Multivariable Calculus Lecture #13 Notes. in each piece. Then the mass mk. 0 σ = σ = σ

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1 Multivariable Calculus Lecture #1 Notes In this lecture, we ll loo at parameterization of surfaces in R and integration on a parameterized surface Applications will include surface area, mass of a surface or total electric charge on a surface given an associated densit function; average value of a function on a surface; and the flu (flow) of a vector field through a surface We will also state the Divergence Theorem (which relates flu of a vector field outward across a closed surface to the divergence of the vector field in the region bounded b this surface) and toes' Theorem (which relates the circulation of a vector field around a closed curve to the curl of that vector field on a surface bounded b this closed curve) We ll dela the proofs of these important theorems until the net lecture A motivating eample uppose a surface in R is composed of a material of varing (surface) densit σ ( z,, ) =σ( ) that is measured in units of mass per area If we wanted to calculate the mass of such a surface we would liel just weigh it, but if weighing it is not an option, we can use Riemann ums to develop an epression for its mass To this end, suppose we partition this surface into small pieces and choose a sample point (,, z ) = in each piece Then the mass m of the -th piece can be approimated as m σ(,, z) =σ( ) where is the surface area of the -th piece umming the masses of all of these pieces we can sa that Mass( ) = m s(,, z ) = s( ) = ma diam( ), If the mesh of this partition is { } we can formall calculate the limit as the mesh approaches zero If this limit eists independent of an choices made in the construction, we define lim (,, ) (,, ) ( ) z z d d σ = σ = σ, the surface integral of this densit function over the surface It is often useful to use the differential shorthand notation dm =σ (,, z) d =σ d when woring with such in integral This enables us to write: Of course, there s nothing special about mass here We could just as easil 1) Mass( ) = dm = s d have used a charge densit to calculate the total charge distributed on the surface or a population densit to find the total population on this surface Indeed, we could calculate the total amount of anthing distributed on the surface b integrating its associated densit function There an even more basic application the surface area of a given surface The idea is the same as above, but without the densit ) Area( ) = d This is a simple epression, but the actual calculation will require us to understand how to epress the element of surface area d in a manner that lends itself to the previous methods we have discussed using iterated single integrals More on that shortl ) We can also define the average value of a function defined on a surface in a manner analogous to how we ve defined this in other contets Generall, we average a function over a geometric object b integrating the function over the object and diving b the geometric content of the object In this contet, the unweighted average of the function f( z,, ) over a surface will be f d f (,, z) d f = = Area( ) Area( ) 1 Revised December 9, 17

2 ) The centroid (geometric center) can be defined as we ve done before b averaging the individual (Cartesian) coordinate functions, ie ( z,, ) o d =, d =, and z = z d Area( ) Area( ) Area( ) We can also define weighted averages as we ve done before if there is an associated densit function that allows us to weigh some areas more than others 5) If is a surface on which a densit function σ ( z,, ) =σ( ) is defined, we define the weighted average of a function f( z,, ) over the surface to be f wtd f dm f sdm f (,, z) s(,, z) d = = = Mass( ) dm s(,, z) d 6) If mass is distributed over a surface with a surface densit function σ ( z,, ), we can determine its center of mass b calculating the weighted averages of the (Cartesian) coordinate functions That is, we define dm dm z dm ( cm, cm, z cm) where cm =, cm =, and zcm = Mass( ) Mass( ) Mass( ) Flu of a vector field across a surface One of the most common applications of surface integrals is in the calculation of the flu of a vector field across (or through) a surface uppose a piecewise smooth surface is orientable in the sense that it s possible to consistentl choose one distinguished side of the surface (Möbius bands and Klein bottles need not appl) Tpicall this might be the outside of a closed surface or, perhaps, the blue side if one side was painted blue and the other side red For such an oriented surface we can determine a unit normal vector n at ever point (ecept where distinct facets of the surface meet) that is perpendicular to this surface in the direction of the distinguished side Let s further suppose that there is a vector field F= F ( z,, ) defined everwhere in the vicinit of this surface If this vector field represented some ind of flow such a water moving with varing direction and velocit, we might be interested in measuring how much water is flowing across some permeable (or virtual) surface, possibl per some unit of time How would we do this? If we partition this surface into small patches, then the flu (flow) through an patch should be approimatel proportional to the area of the patch and to the normal component F N = Fnof the vector field at some sample point in this patch, ie FN If we sum the contributions of all the patches in the partition, we get FN, and if we pass to the limit as the mesh tends to zero we get: 7) Flu of F through = F = F d = Fn d = F d lim N N Here we use the notation d= n d for the vector element of surface area We now turn to the matter of how to calculate surface integrals The primar method is to parameterize the surface and to then use this parameterization to pull bac the integral to the parameter space to get an ordinar double integral which can be calculated using iterated single integrals Integration on urfaces - Toolits The main tools for calculating such integrals are (a) parameterization of the surface (or, equivalentl, finding two coordinates defined on the surface that provide a mesh for the surface), and (b) an epression for the Revised December 9, 17

