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20.9 Surface Integrals

In three dimensions we can integrate over a surface that is sufficiently nice, by dividing it into small pieces and proceeding exactly as for an area in the plane.

A plane is, after all, only a particularly simple and straight example of a surface in three dimensions.

We consider surfaces that for the most part look like planes at small distances so that each tiny piece of surface will have an area d S that is essentially a small piece of a plane.

However, each small piece of surface area, d S , has a normal direction n ⟶ and once again it is appropriate to consider the vector d S ⟶ which is its area, d S times its (outward) normal vector n ⟶ .

You may sum all of these small vectors multiplied by an integrand and define Riemann sums for these to get a surface integral.

The last sentence above refers to multiplying the vector d S ⟶ (remember d S ⟶ is | d S ⟶ | multiplied by the unit outward normal to the surface, n ⟶ ) by the integrand; and there are three obvious ways to do this.

If the integrand is a function f ( x , y , z ) we can multiply, and the sum will be a vector. This is OK but it is the least common thing to do.

The standard thing to do is to have an integrand vector v ⟶ ( x , y ) , and take its dot product with d S ⟶ , and sum that dot product over the pieces. This is the most common form of surface integral.

We denote such an integral by

∫ S v ⟶ ( r ⟶ ) · d S ⟶
or
∫ S v ⟶ ( x , y , z ) · n ⟶ d S

This kind of integral is particular useful in physical applications.

In particular when the vector v ⟶ is a current density, v ⟶ · n ⟶ is then defined to be the amount of whatever v ⟶ is current density of that flows through a surface with normal n ⟶ and surface area d S per unit time.

The integral above then tells how much of that stuff flows through the surface S per unit time.

Current densities are defined for mass, and charge, but surface integrals of this kind are important as well in discussions of electric and magnetic fields.

Gauss's Law , for example states that in electrostatics, the total electric charge within a region R is a constant times the integral over the surface δ R of that region of the component of the electric field normal to the surface

∫ δ R E ⟶ · n ⟶ d S = c Q

(There is a similar relation between the gravitational field and the amount of mass within a region; for magnetic fields, the apparent absence of magnetic charge (monopoles) means that the right hand side of the comparable equation is 0 for magnetic fields.)

Integrals of this kind are usually called Flux integrals.

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