I'm trying to read the paper "Particle flow for nonlinear filters with log-homotopy" by Daum and Huang. ( http://144.206.159.178/ft/CONF/16415230/16415269.pdf )

As far as I understand, they reduce their problem to solving the differential equation

$$\frac{\partial\log(f_\lambda)}{\partial x} \frac{dx}{d\lambda} + \frac{\partial\log(f_\lambda)}{\partial \lambda} = 0$$

However, they then write: For $d=1$, we can solve the equation exactly by writing

$$\frac{dx}{d\lambda} = -\frac{\partial\log(f_\lambda)}{\partial \lambda} / \frac{\partial\log(f_\lambda)}{\partial x}$$

for non-zero gradient and $\frac{dx}{d\lambda} = 0$ otherweise, but for $d>1$, we cannot simply solve the equation by division as above.

I have so far only seen $d$ as the symbol for total derivative, so I have no idea what their explanation is supposed to mean?

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$d$ wouldn't stand for the dimension, would it? – Gerry Myerson Sep 13 '12 at 6:38
Ouch! Of course, you're right. Thank you! – Benno Sep 13 '12 at 12:43