3 element of surface area d defined b the parameterization or coordinates For flu integrals, it s also hand to have an epression for a unit normal vector n for the surface Though there is an all-purpose method for calculating surface integrals for an parameterized surface, it is often easier and more geometricall clear to focus on the special cases of clinders, spheres, and graphs Each situation has its own toolit General method for an parameterized surface Parameterization: uppose is a surface parameterized b a vector-valued function r (,) st = st (,), st (,), zst (,) where the parameters s and t var over some domain in the parameter space D The onl requirement is that the curves in the surface produced b varing one parameter at a time provide a mesh on the surfaces, ie these curves should intersect cleanl (transversall) and produce a patchwor of small cells on the surface that can be used to build Riemann ums on the surface urface area element: If we var s b an amount s, we move an approimate vector displacement along a cross-section of the graph of s, and if we also var t b an amount t, we move an approimate vector displacement along a cross-section of the graph of t These two displacements will span a patch of the surface and the cross-product can be used to determine its approimate area pecificall, s t = s t This is a vector with magnitude approimatel equal to the area of a patch on the graph surface and direction normal to the surface (actuall in the upward normal direction) The magnitude is s t Within a Riemann um epression as the limit of the mesh tends to zero, this ields the surface area element d = dsdt for a general parameterized surface We can also mae use of the vector element of surface area d= n d = dsdt where n denotes the unit normal vector to the surface, oriented in a manner consistent with the cross-product Unit normal vector: From the above calculation, we also see that a unit normal vector to the surface with orientation consistent with the cross product is n = r r In particular, if F = PQR,, is a vector field defined on the surface, the flu of F through can be calculated as d ( ) F = F r D r dsdt [The integrand is the triple scalar product] Revised December 9, 17

4 Clinder Cartesian equation: + = R = Rcosθ = Rsinθ The parameter θ allows movement around the clinder with = z Parameterization: z θ, and the parameter z (which does double-dut as both a coordinate and a parameter) allows movement up and down the clinder urface area element: If we var θ b an amount θ, we move a distance R θ around the clinder, and if we also move an amount z to span a patch of the surface, the area of this patch will be = ( R z)( θ) Within a Riemann um epression as the limit of the mesh tends to zero, this ields the surface area element d = Rdzdθ for a clinder Unit normal vector: We can use gradient methods or observation to see that at an point (, z, ) on the clinder,,, the outward unit normal vector to the surface is n = = cos θ,sin θ, R This method can be modified as necessar for clinders around the -ais or -ais phere Cartesian equation: + + z = R = Rcosθsinφ Parameterization: = Rsinθsinφ The parameter θ (longitude) allows movement z = Rcosφ around the sphere with θ, and the parameter φ (the inclination, related to latitude) allows movement up and down the sphere with φ urface area element: If we var φ b an amount φ, we move a distance R φ along a longitude, and if we also move an amount θ along a latitude (at a radius from the z- ais of r = Rsinφ ), we will move a distance Rsinφ θ along a latitude Together, these will span a patch of the surface with area approimatel ( Rsin φ θ)( R φ) = R sinφ θ φ Within a Riemann um epression as the limit of the mesh tends to zero, this ields the surface area element d = R sinφdφdθ for a sphere Unit normal vector: We can use gradient methods or observation to see that at an point (, z, ) on the sphere, z,, the outward unit normal vector to the surface is n = = cosθsin φ,sinθsin φ,cosφ R Graph z= f (, ) Cartesian equation: z= f (, ) = Parameterization: = or r (, ) = f,, (, ) The variables and here do z= f (, ) double-dut as both parameters and coordinates The var in the -plane over the domain of the function that describes this graph surface Revised December 9, 17

5 urface area element: If we var b an amount, we move an approimate vector displacement along a cross-section of the graph of = 1,, f, and if we also var b an amount, we move an approimate vector displacement along a cross-section of the graph of =,1, f Together, these two displacements will span a patch of the surface and we can use the cross-product to determine its approimate area pecificall, = = 1,, f,1, f = f, f,1 This is a vector with magnitude approimatel equal to the area of a patch on the graph surface and direction normal to the surface (actuall in the upward normal direction) The magnitude is + f + f Within a Riemann um epression as the limit of the mesh tends to zero, this 1 ields the surface area element d + f + f dd for a graph surface We can also mae use of the 1 vector element of surface area d= n d = f, f,1 dd where n denotes the (upward) unit normal vector to the graph surface Unit normal vector: From the above calculation, we see that a unit (upward) normal vector to a graph surface is f, f,1 dd n = An alternative geometric argument also shows that d = = 1+ f + f dd 1+ f + f n Note: [This can be easil adapted to the case where = f(,) z with where = f( z, ) with ddz d = = 1+ f + fz ddz ] n j ddz d = = 1+ f + fz ddz ni Eamples: (1) urface area of a sphere of radius R: Area( ) = d = R sinφdφdθ The inner integral gives R cos φ φ= = R ( 1 1) = R, and the φ = outer integral gives ( R )( ) = R () The flu of the vector field F =, z, outward through a sphere of radius centered at the origin: z,, Flu = F d= d Fn We use the unit outward normal vector n =, so z,, + z + z Fn =, z, = Therefore, Flu = d If we substitute the = cosθsinφ parameterization = sinθsinφ and the area element d = sinφdφdθ, we get z = cosφ Flu = ( cos sin 8sin sin + 16 cos sin ) d d θ φ θ φ φ φ φ θ This ma not be the simplest integral, but it s quite doable The inner integral gives or 5 Revised December 9, 17

