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$(1)$ Please suggest some books regarding the fundamental studies on surface and volume integrals in spherical coordinates.

$(2)$ Are there any books dedicated to only elementary calculus of spherical coordinates? All of the calculus books I know have lengthy discussions regarding Cartesian coordinate system but only some couple of pages regarding spherical coordinates.

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I don't know of any books which solely focus on spherical coordinates, and I would be surprised if there even existed one.

My surprise would stem from the fact that doing anything in spherical coordinates is (almost) exactly the same as integrating in the cartesisan coordinate system, as $$\int_0^{2\pi} \int_0^{\pi} \int_0^1 \rho^2\sin\phi\ d\rho\ d\phi\ d\theta = \int_0^{2\pi} \int_0^{\pi} \int_0^1 x^2\sin y\ dx\ dy\ d\theta$$ by a simple change of variable names. The first can be thought of as integrating $1$ over the unit sphere in spherical coordinates, but as we can see, this is really just equivalent to doing a cartesian integral of $x^2\sin y$ over a subset of $\mathbb{R}^3$.

No doubt bijections like that of spherical coordinates are useful, and this is why coordinate transforms and Jacobians are taught in multivariable calculus, but the theory of integration after we do the coordinate transform and multiply by the Jacobian is exactly the same as for cartesian coordinates.

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