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I want to make sure my understanding of calculating surface integrals of vector fields is accurate. It was never presented this way in a textbook, but I put this together from pieces of knowledge. To my understanding, surface integrals can be calculated in four different ways (depending on how the surface is given). For this, I'll integrate the vector field ##\vec{A}(\vec{r})## over a surface S.

- If one has a parametric surface ##\vec{r}(u,v)##, a surface integral would be calculated as $$\pm \iint_S \vec{A}\left(\vec{r}(u,v)\right)\cdot(\vec{r}_u\times \vec{r}_v)dudv$$where the subscripts represent partial derivatives, and the sign is determined by the direction of the cross product relative to the orientation of the surface.
- If one has a surface in the form ##F(x,y,z)=0##, a surface integral would be calculated as $$\pm \iint_S \frac{\vec{A}\cdot\nabla F}{\left|F_z\right|}dxdy$$where the subscript represents a partial derivative, and the three variables can be interchanged. One must also substitute into get the integrand in terms of only x and y (or whatever the variables of integration are). The sign is determined by the direction of the gradient relative to the orientation of the surface.
- If one has a surface in the form ##z=f(x,y)## (or alternatively ##y=f(x,z)## or ##x=f(y,z)## with the appropriate adjustments made), the surface integral would be calculated as $$\pm \iint_S \vec{A}\left(x,y,f(x,y) \right)\cdot(-f_x\hat{i}-f_y\hat{j}+\hat{k})dxdy$$where the subscripts represent partial derivatives. The sign is plus if the surface is oriented upwards, and minus if it is oriented downwards.
- For this last one, one needs a surface that can be expressed relatively easily as ##x=f(y,z)##, ##y=g(x,z)## AND ##z=h(x,y)##. The vector field will be expressed as ##\vec{A}=A_1 \hat{i}+A_2 \hat{j}+A_3 \hat{k}##. The surface integral can be computed as $$\iint_S \left[ \pm A_1\left(f(y,z),y,z \right)dydz \pm A_2\left(x,g(x,z),z \right)dxdz \pm A_3\left(x,y,h(x,y) \right)dxdy\right]$$where the three signs are respectively determined by the orientation of the surface in the x, y, and z directions (plus for oriented in the positive direction, minus for oriented in the negative direction).

I'm not 100% sure about the last one, but I think I have seen something like it before. It also appears as if it would be useful for the divergence theorem. Are all four of these valid ways to compute surface integrals?

Thank you in advance!