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I just have three little questions relating to the physical interpretation of integrating surface integrals of scalar-valued functions over a surface & then integrating the gradient of that scalar-valued function over the surface (i.e. a surface integral of a vector-valued function), whether there is some form of duality of interpretations or not.

First off, what is the physical interpretation of the surface integral of a scalar-valued temperature function over the surface of a triangle?

If I set up a scalar surface integral of a temperature function $T(x,y,z) = x^2 + y^2 + z^2$ over the surface of a triangle w/ vertices $A:(1,0,0)$, $B:(0,1,0)$, $C:(0,0,1)$ I'm integrating $T$ over the surface $S = [ (x,y,z) \in \mathbb{R}^3 | x + y + z - 1 = 0]$ (how do you put in curly braces {} ?).

But what does it actually mean to sum up the values of this temperature function over all values of the surface of the triangle?

If you (lazily) write $T(x_i,y_i,1-x_i-y_i) \Delta S_i$ this gives the value of the temperature in the surface element $\Delta S_i$, however it's scaled down 'infinitesimally' by a factor of $\Delta S_i$ since $\Delta S_i$ is extremely small, & I can't see how anybody could attribute any physical meaning to summing infinitesimal versions of the temperature at a point over an entire surface. If I then try to imagine that $T(x_i,y_i,1-x_i-y_i) \Delta S_i$ represents the volume of something (length $\times$ area) that makes no sense to me either.

If I then think of surface integrals of scalar functions, in this context, as merely being an accessory to finding average values, say the average temperature as the surface integral divided by the area of the surface, then I get a nice physical interpretation but I'm still led to ask what the surface integral itself means before we divide by the area? I know that integrating the gradient of the temperature function over the surface gives the flow per unit time of temperature through the triangle, so there's half of a duality of interpretation in this context.

My second question is the same as the first one, replacing temperature with electric potential & gradient of temperature with electric field, I just don't see what the physical interpretation of integrating the potential is, again there's half of a duality of interpretation in this context in that integrating the gradient of a potential function over a surface makes sense but I can't interpret integrating that potential function itself & how it relates to integrating it's gradient.

Finally if I take the surface integral of a density function over the triangle I'll get the mass of that triangle, thus a physical interpretation of a surface integral in this case is that it finds the mass. However what does it mean to integrate the gradient of the density over the surface of the triangle? Does it mean anything - why or why not? In this context the direction of interpretation is reversed, this time I can't interpret integrating the gradient yet I can interpret integrating the potential function...

Just hoping somebody has some nice thoughts on the matter, thanks for reading!

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