1. Surface Area and Surface Integrals

2. Surface Area
Given some surface in 3 space, we want to calculate its surface area
Just as before, a double integral can be used to calculate the area of a surface
We are going to look at how to calculate the surface area of a parameterized surface over a given region

3. Given the vector parameterization the surface area is given by
Let’s take a look at where this comes from
Example
Find the surface area of a cone with a height of 1
The parameterization is
Let’s check it out in maple

4. Alternative Notation
If we want to find the surface area of a function, z = f(x,y), than we can simplify the cross product
Then
and

5. Alternative Notation
If we want to find the surface area of a function, z = f(x,y), than we can use the following
Example
Find the surface area of the plane
z = 6 – 3x – 2y that lies in the first octant

6. We can calculate the surface area over any given region Example
Find the surface area of the function z = xy between the two cylinders

7. Surface Integrals
A surface integral involves integrating a function over some surface in 3 space
We have calculated integrals of functions over regions in the xy plane and over 3 dimensional figures, now we want to integrate over a 2 dimensional surface in 3 space
Thus if the function represents a density, the surface integral would calculate the total mass of the 2D plate that has the shape of the surface

8. Surface Integrals
To calculate a surface integral of g over the surface D if the surface is defined parametrically we have
Example
Calculate the surface integral of f(x,y) = xy over the cone of radius 1 in the first octant from the previous example

9. Surface Integrals
To calculate a surface integral of g over the surface D if the surface is given by z = f(x,y) we can use
Example
Find the surface integral of the function g(x,y,z) = xyz over the plane z = 6 – 3x – 2y that lies in the first octant

10. Surface Integrals of Vector Fields
Recall that a line integral of a vector field could be interpreted as work done by the force field on a particle moving along the path
If the vector field    represents the flow of a fluid, then the surface integral of   will represent the amount of fluid flowing through the surface (per unit time)
In this case the amount of fluid flowing through the per unit time is called the flux
Surface integrals of a vector field are sometimes referred to as flux integrals

11. Surface Integrals of Vector Fields
The term flux comes from physics
It is used to denote the rate of transfer of:
Fluid
liquid flow density
Particles
Electromagnetism
Energy across a surface
Total charge of a surface

12. Surface Integrals of Vector Fields
Imagine water flowing through a surface
If the flow of water is perpendicular to the surface a lot of water will flow through and the flux will be large
If the flow of water is parallel to the surface then no water will flow through the surface and the flux will be zero
In order to calculate the flux we must add up the component of that is perpendicular to the surface

13. Surface Integrals of Vector Fields
Let represent a unit normal vector to the surface
Than in order to find the component of that is perpendicular we can use our dot product
This is 0 if and are perpendicular
Positive if and are in the same direction
Negative if and are in opposite directions
Given some fluid flow , integrating will determine the total flux of fluid through a surface
It will be positive if it is in the same direction as
Negative if it is in the opposite direction of

14. Surface Integrals of Vector Fields
Now we must sum over our surface so we will combine our dot product with our formula for a surface integral from before
and we get the following
This can be simplified!

15. Surface Integrals of Vector Fields
The formula for a unit normal vector given our surface parameterization is
Inserting that into our surface integral
we get

16. Surface Integrals of Vector Fields
We can cancel scalars
to get
Example
The surface will be the parabaloid , 0 ≤ z ≤ 1 with the vector field
Should our integral be positive or negative?
How can we tell?

17. Surface Integrals of Vector Fields
In order for a surface to have an orientation the surface must have two sides
Thus every point will have two normal vectors,
The set we choose determines the orientation which is described as the positive orientation
You should be able to choose a normal vector in a way so that if it varies in a continuous way over the surface, when you return to the initial position it still points in the same direction

18. The Möbius band is not orientable
No matter where you start to construct a continuous unit normal field, moving the vector continuously around the surface will return it to the starting point with a direction opposite to the one it had when it started.

19. Surface Integrals of Vector Fields
As mentioned before, a surface integral over a vector field is positive if the normal of the surface and flow are in the same direction, negative if they are in opposite directions and 0 if they are perpendicular
How do we know which normal to use for a surface?
A surface is closed if it is the boundary of some solid region
For example the surface of a sphere is closed
A closed surface has a positive orientation if we choose the set of normal vectors that point outward from the region
A closed surface has a negative oritenation if we choose the set of normal vectors that point inward toward the region
This convention is only used for closed surfaces
The surface in our previous example was not closed so this does not apply

20. Surface Integrals of Vector Fields
In order to calculate our surface integral we use
Since is a normal vector to the surface we can rewrite the integral as
Now if our surface is given by a function z = f(x,y) then
where f(x,y,z) = f(x,y) - z and our integral becomes
Let’s try our previous example again with this method

21. Surface Integrals of Vector Fields
Calculate the flux of of the surface S which is a hemisphere given by the following
In this case we have a closed bounded region so our surface has a positive orientation that is pointing outwards
Should we expect our integral to be positive or negative?
In order to calculate this integral we will have to break S into 2 separate regions

22. Relationship between Surface Integrals and Line Integrals
To calculate a line integral we use
which summed up the components of the vector field that were tangent to the path given by
To calculate a surface integral we use
which sums up the components of the vector field that are in the normal direction given by