In this example, what does the integrand signify? I thought the purpose of a surface integral would be to find the area of S, but then what is the 6xy for? Geometrically, what would it mean if it was lets say x + z instead of 6xy?
#What does it mean to evaluate a double integral over a surface
uneven lagoon
cerulean elk
short ore
this integral would indeed yield the area of the surface if we were integrating just 1 instead of 6xy. In every other case it would require a four-dimensional space to visualize it
you may try to imagine 6xy going up and down in some mystery dimension as it takes on different values of S
uneven lagoon
you may try to imagine 6xy going up and down in some mystery dimension as it tak...
so what's the purpose of integrating a function over a surface?
short ore
engineer/physicist moment
it is hard for me to imagine a real world use for this but luckily that's not what math is for
uneven lagoon
oh
i'm assuming this is just a pedagogical tool for when you're gonna be calculating the flux through a surface
short ore
who knows, maybe there is a use for this stuff in relativity or something since we live in 3 + 1 dimensions
that's what i'd guess if you're studying engineering or physics
i may've actually been speaking out of my arse here since S is two-dimensional so you do have an extra dimension for visualization