Consider the following simple 1D example -
Let $f(x)$ be a scalar valued function. $f'(x)$ is its derivative.
$\int_{x_1}^{x_2}f'(x)dx = f(x_2)-f(x_1)$.
So if I have the analytical expression for $f(x)$ and I have what is claimed to be the analytical expression for $f'(x)$, I can easily see if the analytical expression of $f'(x)$ is indeed correct by integrating it.
I was wondering if there is a similar thing that can be done in case of multivariate functions.
If I have a function $g(n_1,n_2,...n_m)$ and what is claimed to be an analytical expression for the function's Jacobian $J$, is there any analogous way to "integrate" $J$ and see if it is correct?
If $g\colon\Bbb R^m\to\Bbb R^k$, its Jacobian $Dg(x)$ is a $k\times m$ matrix at each point $x$. The chain rule tells us that if $\phi\colon [a,b]\to\Bbb R^m$ is a differentiable path, then $$(g\circ\phi)'(t) = Dg(\phi(t))\phi'(t).$$ In particular, we get $$g(\phi(b))-g(\phi(a)) = \int_a^b (g\circ\phi)'(t)\,dt = \int_a^b Dg(\phi(t))\phi'(t)\,dt.$$ If you take $\phi$ to be a straight-line path, then $\phi'(t)=v$ is constant, and you're doing the integral of the vector-valued function $Dg(\phi(t))v$ on $[a,b]$.