Let $\widehat\Gamma$ and $\Gamma$ be differentiable manifolds, and let $\Phi: \widehat\Gamma \to \Gamma$ be a smooth, bijective, orientation-preserving map.
I understand that for a real valued function $\omega$ on $\Gamma$, then $$\int_\Gamma = \int_{\widehat\Gamma}\Phi^*\omega = \int_{\widehat\Gamma}\omega\circ\Phi \tag{i}$$ where $\Phi^*$ is the pullback operator such that $\Phi^*\omega \equiv \omega\circ\Phi$. (Is this right?)
Please I need help with how to approach an integrand that has functions defined on both $\Gamma$ and $\widehat\Gamma$ (these are identified by a hat). In particular, I have these integrals on $\widehat\Gamma$ and need to transform them to integrals on $\Gamma$. The integrals look something like these:
$$I_1 = \int_{\widehat\Gamma}(\widehat{n}\cdot \widehat{J}T\widehat{F}\widehat{n})(\widehat{u}\cdot\widehat{n})\;d\widehat{s}$$ $$I_2 = \int_{\widehat\Gamma}(\widehat{a}\cdot\widehat{n})(\widehat{u}\cdot\widehat{n})\;d\widehat{s}$$ $$I_3 = \int_{\widehat\Gamma}\widehat{a}\cdot\widehat{n}\;d\widehat{s}$$ where $\widehat{F} = \widehat\nabla\Phi$, $J=\textrm{Det}\widehat{F} > 0$, and $\widehat{n}$ is a unit normal vector to the surface element $d\widehat{s}$ of $\widehat\Gamma$, $\widehat{a}$ and $\widehat{u}$ are vectors defined on $\widehat{\Gamma}$; and $T$ is a second order tensor defined on $\Gamma$.
For $I_2$ and $I_3$ here is what I am thinking: Define the pushforward operator $\varphi = \Phi^{-1}$ then $$I_2 = \int_{\widehat\Gamma}\biggl(\left((\widehat{a}\circ\varphi)\circ\Phi\right)\cdot\left((\widehat{n}\circ\varphi)\circ\Phi\right)\biggr)\biggl(\left((\widehat{u}\circ\varphi)\circ\Phi\right)\cdot\left((\widehat{n}\circ\varphi)\circ\Phi\right)\biggr)\;d\widehat{s}$$ $$=\int_\Gamma \Bigl((\widehat{a}\circ\varphi)\cdot(\widehat{n}\circ\varphi)\Bigr)\Bigl((\widehat{u}\circ\varphi)\cdot(\widehat{n}\circ\varphi)\Bigr)\;ds \tag{ii}$$ by applying (i). And similarly, $$I_3 = \int_\Gamma (\widehat{a}\circ\varphi)\cdot(\widehat{n}\circ\varphi)\;ds \tag{iii}$$
Further, the pushforward operator on a contravariant vector $\widehat{a}$ and a second-order tensor $\widehat{T}$ (which maps contravarian vectors to contravariant vectors) are respectively defined: $$\varphi^*{\widehat a} = \widehat{a}\circ\varphi = \widehat{F}\widehat{a}, \qquad\qquad\qquad \varphi^*{\widehat T} = \widehat{T}\circ\varphi = \widehat{F}^{-T}\widehat{T}\widehat{F}^{-1},$$ then I can simply calculate each pushed-forward term in the integral e.g. $\widehat{a}\circ\varphi$ and substitute into (ii) and (iii). Is any of this correct?
For $I_1$ I am not sure how to proceed since $T$ in the integrand is defined on $\Gamma$ and not $\widehat\Gamma$.
I apologize if my statements are not rigorous. I'm a programmer, I don't have a math background.