I'm reading(adsbygoogle = window.adsbygoogle || []).push({}); Div, Grad Curl, and All That, and in coming up with a formula for the divergence, H.M. Schey starts with a small cube centered at (x,y,z), labels the face parallel to the yz-plane as S_{1}and calculates

[tex] \int\int_{S_1}\mathbf{F}\cdot\hat{\mathbf{n}}dS=\int\int_{S_1}F_x(x,y,z)dS [/tex]

He then says that because the cube is small, the integral is equal to F_{x}evaluated at the center of S_{1}times the area of S_{1}. The justification apparently comes from some mean value theorem that says "the integral of F_{x}over S_{1}is equal to the area of S_{1}multiplied by the function evaluated somewhere on S_{1}."

I tried looking up this theorem to no avail. Could someone point me to a proof or maybe outline why it's true?

Thanks in advance.

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# Mean Value Theorem in Surface Integrals

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