PDEMOC9

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Show that if $z=u(x,y)\,$ is an integral surface of $V=<a,b,c>\,$ containing a point $P\,$ , then the surface contains the characteristic curve $\chi\,$ passing through $P\,$ . (Assume the vector field $V\,$ is $C^1\,$ , which means that the first derivative exists everywhere).

It is required to prove that $\chi \subset u\,$ .

Let $\chi:(f(t),g(t),h(t))\,$ and $P:(f(0),g(0),h(0))\,$ .

Now, show that $u(f(t),g(t))=h(t)\,$ .

$\frac{d}{dt} \left[ u(x(t),y(t))-h(t)\right]=u_xx'(t)+u_yy'(t)-h'(t)\,$

$=u_xa+u_yb-c(x(t),y(t),z(t))=c(x(t),y(t),u)-c(x(t),y(t),z(t))\,$

From the mean value theorem, at some point the last equation equals $\frac{\partial c}{\partial z} (u-z)\,$ .

This is true iff $\frac{d(u-z)}{dt} = \frac{\partial c}{\partial z}(u-z)\,$

$\implies (u-z)(t) = (u-z)(0)e^{\int_0^t \frac{\partial c}{\partial z} ds}=0\,$

since $u(x(0),y(0))=z(0)\,$ .

Main Page : Partial Differential Equations : Method of Characteristics

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