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The definition of Line Integral can be this:

[tex]\int_s\vec{f}\cdot d\vec{r}=\int_s(f_1dx+f_2dy+f_3dz)[/tex]

And the definition of Surface Integral can be this:

[tex]\int\int_S(f_1dydz+f_2dzdx+f_3dxdy)[/tex]

However, in actually:

[tex]\\dx=dy\wedge dz \\dy=dz\wedge dx \\dz=dx\wedge dy[/tex]

What do the Surface Integral be equal to:

[tex]\int\int_S(f_1dy\wedge dz+f_2dz\wedge dx+f_3dx\wedge dy)=\int\int_S(f_1dx+f_2dy+f_3dz)=\int\int_S\vec{f}\cdot d\vec{r}[/tex]

I know, I know... I know that, generally, the definition to Integral Surface is:

[tex]\int\int_S\vec{f}\cdot \hat{n}\;dS[/tex]

I until like this definition when compared to its respective Line Integral:

[tex]\int_s\vec{f}\cdot \hat{t}\;ds[/tex]

But, is correct to definite the Surface Integral as:

[tex]\int\int_S\vec{f}\cdot d\vec{r}[/tex]

being

[tex]d\vec{r}=(dx,dy,dz)[/tex]

?

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# Definition for Surface Integral

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