2 Surface AreaGiven some surface in 3 space, we want to calculate its surface areaJust as before, a double integral can be used to calculate the area of a surfaceWe 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 fromExampleFind the surface area of a cone with a height of 1The parameterization isLet’s check it out in maple
4 Alternative NotationIf we want to find the surface area of a function, z = f(x,y), than we can simplify the cross productThenand
5 Alternative NotationIf we want to find the surface area of a function, z = f(x,y), than we can use the followingExampleFind the surface area of the planez = 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 IntegralsA surface integral involves integrating a function over some surface in 3 spaceWe 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 spaceThus 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 IntegralsTo calculate a surface integral of g over the surface D if the surface is defined parametrically we haveExampleCalculate the surface integral of f(x,y) = xy over the cone of radius 1 in the first octant from the previous example
9 Surface IntegralsTo calculate a surface integral of g over the surface D if the surface is given by z = f(x,y) we can useExampleFind 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 pathIf 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 fluxSurface integrals of a vector field are sometimes referred to as flux integrals
11 Surface Integrals of Vector Fields The term flux comes from physicsIt is used to denote the rate of transfer of:Fluidliquid flow densityParticlesElectromagnetismEnergy across a surfaceTotal charge of a surface
12 Surface Integrals of Vector Fields Imagine water flowing through a surfaceIf the flow of water is perpendicular to the surface a lot of water will flow through and the flux will be largeIf the flow of water is parallel to the surface then no water will flow through the surface and the flux will be zeroIn 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 surfaceThan in order to find the component of that is perpendicular we can use our dot productThis is 0 if and are perpendicularPositive if and are in the same directionNegative if and are in opposite directionsGiven some fluid flow , integrating will determine the total flux of fluid through a surfaceIt will be positive if it is in the same direction asNegative 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 beforeand we get the followingThis can be simplified!
15 Surface Integrals of Vector Fields The formula for a unit normal vector given our surface parameterization isInserting that into our surface integralwe get
16 Surface Integrals of Vector Fields We can cancel scalarsto getExampleThe surface will be the parabaloid , 0 ≤ z ≤ 1 with the vector fieldShould 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 sidesThus every point will have two normal vectors,The set we choose determines the orientation which is described as the positive orientationYou 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 perpendicularHow do we know which normal to use for a surface?A surface is closed if it is the boundary of some solid regionFor example the surface of a sphere is closedA closed surface has a positive orientation if we choose the set of normal vectors that point outward from the regionA closed surface has a negative oritenation if we choose the set of normal vectors that point inward toward the regionThis convention is only used for closed surfacesThe 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 useSince is a normal vector to the surface we can rewrite the integral asNow if our surface is given by a function z = f(x,y) thenwhere f(x,y,z) = f(x,y) - z and our integral becomesLet’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 followingIn this case we have a closed bounded region so our surface has a positive orientation that is pointing outwardsShould 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 usewhich summed up the components of the vector field that were tangent to the path given byTo calculate a surface integral we usewhich sums up the components of the vector field that are in the normal direction given by