#### Family of Curves

Let S: \(\vec{r}=\vec{r}(u,v)\) be a surface and

\( \phi(u,v)=c\) (i)

be a single valued function of \((u,v)\) having continuous derivatives \( \phi_1\) and \( \phi_2\) which do not vanish together. Then an equation (i) where c is real parameter gives a family of curves lying on the surface.

For different values of c, (i) gives different curves on the surface.

#### Example

- As we know \( x^2+y^2=c\); where c is a real parameter represent a family of circles

Similarly, \( u^2+v^2=c\); where c is a real parameter represent a family of circles on the surface\)

- As we know \( y=mx\); where m is real parameter represent a family of straight

Similarly, \( v=mu\); where m is a real parameter represent a family of curves on the surface

- As we know, \(xy=c;c \ne 0\); where c is real parameter represent a family hyperbola

Similarly, \( uv=c\); where c is a real parameter represent a family of curves on the surface

- As we know, \(x^2-y^2=c;c\neq 0\); where c is real parameter represent another family hyperbolas

Similarly, \( u^2-v^2=c\); where c is a real parameter represent a family of curves on the surface

#### Differential equation of family of curves

Let S: \(\vec{r}=\vec{r}(u,v)\) be a surface and \( \phi(u,v)=c\) be a family of curves, then

\( \phi(u,v)=c\)

or\( \phi_1 du+\phi_2 dv=0 \)

or\(\frac{du}{dv}=- \frac{\phi_2}{\phi_1} \)

or\(\frac{du}{dv}=- \frac{Q}{P} \) where \((-\phi_2,\phi_1) \) are proportional to (-Q,P)

is called differential equation of family of curves.

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