From Math 2220 Class 36 Integrals Other Problems Dr. Allen Back - PowerPoint PPT Presentation

From Math 2220 Class 36 V2c Surface Integrals Why Surface From Math 2220 Class 36 Integrals Other Problems Dr. Allen Back Why Green’s Greens Problems Flow Lines Nov. 19, 2014 Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Surface Integrals From Math Picture of � T u , � T v for a Lat/Long Param. of the Sphere. 2220 Class 36 V2c Surface Integrals Why Surface Integrals Other Problems Why Green’s Greens Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Surface Integrals From Math Basic Parametrization Picture 2220 Class 36 V2c Surface Integrals Why Surface Integrals Other Problems Why Green’s Greens Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Surface Integrals From Math Parametrization Φ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v )) 2220 Class 36 V2c Tangents T u = ( x u , y u , z u ) T v = ( x v , y v , z v ) Surface Integrals Area Element dS = � � T u × � T v � du dv Why Surface Normal � N = � T u × � Integrals T v Other T u × � � T v | Problems Unit normal ˆ n = ± � � T u × � Why Green’s T v � (Choosing the ± sign corresponds to an orientation of the Greens Problems surface.) Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Surface Integrals From Math Two Kinds of Surface Integrals 2220 Class 36 Surface Integral of a scalar function f ( x , y , z ) : V2c Surface �� Integrals f ( x , y , z ) dS Why Surface S Integrals Other Surface Integral of a vector field � F ( x , y , z ) : Problems Why Green’s �� � Greens F ( x , y , z ) · ˆ n dS . Problems S Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Surface Integrals Surface Integral of a scalar function f ( x , y , z ) calculated by From Math 2220 Class 36 �� �� V2c f (Φ( u , v )) � � T u × � f ( x , y , z ) dS = T v � du dv S D Surface Integrals where D is the domain of the parametrization Φ . Why Surface Surface Integral of a vector field � F ( x , y , z ) calculated by Integrals �� Other � Problems F ( x , y , z ) · ˆ n dS Why Green’s S � � � T u × � Greens �� T v | � � � T u × � Problems = ± F (Φ( u , v )) · T v � du dv � � T u × � T v � Flow Lines D Div and Curl where D is the domain of the parametrization Φ . Surface of Revolution Case Graph Case Surface Parametriza- tion

Surface Integrals From Math 3d Flux Picture 2220 Class 36 V2c Surface Integrals Why Surface Integrals Other Problems Why Green’s Greens Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Surface Integrals The preceding picture can be used to argue that if � F ( x , y , z ) is From Math 2220 Class 36 the velocity vector field, e.g. of a fluid of density ρ ( x , y , z ), V2c then the surface integral �� Surface Integrals ρ� F · ˆ n dS Why Surface S Integrals Other (with associated Riemann Sum Problems Why Green’s � k ) � ρ ( x ∗ i , y ∗ j , z ∗ F ( x ∗ i , y ∗ j , z ∗ n ( x ∗ i , y ∗ j , z ∗ k ) · ˆ k ) ∆ S ijk ) Greens Problems Flow Lines represents the rate at which material (e.g. grams per second) Div and Curl crosses the surface. Surface of Revolution Case Graph Case Surface Parametriza- tion

Surface Integrals From Math 2220 Class 36 V2c Surface From this point of view the orientation of a surface simple tells Integrals us which side is accumulatiing mass, in the case where the Why Surface Integrals value of the integral is positive. Other Problems Why Green’s Greens Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Surface Integrals From Math 2d Flux Picture 2220 Class 36 V2c Surface Integrals Why Surface Integrals Other Problems Why Green’s Greens Problems Flow Lines Div and Curl Surface of Revolution There’s an analagous 2d Riemann sum and interp of Case Graph Case � � F · ˆ n ds . Surface Parametriza- C tion

