From Math 2220 Class 39 and Gauss Conservative Vector Fields Dr. - PowerPoint PPT Presentation

Why Green’s and Gauss’ From Math 2220 Class 39 V2 The differential forms point of view introduced in section 8.5 Stokes and makes all these theorems one theorem, usually called Stokes’, Gauss and the proof becomes a combination of more advanced linear Why Green’s and Gauss’ algebra constructions (differential forms) together with the one Conservative variable Fundamental Theorem of Calculus. Vector Fields Time permitting, we’ll talk a bit about this on Monday. Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Conservative Vector Fields A vector field � From Math F ( x , y , z ) which can be written as 2220 Class 39 V2 � F = ∇ f Stokes and Gauss Why Green’s and Gauss’ is called conservative. We already know Conservative Vector Fields � � F · d � s = 0 Systematic Method of C Finding a Potential dTheta for any closed curve. Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Conservative Vector Fields From Math 2220 Class 39 V2 Stokes and The origin of the term is physics (I think) where in the case of Gauss � F a force, it does no work (and so saps/adds no energy) as a Why Green’s and Gauss’ particle traverses the closed curve. Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Conservative Vector Fields A vector field � F ( x , y , z ) which can be written as From Math 2220 Class 39 � F = ∇ f V2 Stokes and Gauss is called conservative. We already know Why Green’s and Gauss’ � � F · d � s = 0 Conservative Vector Fields C Systematic Method of for any closed curve. Finding a Potential In physics, the convention is to choose φ so that dTheta � F = −∇ φ Integral Theorem Problems Surface of and φ is referred to as the potential energy. Revolution Case Graph Case Surface Integrals

Conservative Vector Fields From Math 2220 Class 39 V2 Stokes and Conservation of energy (in e.g mechanics) becomes a theorem Gauss in multivariable calculus combining the definition of a flow line Why Green’s and Gauss’ with the computation of a line integral. Newton’s 2nd law ( � Conservative F = m � a and other versions) is also key . . . . Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Conservative Vector Fields From Math 2220 Class 39 V2 Stokes and Gauss The concept of voltage arises here too; it is just a potential Why Green’s energy per unit charge. and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Conservative Vector Fields From Math 2220 Class 39 The theorem (vector identity) curl ( ∇ f ) = 0 means the V2 curl ( � F ) = 0 Stokes and Gauss Why Green’s is a necessary condition for the existence of a function f and Gauss’ satisfying Conservative Vector Fields Systematic ∇ f = � F . Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Conservative Vector Fields From Math 2220 Class 39 It turns out that for vector fields defined on e.g. all of R 2 or V2 R 3 , the converse of the theorem curl ( ∇ f ) = 0 is true. (For R 2 , Stokes and we’re thinking of the scalar curl.) Gauss F ) = 0, (for a C 1 vector In other words, in such a case, if curl ( � Why Green’s and Gauss’ field), there is guaranteed to be a function f ( x , y , z ) such that Conservative Vector Fields ∇ f = � F . Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Conservative Vector Fields From Math 2220 Class 39 V2 Stokes and While this hold for vector fields with domains R 2 , R 3 , or more Gauss generally any “simply connected” region, the example d θ below Why Green’s and Gauss’ shows this converse does not hold in general. Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Conservative Vector Fields From Math 2220 Class 39 V2 Stokes and Gauss Time permitting, we’ll talk about simple connectivity next Why Green’s week. and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Systematic Method of Finding a Potential From Math 2220 Class 39 V2 Stokes and Finding a potential by inspection is fine when you can, but it is Gauss Why Green’s not systematic. and Gauss’ I often ask on a final exams for this. Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Systematic Method of Finding a Potential From Math 2220 Class 39 V2 Problem: Use a systematic method to find a function f ( x , y , z ) Stokes and Gauss for which Why Green’s ∇ f = (2 xy , x 2 + z 2 , 2 yz + 1) . and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Systematic Method of Finding a Potential From Math 2220 Class 39 V2 Problem: Use a systematic method to find a function f ( x , y , z ) for which Stokes and Gauss ( y 2 ze xyz + 1 y , (1 + xyz ) e xyz − x Why Green’s ∇ f = y 2 , and Gauss’ Conservative − (cos 2 ( xyz )) e z + xy 2 e xyx − e z sin 2 ( xyz )) . Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Systematic Method of Finding a Potential From Math 2220 Class 39 V2 Problem: Use a systematic method to find a function f ( x , y , z ) Stokes and Gauss for which Why Green’s ∇ f = (2 xz , 2 y , x 2 + 6 e z ) . and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