6 ( cos (1 cos ) sin 8sin (1 cos ) sin + 16 cos sin ) θ φ φ θ φ φ φ φ dφ cos φ = [ cos θ 8sin θ] cosφ+ cos φ = [ cos θ 8sin θ] [] 1 cos 1 cos θ 8 θ = + = ( cos θ) The outer integral is then 6 (8 16cos ) d + θ θ = = Note: This integral can also be (more simpl) done using the Divergence Theorem We calculate div F = 1+ z = + z, and Flu = F d= (div F) dv = ( + z) dv = dv = Vol( B) = ( ) = = Bnd( B) B B B () urface area of the paraboloid z = + ling over the dis + in the -plane There are several good approaches If were to use (, ) as parameters, we might describe the paraboloid = parametricall b = or r (, ) =,, + The methods described above (with z= f(, )) z = + give us that d = 1+ f + f dd = 1 + ( ) + ( ) dd = 1+ ( + ) dd, so the surface area would be D 1 ( ) + + dd The sensible thing is to change to polar coordinates to calculate this integral over the dis This gives r rdrdθ = = ( ) = rcosθ We could also have begun b using (, r θ ) as parameters This would give = rsinθ or z = r r r r ( r, θ) = rcos θ, rsin θ, r We calculate = cos θ,sin θ, r and = rsin θ, r cos θ,, so θ d = = r cos θ, r sin θ, r = r rcos θ, rsin θ,1 and θ d = drdθ = r r cos θ, r sin θ,1 drdθ = r r (cos θ + sin θ) + 1 drdθ = r 1+ r drdθ θ we again get θ 6 Area( ) = d = 1+ r rdrd = = ( ) Divergence and curl of vector fields in R Recall that if F ( z,, ) = Pz (,, ), Qz (,, ), Rz (,, ) defines a vector field in some region in R where the component functions Pz (,, ), Qz (,, ) and Rz (,, ) are differentiable, we define: P Q R div( F ) = + + (divergence of F) z R Q P R Q P curl( F ) =,, (curl of F) z z 6, so 6 Revised December 9, 17

7 Divergence Theorem (also nown as Gauss Theorem) uppose is a closed surface in R that bounds a solid region B and that this boundar is oriented via an outward unit normal vector n Further suppose that F = Pz (,, ), Qz (,, ), Rz (,, ) is a vector field defined and differentiable throughout B (and its boundar) with continuous first partial derivatives Then: net flu of F outward = FN d = d = = div( ) dv across = B = B Fn = B Fd = B F B This theorem provides some eplanation for the interpretation of the divergence of a vector field as a source densit Essentiall, the total amount of stuff flowing outward across the boundar of a closed region should measure the total amount of the source of that stuff emanating from within the region toes Theorem uppose is an oriented surface in R (with unit normal vector n defined on the surface to choose a side ) with boundar curve C oriented in the counterclocwise sense, ie if the unit normal vector n represents up, then ou traverse the boundar in such a wa that the surface is to our left [ Bnd( ) = = C ] Further suppose that F = Pz (,, ), Qz (,, ), Rz (,, ) is a vector field defined and differentiable throughout (and its boundar) with continuous first partial derivatives Then: circulation of = d = curl( ) = [curl( ) ] d around C = F r C= F d F n F This theorem provides some eplanation for the interpretation of the curl of a vector field as a circulation densit, ie a measure of local rotation of the vector field Essentiall, the circulation of the vector field around the perimeter is the same as the integral of the circulation densit over the surface In the net lecture we ll prove these theorems and appl them in a variet of was both theoretical and practical We ll also show that Green s Theorem is a corollar of toes Theorem We ll start b giving purel geometric, coordinate-free definitions of the divergence and curl of a vector field and show that the theorems are reall just corollaries (integral versions) of these definitions We ll then show that these geometric definitions ield the previousl stated algebraic definitions of divergence and curl Notes b Robert Winters and Renée Chipman 7 Revised December 9, 17