"The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve." This is from https://en.wikipedia.org/wiki/Gradient_theorem
My question is the following: Is there a generalization of gradient theorem for vector valued functions, namely:
Let $f:\mathbb{R}^n \to \mathbb{R}^n $ be a continuously differentiable vector valued function and $\gamma$ be any curve from $\mathbf{a}$ to $\mathbf{b}$.
Then
$$ \int_{\gamma} \mathbf{J}[f] d\mathbf{r} = f(\mathbf{b}) - f(\mathbf{a}) $$
where $\mathbf{J}$ is the Jacobian.
If yes, is there a generalization of conservative vector field to matrices?