d θ From Math 2220 Class 39 V2 Stokes and Gauss � − y dx + x dy Why Green’s and Gauss’ C Conservative for C the unit circle traversed counterclockwise. Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

d θ From Math 2220 Class 39 V2 Stokes and Gauss � x dx + y dy Why Green’s and Gauss’ C Conservative for C the unit circle traversed counterclockwise. Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

d θ From Math 2220 Class 39 � − y dx + x dy V2 x 2 + y 2 C Stokes and Gauss is still nonzero for C a circle of radius R centered at the origin Why Green’s and Gauss’ traversed counterclockwise. Conservative This is remarkable since Vector Fields Systematic ∇ tan − 1 y 1 Method of Finding a x = x 2 + y 2 ( − y , x )) . Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

d θ From Math 2220 Class 39 V2 Stokes and Gauss � Why does this not contradict C ∇ f · d � s = 0 for a closed curve Why Green’s and Gauss’ (i.e. start=end) C ? Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Integral Theorem Problems Problem: Let From Math 2220 Class 39 F ( x , y , z ) = ( y 2 + z 2 , x 2 + z 2 , x 2 ) . V2 � Stokes and Gauss Why Green’s Find and Gauss’ � � F · d � s Conservative Vector Fields C Systematic Method of where C is the boundary of the plane x + 2 y + 2 z = 2 Finding a Potential intersected with the first octant, oriented counterclockwise dTheta from above. Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Integral Theorem Problems From Math 2220 Class 39 Problem: Find the flux of the vector field V2 Stokes and � F ( x , y , z ) = ( xy , yz , xz ) Gauss Why Green’s and Gauss’ Conservative through the boundary of the unit cube Vector Fields 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 , 0 ≤ z ≤ 1 where the boundary of the Systematic Method of cube has its usual outward normal. Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Integral Theorem Problems From Math 2220 Class 39 Problem: Find V2 �� � F · ˆ n dS Stokes and Gauss S Why Green’s and Gauss’ 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 Conservative Vector Fields orientation chosen to be a radial outward (from the axis of the Systematic Method of cylinder) normal. Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Integral Theorem Problems From Math 2220 Class 39 V2 1 � C x dy − y dx for C the boundary of the ellipse 2 Stokes and Gauss 3 2 + y 2 x 2 Why Green’s 4 2 = 1 and Gauss’ Conservative Vector Fields oriented counterclockwise. Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Integral Theorem Problems From Math 2220 Class 39 Let V2 1 � F = x 2 + y 2 ( − y , x ) . Stokes and Gauss Why Green’s and Gauss’ If C 1 and C 2 are two simple closed curves enclosing the origin Conservative (and oriented with the usual inside to the left), can you say Vector Fields C 1 � C 2 � � � whether one of F · d � s and F · d � s is bigger than the Systematic Method of other? Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface of Revolution Case From Math 2220 Class 39 This is not worth memorizing! V2 If one rotates about the z -axis the path (curve) z = f ( x ) in the Stokes and Gauss xz -plane for 0 ≤ a ≤ x ≤ b , one obtains a surface of revolution Why Green’s with a parametrization and Gauss’ Conservative Vector Fields Φ( u , v ) = ( u cos v , u sin v , f ( u )) Systematic Method of and dS =? Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface of Revolution Case From Math 2220 Class 39 V2 Stokes and Gauss � 1 + ( f ′ ( u )) 2 du dv . dS = u Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Graph Case From Math 2220 Class 39 V2 This is not worth memorizing! Stokes and For the graph parametrization of z = f ( x , y ), Gauss Why Green’s and Gauss’ Φ( u , v ) = ( u , v , f ( u , v )) Conservative Vector Fields and dS =? Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Graph Case From Math 2220 Class 39 V2 Stokes and Gauss � 1 + f 2 u + f 2 dS = v du dv . Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Graph Case For such a graph, the normal to the surface at a point From Math 2220 Class 39 ( x , y , f ( x , y )) (this is the level set z − f ( x , y ) = 0) is V2 ( − f x , − f y , 1) Stokes and Gauss Why Green’s and Gauss’ so we can see that Conservative Vector Fields 1 cos γ = Systematic � Method of 1 + f 2 x + f 2 y Finding a Potential dTheta determines the angle γ of the normal with the z -axis. Integral And so at the point ( u , v , f ( u , v )) on a graph, Theorem Problems 1 Surface of dS = cos γ du dv . Revolution Case Graph Case (Note that du dv is essentially the same as dx dy here.) Surface Integrals

