- #1

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- TL;DR Summary
- doubt in the derivation of the expression for calculating the surface integral for a function F.

this method of derivation is approximating the function using a polyhedron.

concentrating on one of the surfaces(say the L'th surface which has an area ##\Delta S_l## and let ##(x_l,y_l,z_l)## be the coordinate of the point at which the face is tangent to the surface and let ##\hat n## be the unit normal vector)

the area of this surface is given by ##F(x_l, y_l, z_l).\hat n_l \Delta S_l##

on adding all of these we get,

my doubt in this derivation is why is there a dot product being taken between the vectors and then multiplied by the area(basically I don't understand why this equation, ##F(x_l, y_l, z_l).\hat n_l \Delta S_l## give the area as shouldn't the summation of all the ##\Delta S_l## simply be equal to the surface area of the function??)

concentrating on one of the surfaces(say the L'th surface which has an area ##\Delta S_l## and let ##(x_l,y_l,z_l)## be the coordinate of the point at which the face is tangent to the surface and let ##\hat n## be the unit normal vector)

the area of this surface is given by ##F(x_l, y_l, z_l).\hat n_l \Delta S_l##

on adding all of these we get,

my doubt in this derivation is why is there a dot product being taken between the vectors and then multiplied by the area(basically I don't understand why this equation, ##F(x_l, y_l, z_l).\hat n_l \Delta S_l## give the area as shouldn't the summation of all the ##\Delta S_l## simply be equal to the surface area of the function??)