Can someone explain what is the purpose of surface integrals? How are they different from flux integrals?
Edit: the explanations already on this sub did not help that much sorry
In: Mathematics
Flux is to find the flow of a vector field through a surface. Specifically you will see the dot product of vector F and dS vector. A surface integral doesn’t necessarily have to calculate flux. It could calculate the surface area instead or the total force on a scrap of metal in space due to gravity.
Below I use F as a general vector field, it doesn’t necessarily represent force.
Below is latex
Flux
$oiint_{S}{vec{F}cdotvec{dS}}$
Surface area
$oiint_{S}{dS}$
General Surface Integral
$oiint_{S}{vec{F} dS}$
A flux integral is a special case of a surface integral.
A surface integral is integrating the value of *any* function over a surface. A regular one-dimension line integral can be thought of as a “cumulative” value, like a running sum. A surface integral is the same: the cumulative value of a function over an area. A trivial example of a surface is to calculate the surface area itself.
A “flux integral” is a surface integral, where the value being integrated represents the movement of a conserved value through the surface. The flow of fluid through a filter, or a magnetic field passing through the surface of an object.
In particular, the “flux” function being integrated needs to be a *vector-valued* function. Surface integrals can also apply to scalar-valued functions.
ELI5 isn’t always the best place for calculus questions, especially if you already understand some calculus topics. You might try in /r/AskScience.
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