Surface Integrals From Math Picture of � T u , � T v for a Lat/Long Param. of the Sphere. 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface Integrals From Math Basic Parametrization Picture 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface Integrals From Math Parametrization Φ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v )) 2220 Class 39 V2 Tangents T u = ( x u , y u , z u ) T v = ( x v , y v , z v ) Stokes and Gauss Area Element dS = � � T u × � T v � du dv Why Green’s Normal � N = � T u × � and Gauss’ T v Conservative T u × � � T v | Vector Fields Unit normal ˆ n = ± � � T u × � Systematic T v � Method of (Choosing the ± sign corresponds to an orientation of the Finding a Potential surface.) dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface Integrals From Math Two Kinds of Surface Integrals 2220 Class 39 Surface Integral of a scalar function f ( x , y , z ) : V2 Stokes and �� Gauss f ( x , y , z ) dS Why Green’s S and Gauss’ Conservative Surface Integral of a vector field � F ( x , y , z ) : Vector Fields Systematic Method of �� � Finding a F ( x , y , z ) · ˆ n dS . Potential S dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface Integrals Surface Integral of a scalar function f ( x , y , z ) calculated by From Math 2220 Class 39 �� �� V2 f (Φ( u , v )) � � T u × � f ( x , y , z ) dS = T v � du dv S D Stokes and Gauss where D is the domain of the parametrization Φ . Why Green’s Surface Integral of a vector field � F ( x , y , z ) calculated by and Gauss’ �� Conservative � Vector Fields F ( x , y , z ) · ˆ n dS Systematic S � � Method of � T u × � �� T v | Finding a � � � T u × � = ± F (Φ( u , v )) · Potential T v � du dv � � T u × � T v � D dTheta Integral where D is the domain of the parametrization Φ . Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface Integrals From Math 3d Flux Picture 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface Integrals The preceding picture can be used to argue that if � F ( x , y , z ) is From Math 2220 Class 39 the velocity vector field, e.g. of a fluid of density ρ ( x , y , z ), V2 then the surface integral �� Stokes and Gauss ρ� F · ˆ n dS Why Green’s S and Gauss’ Conservative (with associated Riemann Sum Vector Fields Systematic � k ) � ρ ( x ∗ i , y ∗ j , z ∗ F ( x ∗ i , y ∗ j , z ∗ n ( x ∗ i , y ∗ j , z ∗ k ) · ˆ k ) ∆ S ijk ) Method of Finding a Potential dTheta represents the rate at which material (e.g. grams per second) Integral crosses the surface. Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface Integrals From Math 2220 Class 39 V2 Stokes and From this point of view the orientation of a surface simple tells Gauss us which side is accumulatiing mass, in the case where the Why Green’s and Gauss’ value of the integral is positive. Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface Integrals From Math 2d Flux Picture 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems There’s an analagous 2d Riemann sum and interp of Surface of Revolution � Case � F · ˆ n ds . Graph Case C Surface Integrals

Surface Integrals From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface Integrals From Math 2220 Class 39 Problem: Calculate V2 �� Stokes and � F ( x , y , z ) · ˆ n dS Gauss S Why Green’s and Gauss’ for the vector field � F ( x , y , z ) = ( x , y , z ) and S the part of the Conservative paraboloid z = 1 − x 2 − y 2 above the xy -plane. Choose the Vector Fields Systematic positive orientation of the paraboloid to be the one with normal Method of Finding a pointing downward. Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Surface Integrals From Math 2220 Class 39 V2 Stokes and Gauss Problem: Calculate the surface area of the above paraboloid. Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Paraboloid z = x 2 + 4 y 2 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s The graph z = F ( x , y ) can always be parameterized by and Gauss’ Conservative Vector Fields Φ( u , v ) = < u , v , F ( u , v ) > . Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Paraboloid z = x 2 + 4 y 2 From Math 2220 Class 39 V2 Stokes and Gauss The graph z = F ( x , y ) can always be parameterized by Why Green’s and Gauss’ Conservative Φ( u , v ) = < u , v , F ( u , v ) > . Vector Fields Systematic Method of Finding a Potential Parameters u and v just different names for x and y resp. dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Paraboloid z = x 2 + 4 y 2 From Math 2220 Class 39 V2 Stokes and Gauss The graph z = F ( x , y ) can always be parameterized by Why Green’s and Gauss’ Conservative Φ( u , v ) = < u , v , F ( u , v ) > . Vector Fields Systematic Method of Finding a Potential Use this idea if you can’t think of something better. dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Paraboloid z = x 2 + 4 y 2 From Math The graph z = F ( x , y ) can always be parameterized by 2220 Class 39 V2 Φ( u , v ) = < u , v , F ( u , v ) > . Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Paraboloid z = x 2 + 4 y 2 The graph z = F ( x , y ) can always be parameterized by From Math 2220 Class 39 Φ( u , v ) = < u , v , F ( u , v ) > . V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Note the curves where u and v are constant are visible in the Graph Case wireframe. Surface Integrals

