I'm doing a multivariable calculus course at the moment. I've seen path integrals in the cartesian coordinate system as the following definition:
Definition. The path integral of $f(x,y,z)$ along the curve $C$ is
$$ \int_C f ds = \int_a^bf(\mathbf{x}(u))||\mathbf{x}'(u)||du$$
where $\mathbf{x}:[a,b]\to\mathbb R^3$ is the parametric representation of $C$. The definition didn't make any reference to the coordinate system.
Question. Suppose that $ (x,y,z)=\mathbf \Phi(\xi_1,\xi_2,\xi_3) $ where the transformation $\mathbf \Phi:U\to V$ is sufficiently differentiable and has a inverse $\mathbf{\Phi^{-1}}:V\to U$ where $U,V$ are open subsets of $\mathbb R^3$.
What would be the definition of a path integral over $C$ when $f$ is a function of curvilinear coordinates ($\xi_1,\xi_2, \xi_3$), and the curve was parameterized in curvilinear coordinates as $\mathbf{\xi}=\mathbf{\xi}(u)$ for $a\le > u\le b$?
I couldn't find a definition literally anywhere online. The definition on wikipedia https://en.wikipedia.org/wiki/Curvilinear_coordinates#Tensors_in_curvilinear_coordinates has $f$ as a function of cartesian coordinates.
Why can't you simply evaluate the integral as $$\int_a^bf(\mathbf \xi)||\mathbf{\xi}'(u)||du ?$$
Is the definition the same as a regular "cartesian" path integral? If so, can you please derive how you would get a path integral for a function in curvilinear coordinates from the definition above for cartesian?
Note that the curve $f$ in Cartesian coordinates is given by $\tilde f = f \circ \Phi^{-1}$ and similarly the curve parametrization is given by $\tilde x = \Phi \circ \xi$. So in the Cartesian coordinates the path integral would be
$$ \begin{align} \int_a^b \tilde f (\tilde x(u)) \| \tilde x'(u) \| du &= \int_a^b (f \circ \Phi^{-1} \circ \Phi \circ \xi)(u) \| (\Phi \circ \xi)'(u)\|du \\ &=\int_a^b f(\xi(u)) \|(D\Phi)(\xi(u)) \xi' (u)\|du \end{align}$$
So yes there is no new definition, you just have to write it in terms of a cartesian path integral. You see that the difference is in the Jacobian of the coordinate transformation that we have to respect.