Every note and book I read about surface integrals of vector fields only show how to solve these integrals when the vector field is in Cartesian coordinates. I'm curious about what would be the right procedure to solve these integrals when talking about a vector field that is described in another coordinate system.
For example, if I describe the following vector field:
$$ \newcommand{\uvec}[1]{\boldsymbol{\hat{\textbf{#1}}}} \vec{F} = x\uvec{j} $$
Using spherical coordinates, I'd get:
$$ \vec{F} = r\sin(\theta)\cos(\theta)\sin^2(\phi)\uvec{r} + r\cos(\theta)\cos(\phi)\sin(\phi)\sin(\theta)\uvec{φ} + r\cos^2(\theta)\sin(\phi)\uvec{θ} $$
Now if I want to find the flux of this vector field passing through the part of the sphere with radius 1 in the first octant, I would use
$$\iint_S \vec{F} \cdot d\vec{S}$ = $\iint_S \vec{F} \cdot \vec{n} dS$$
In spherical coordinates, the normal vector of that sphere is:
$$\vec{n} = \uvec{r}$$
Both the vector field and the normal vector are expressed in spherical coordinates, the dot product results:
$$\vec{F} \cdot \vec{n} = \vec{F} \cdot \uvec{r} = r\sin(\theta)\cos(\theta)\sin^2(\phi)$$
So:
$$\iint_S \vec{F} \cdot \vec{n} dS = \iint_S r\sin(\theta)\cos(\theta)\sin^2(\phi) dS$$
The surface equation is:
$$r=1$$
Then:
$$\int_{0}^{\pi/2}\int_{0}^{\pi/2} 1\sin(\theta)\cos(\theta)\sin^2(\phi)\cdot (1^2\sin(\phi)) d\theta d\phi $$
and
$$\int_{0}^{\pi/2}\int_{0}^{\pi/2} \sin(\theta)\cos(\theta)\sin^3(\phi) d\theta d\phi = \frac 1{3}$$
However, this is not the correct result.
When the field and the surface are both in Cartesian coordinates, the procedure I used above to evaluate the integral would usually work. Could someone explain me why this not work and how would be the correct procedure to evaluate this kind of integral?
Thanks in advance