Why is dx/dt = -(∂u/∂t) / (∂u/∂x)?

Question

I found that $\frac{dx}{dt} = -\cfrac{ \frac{\partial u}{\partial t} }{ \frac{\partial u}{\partial x}}$ on the internet. I can´t figure out if it is true and why.

Answer 1

This is a result of the implicit function theorem. In fact, in some presentations (i.e. Rudin) this is included in the theorem. Look there for definitions/proofs.

Answer 2

This expression does not mean anything. So first we must define the variables. I assume that you meant: let $u(\alpha,\beta)$ be a smooth function of two real variables and let $x$ be a smooth univariate function with real values such that $u(t,x(t))=0$. Then, $x’(t)=-\frac{(\partial u)/(\partial \alpha)(t,x(t))}{(\partial u)/(\partial \beta)(t,x(t))}$.

This is straightforward when you differentiate wrt $t$ the equation $u(t,x(t))=0$.