Surface Integrals of Vector Fields
Consider a vector field v on S, that is, for each x in S, v(x) is a vector.
The surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector. This applies for example in the expression of the electric field at some fixed point due to an electrically charged surface, or the gravity at some fixed point due to a sheet of material.
Alternatively, if we integrate the normal component of the vector field, the result is a scalar. Imagine that we have a fluid flowing through S, such that v(x) determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S in unit amount of time.
This illustration implies that if the vector field is tangent to S at each point, then the flux is zero, because the fluid just flows in parallel to S, and neither in nor out. This also implies that if v does not just flow along S, that is, if v has both a tangential and a normal component, then only the normal component contributes to the flux. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, and integrate the obtained field as above. We find the formula
The cross product on the right-hand side of this expression is a surface normal determined by the parametrization.
This formula defines the integral on the left (note the dot and the vector notation for the surface element).
Read more about this topic: Surface Integral
Famous quotes containing the words surface and/or fields:
“But the surface of the Earth was meant for man. He wasn’t meant to live in a hole in the ground.”
—Edward L. Bernds (b. 1911)
“Or seen the furrows shine but late upturned,
And where the fieldfare followed in the rear,
When all the fields around lay bound and hoar
Beneath a thick integument of snow.
So by God’s cheap economy made rich
To go upon my winter’s task again.”
—Henry David Thoreau (1817–1862)