\mathop{ cos}\nolimits (2t).

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