Derivative of integral in one direction Suppose I have a 3D scalar field $\phi(x,y,z)$, where $x,y,z$ are cartesian coordinates in $\mathbb{R}^3$. I can define another 3D field, which is the integral of the field with respect to one dimension
$$ \Phi(x,y,z) = \int_{z}^{\eta} \phi(x,y,z^\prime) \,d z^\prime$$
This has the interpretation of integrating from a depth $z$ to a surface $\eta$, where $\eta(x,y)$ is known and fixed for every $x$ and $y$. If $\phi$ has the interpretation of a density*gravity, this is something like the weight of a material above a point in space.  I am interested in 3 questions:

*

*What is the partial derivative $\frac{\partial}{\partial z} \Phi = \frac{\partial}{\partial z} \int_{z}^{\eta} \phi(x,y,z^\prime) \,d z^\prime$ ?

*What is the total derivative $\frac{d}{dz} \Phi = \frac{d}{dz} \int_{z}^{\eta} \phi(x,y,z^\prime) \,d z^\prime$ ? Is it the same as the partial derivative in this case?

*How could I write the line integral $\Phi(z)$ (from a depth $z$ upwards, that is from $(x_0,y_0,z)\to(x_0,y_0,\eta)$) as a volume integral over all 3 spatial coordinates? Something like $\Phi(z) = \int_{z}^{\eta}\int_{A} \phi(x,y,z^\prime) D(x,y,z)\,dA \,dz^\prime$ where $D(x,y,z)$ is something like a delta function. I ask because I need to approximate the integral over predefined 3D blocks where I know 3D quadrature rules.

Any insight for any of the 3 questions would be appreciated!
edit: changed the order of the questions to ask partial derivative first

1)) If for fixed $x$ and $y$, $f(z)=\phi(x,y,z)$ is a continuous function of $z$ then for any primitive $F(z)$ of $f(z)$ we have $$\Phi(x,y,z)=\int_{z}^{\eta} \phi(x,y,z^\prime) \,d z^\prime=F(\eta)-F(z)$$ by Newton-Leibnitz' formula. Then
$$\frac{\partial}{\partial z} \Phi(x,y,z)=(F(\eta)-F(z))’=-f(z)=-\phi(x,y,z).$$
