- #1

binbagsss

- 1,233

- 10

I'm looking at the Jacobian condition which is ## J= a \frac{dy_{0}}{ds}-b\frac{dx_0}{ds}##

where the pde takes the form ##c= a\frac{\partial u}{\partial x} + b \frac{\partial u}{\partial y} ##, where ##a=\frac{\partial x}{\partial \tau } ##, ##b=\frac{\partial y}{\partial \tau }##, ##c=\frac{\partial u}{\partial \tau} ##

If ## J=0 ##, there are two possibilities: zero solutions, or an infinite number of solutions.

The zero solution case corresponds to the initial curve not being a charecteristic, and the infinite solution case to the initial curve being a characteristic.

**Question:**

I have in my notes that there exsists a solution if ## \frac{dx_{0}}{ ds}\frac{1}{a}= \frac{du_{0}}{ds}\frac{1}{c}## holds and no solution if this does not hold.

So I conclude that

**this condition holding must be equivalnet to the intial curve being a characteristic. I'm just stuggling to see how it says this?**

**What I know:**

##J## is a conidtion ensuring the characteristics are not tangential to the curve where initial conditions are prescribed.

**Any help in seeing how the 2 are equivalent, greatly appreciated.**