If $\frac{\partial u}{\partial \phi} = - \frac{\partial v}{\partial \lambda}$, is $u$ = $-v$

Question

Under the continuity equation in atmospheric dynamics:

$\frac{\partial u}{\partial \phi} + \frac{\partial v}{\partial \lambda} = 0$

Therefore, making:

$\frac{\partial u}{\partial \phi} = - \frac{\partial v}{\partial \lambda}$

So, my mathematics related question is: if $\frac{\partial u}{\partial \phi} = - \frac{\partial v}{\partial \lambda}$, is it correct to assume under these circumstances than $u$ = $-v$. If not, why not?

Any help would be appreciated greatly.

Clarification:

The equation for $u$ is the following:

$u = - \frac{g}{f} \cdot \frac{1}{a} \cdot \frac{\partial \Phi}{\partial \phi}$

where

$\Phi = z \cdot (g - R\omega^2cos^2(\phi))$.

I also know the equation for $v$ is:

$v = \frac{g}{f} \cdot \frac{1}{a \cdot cos(\phi)} \cdot \frac{\partial \Phi}{\partial \lambda}$

Some further information on this question can be found here:

https://earthscience.stackexchange.com/questions/16825/computation-of-geostrophic-wind

Answer 1

It is not correct. See this counterexample $u = \phi + \lambda^5$ and $v = -\lambda + \phi ^5$. The differential equation is true but it it clear that $u\neq v$.

We have the statement

$$u(\phi, \lambda) = -\int \dfrac{\partial v(\phi, \lambda)}{\partial \lambda} d \phi + f(\lambda)$$

in which $f(\lambda)$ takes a similar role as the constant of integration for univariate integrals.