Surface Integrals From Math 2220 Class 36 V2c Surface Integrals Why Surface Integrals Other Problems Why Green’s Greens Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Surface Integrals From Math 2220 Class 36 Problem: Calculate V2c �� Surface � F ( x , y , z ) · ˆ n dS Integrals S Why Surface Integrals for the vector field � F ( x , y , z ) = ( x , y , z ) and S the part of the Other paraboloid z = 1 − x 2 − y 2 above the xy -plane. Choose the Problems Why Green’s positive orientation of the paraboloid to be the one with normal Greens pointing downward. Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Surface Integrals From Math 2220 Class 36 V2c Surface Integrals Problem: Calculate the surface area of the above paraboloid. Why Surface Integrals Other Problems Why Green’s Greens Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Why Surface Integrals From Math The Vector Case 2220 Class 36 V2c 1 Electromagnetism. (Two of Maxwell’s equations involve Surface surface integrals.) Integrals 2 Partial differential equations. (Integration by parts is an Why Surface Integrals important technique here, and is based on the chapter 8 Other integral theorems.) Problems 3 Heat, fluid flow and difffusion. Why Green’s Greens 4 (Related to Markov chains in probability.) Problems 5 At an advanced level, financial mathematics. (e.g. Flow Lines Black-Scholes is similar to the heat equation.) Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Why Surface Integrals From Math 2220 Class 36 V2c Surface Integrals Why Surface Integrals Other Problems Why Green’s Greens Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Why Surface Integrals From Math 2220 Class 36 V2c The Scalar Case Surface Integrals Why Surface 1 As with multiple integrals, going from a density of Integrals something (this time per unit surface area) to a total. Other Problems 2 Vector surface integrals are surface integrals of the scalar Why Green’s function “normal component.” Greens Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Why Surface Integrals From Math 1d-Heat Equation 2220 Class 36 ∂ t = c ∂ 2 T ∂ T V2c ∂ x 2 Surface ( T ( x , t ) is the temperature at position x and Integrals time t along a rod.) Why Surface Integrals Black-Scholes Other ∂ t + cS 2 ∂ 2 V ∂ V ∂ S 2 + rS ∂ V Problems ∂ S − rV = 0 . Why Green’s ( V ( S , t ) is the price of a derivative as a function Greens Problems of an underlying stock price S at time t . ) Flow Lines (See e.g. “Black-Scholes model” in Wikipedia.) Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Other Problems From Math 2220 Class 36 Problem: Find V2c �� � F · ˆ n dS Surface Integrals S Why Surface Integrals for � F ( x , y , z ) = (0 , yz , z 2 ) and S the portion of the cylinder y 2 + z 2 = 1 with 0 ≤ x ≤ 1, z ≥ 0, and the positive Other Problems orientation chosen to be a radial outward (from the axis of the Why Green’s cylinder) normal. Greens Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Other Problems From Math 2220 Class 36 Problem: Letting � r denote the vector field ( x , y , z ), find the V2c value of the surface integral Surface Integrals �� � r Why Surface r � 3 · ˆ n dS Integrals � � S Other Problems where S is the sphere of radius R about the origin, oriented Why Green’s with an outward normal. Greens Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Other Problems From Math Spherical coordinates (letting ρ = R be constant) gives a 2220 Class 36 natural parametrization of the sphere of radius R centered at V2c the origin. Surface It is geometrically to be expected (and you’ve done some Integrals verifications like this at least in look at problems) that the Why Surface Integrals Jacobian result for spherical coordinates translates to the area Other element on such a sphere to be Problems Why Green’s dS = R 2 sin φ d φ d θ. Greens Problems Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion

Why Green’s From Math 2220 Class 36 Green’s theorem says that for simple closed (piecewise smooth) V2c curve C whose inside is a region R , we have Surface Integrals � �� ∂ Q ∂ x − ∂ P P ( x , y ) dx + Q ( x , y ) dy = ∂ y dx dy Why Surface Integrals C R Other Problems as long as the vector field � F ( x , y ) = ( P ( x , y ) , Q ( x , y )) is C 1 Why Green’s on the set R and C is given its usual “inside to the left” Greens Problems orientation. Flow Lines Div and Curl Surface of Revolution Case Graph Case Surface Parametriza- tion