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I am studying for a Calculus exam, and one of the topics I should know about is surface integrals. Now, I am using Stewart 6e, and in there I have found several equations for computing surface integrals. These are the equations I found:

$\iint_{S}{f(x, y, z)\:dS}=\iint_{D}{f(\vec{r}(u,v))*\left|\vec{r}_{u}\times\vec{r}_{v}\right|\:dA} \\ \iint_{S}{f(x, y, z)\:dS}=\iint_{D}{f(x, y, g(x, y))*\sqrt{\left(\frac{dg}{dx}\right)^{2}+\left(\frac{dg}{dy}\right)^{2}+1}\:dA} \\ \iint_{S}{\vec{F}\:d\vec{S}}=\iint_{D}{\vec{F}*\left(\vec{r}_{u}\times\vec{r}_{v}\right)\:dA} \\ \iint_{S}{\vec{F}\:d\vec{S}}=\iint_{D}{\left(-P\left(\frac{dg}{dx}\right)-Q\left(\frac{dg}{dy}\right)+R\right)\:dA}$

So, my question is: How do I know which one to use for a given exercise? Obviously the bottom two should be used if I'm dealing with a vector field, and the top two if I'm dealing with just an equation. Also, I noticed that the second and last one both require a seperate function g(x,y). But still, it isn't quite clear to me when to use which one. Please help me out!

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    Ok, this gives me everything I need on surf$a$ce integrals for my Calculus exam. Thank you very much!2012-06-24

2 Answers 2

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The comments above by Arthur were very helpful. My question has been answered.

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When there is $ \hat n $ involved, use $ \iint_s \vec F(x,y,g(x,y))\cdot \frac{\nabla u(x, y)}{|\nabla u(x, y)|} \sqrt{ 1 + \left ( \partial g \over \partial x\right )^2 + \left ( \partial g \over \partial y\right )^2} dxdy $ $u$ being surface. Or, $ \iint_{S} \vec F(x, y, z). \hat n dS=\iint_{D}{ \vec F(\vec{r}(u,v))*\left(\vec{r}_{u}\times\vec{r}_{v}\right)\:dudv} $

If there is not $\hat n$

$ \iint_s \vec F(x,y,g(x,y)) \sqrt{ 1 + \left ( \partial g \over \partial x\right )^2 + \left ( \partial g \over \partial y\right )^2} dxdy $ Or, $ \iint_{S} \vec F(x, y, z) dS=\iint_{D}{ \vec F(\vec{r}(u,v))\left|\vec{r}_{u}\times\vec{r}_{v}\right|\:dudv} $ Both are equivalent. check this out surface integral of vector along the curved surface of cylinder and parametrization of surface element in surface integrals.