Paraboloid z = x 2 + 4 y 2 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ A trigonometric parametrization will often be better if you have Conservative Vector Fields to calculate a surface integral. Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Paraboloid z = x 2 + 4 y 2 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s A trigonometric parametrization will often be better if you have and Gauss’ to calculate a surface integral. Conservative Vector Fields Φ( u , v ) = < 2 u cos v , u sin v , 4 u 2 > . Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Paraboloid z = x 2 + 4 y 2 From Math A trigonometric parametrization will often be better if you have 2220 Class 39 to calculate a surface integral. V2 Stokes and Φ( u , v ) = < 2 u cos v , u sin v , 4 u 2 > . Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Paraboloid z = x 2 + 4 y 2 From Math A trigonometric parametrization will often be better if you have 2220 Class 39 to calculate a surface integral. V2 Stokes and Φ( u , v ) = < 2 u cos v , u sin v , 4 u 2 > . Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Algebraically, we are rescaling the algebra behind polar Integral Theorem coordinates where Problems x = r cos θ Surface of Revolution y = r sin θ Case leads to r 2 = x 2 + y 2 . Graph Case Surface Integrals

Paraboloid z = x 2 + 4 y 2 From Math A trigonometric parametrization will often be better if you have 2220 Class 39 to calculate a surface integral. V2 Stokes and Φ( u , v ) = < 2 u cos v , u sin v , 4 u 2 > . Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Here we want x 2 + 4 y 2 to be simple. So Integral Theorem Problems x = 2 r cos θ Surface of Revolution y = r sin θ Case Graph Case will do better. Surface Integrals

Paraboloid z = x 2 + 4 y 2 A trigonometric parametrization will often be better if you have From Math 2220 Class 39 to calculate a surface integral. V2 Φ( u , v ) = < 2 u cos v , u sin v , 4 u 2 > . Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential Here we want x 2 + 4 y 2 to be simple. So dTheta Integral Theorem x = 2 r cos θ Problems Surface of y = r sin θ Revolution Case will do better. Graph Case Plug x and y into z = x 2 + 4 y 2 to get the z-component. Surface Integrals

Parabolic Cylinder z = x 2 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Graph parametrizations are often optimal for parabolic Conservative Vector Fields cylinders. Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Parabolic Cylinder z = x 2 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Φ( u , v ) = < u , v , u 2 > Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Parabolic Cylinder z = x 2 From Math 2220 Class 39 Φ( u , v ) = < u , v , u 2 > V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Parabolic Cylinder z = x 2 From Math 2220 Class 39 Φ( u , v ) = < u , v , u 2 > V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems One of the parameters (v) is giving us the “extrusion” Surface of Revolution direction. The parameter u is just being used to describe the Case curve z = x 2 in the zx plane. Graph Case Surface Integrals

Elliptic Cylinder x 2 + 2 z 2 = 6 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative The trigonometric trick is often good for elliptic cylinders Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Elliptic Cylinder x 2 + 2 z 2 = 6 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative √ √ √ √ √ Vector Fields Φ( u , v ) = < 3 · 2 cos v , u , 3 sin v > = < 6 cos v , u , 3 sin v > Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Elliptic Cylinder x 2 + 2 z 2 = 6 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ √ √ Conservative Φ( u , v ) = < 6 cos v , u , 3 sin v > Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Elliptic Cylinder x 2 + 2 z 2 = 6 From Math √ √ 2220 Class 39 Φ( u , v ) = < 6 cos v , u , 3 sin v > V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Elliptic Cylinder x 2 + 2 z 2 = 6 From Math 2220 Class 39 V2 Stokes and Gauss √ √ Why Green’s Φ( u , v ) = < 6 cos v , u , 3 sin v > and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Elliptic Cylinder x 2 + 2 z 2 = 6 From Math √ √ 2220 Class 39 Φ( u , v ) = < 6 cos v , u , 3 sin v > V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic What happened here is we started with the polar coordinate Method of Finding a idea Potential x = r cos θ dTheta z = r sin θ Integral Theorem but noted that the algebra wasn’t right for x 2 + 2 z 2 so shifted Problems Surface of to √ Revolution x = 2 r cos θ Case Graph Case z = r sin θ Surface Integrals

Elliptic Cylinder x 2 + 2 z 2 = 6 From Math 2220 Class 39 √ √ V2 Φ( u , v ) = < 6 cos v , u , 3 sin v > Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential √ dTheta x = 2 r cos θ Integral Theorem z = r sin θ Problems makes the left hand side work out to 2 r 2 which will be 6 when Surface of Revolution √ Case r = 3 . Graph Case Surface Integrals

Ellipsoid x 2 + 2 y 2 + 3 z 2 = 4 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ A similar trick occurs for using spherical coordinate ideas in Conservative Vector Fields parameterizing ellipsoids. Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Ellipsoid x 2 + 2 y 2 + 3 z 2 = 4 From Math 2220 Class 39 V2 Stokes and Gauss A similar trick occurs for using spherical coordinate ideas in Why Green’s and Gauss’ parameterizing ellipsoids. Conservative Vector Fields √ � 4 Systematic Φ( u , v ) = < 2 sin u cos v , 2 sin u sin v , 3 cos u > Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Ellipsoid x 2 + 2 y 2 + 3 z 2 = 4 From Math 2220 Class 39 √ � 4 Φ( u , v ) = < 2 sin u cos v , 2 sin u sin v , 3 cos u > V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Hyperbolic Cylinder x 2 − z 2 = − 4 From Math 2220 Class 39 V2 Stokes and Gauss You may have run into the hyperbolic functions Why Green’s and Gauss’ e x + e − x Conservative cosh x = Vector Fields 2 Systematic e x − e − x Method of sinh x = Finding a Potential 2 dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Hyperbolic Cylinder x 2 − z 2 = − 4 From Math 2220 Class 39 V2 You may have run into the hyperbolic functions Stokes and Gauss e x + e − x Why Green’s and Gauss’ cosh x = 2 Conservative e x − e − x Vector Fields sinh x = Systematic 2 Method of Finding a Potential Just as cos 2 θ + sin 2 θ = 1 helps with ellipses, the hyperbolic dTheta version cosh 2 θ − sinh 2 θ = 1 leads to the nicest hyperbola Integral Theorem parameterizations. Problems Surface of Revolution Case Graph Case Surface Integrals

Hyperbolic Cylinder x 2 − z 2 = − 4 From Math 2220 Class 39 V2 Stokes and Just as cos 2 θ + sin 2 θ = 1 helps with ellipses, the hyperbolic Gauss version cosh 2 θ − sinh 2 θ = 1 leads to the nicest hyperbola Why Green’s and Gauss’ parameterizations. Conservative Vector Fields Systematic Method of Finding a Potential Φ( u , v ) = < 2 sinh v , u , 2 cosh v > dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Hyperbolic Cylinder x 2 − z 2 = − 4 From Math 2220 Class 39 V2 Φ( u , v ) = < 2 sinh v , u , 2 cosh v > Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Saddle z = x 2 − y 2 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative The hyperbolic trick also works with saddles Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Saddle z = x 2 − y 2 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Φ( u , v ) = < u cosh v , u sinh v , u 2 > Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Saddle z = x 2 − y 2 From Math 2220 Class 39 Φ( u , v ) = < u cosh v , u sinh v , u 2 > V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Hyperboloid of 1 Sheet x 2 + y 2 − z 2 = 1 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ The spherical coordinate idea for ellipsoids with sin φ replaced Conservative Vector Fields by cosh u works well here. Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Hyperboloid of 1 Sheet x 2 + y 2 − z 2 = 1 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Φ( u , v ) = < cosh u cos v , cosh u sin v , sinh u > Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Hyperboloid of 1 Sheet x 2 + y 2 − z 2 = 1 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Hyperboloid of 2-Sheets x 2 + y 2 − z 2 = − 1 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Φ( u , v ) = < sinh u cos v , sinh u sin v , cosh u > Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Hyperboloid of 2-Sheets x 2 + y 2 − z 2 = − 1 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Top Part of Cone z 2 = x 2 + y 2 From Math 2220 Class 39 V2 Stokes and Gauss Why Green’s and Gauss’ Conservative x 2 + y 2 . � So z = Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Top Part of Cone z 2 = x 2 + y 2 From Math 2220 Class 39 V2 Stokes and Gauss x 2 + y 2 . � Why Green’s So z = and Gauss’ The polar coordinate idea leads to Conservative Vector Fields Φ( u , v ) = < u cos v , u sin v , u > Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals

Top Part of Cone z 2 = x 2 + y 2 x 2 + y 2 . � So z = From Math 2220 Class 39 The polar coordinate idea leads to V2 Stokes and Φ( u , v ) = < u cos v , u sin v , u > Gauss Why Green’s and Gauss’ Conservative Vector Fields Systematic Method of Finding a Potential dTheta Integral Theorem Problems Surface of Revolution Case Graph Case Surface Integrals