From Math 2220 Class 30 V2c Schedule Left and Right Inverses From Math 2220 Class 30 Change of Coordinates Polar/Sph/Cyl Dr. Allen Back Problems Inverses from Algebra Why Cont. Fcns are Nov. 5, 2014 Integrable
Schedule From Math 2220 Class 30 V2c Schedule Left and Right 12 lectures, 4 recitations left including today. Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Schedule From Math 2220 Class 30 a Most of what remains is vector integration and V2c the integral theorems. Schedule b We’ll start 7.1, 7.2,4.2 on Friday. Left and Right Inverses c If you are not in physics, please bear with us on Change of Coordinates this material. There are good reasons why it is Polar/Sph/Cyl always in multivariable calculus courses, though it Problems will not be that evident at an elementary level. Inverses from Algebra Why Cont. Fcns are Integrable
Left and Right Inverses From Math 2220 Class 30 V2c Right inverses and existence of solutions:s Schedule Set theoretically, for f : A → B , a function g : B → A Left and Right Inverses satisfying Change of f ◦ g = Id B Coordinates Polar/Sph/Cyl Problems (i.e. f ( g ( q )) = q for all q ∈ B ) is EQUIVALENT to the Inverses from Algebra equation Why Cont. f ( p ) = q Fcns are Integrable having a solution p ∈ A for any q ∈ B .
Left and Right Inverses From Math 2220 Class 30 V2c Left inverses and uniqueness of solutions: Schedule Set theoretically, for f : A → B , a function g : B → A Left and Right satisfying Inverses g ◦ f = Id A Change of Coordinates Polar/Sph/Cyl Problems (i.e. g ( f ( p )) = p for all p ∈ A ) is EQUIVALENT to the Inverses from equation Algebra Why Cont. f ( p ) = q Fcns are Integrable whenever (i.e. for which q ) it has a solution, having that solution be unique among elements of A .
Left and Right Inverses From Math 2220 Class 30 V2c Schedule Left and Right Inverses Change of Of course in linear algebra, we learn that for linear Coordinates transformations f : R n → R n given by f ( x ) = Ax , either of the Polar/Sph/Cyl Problems above are equivalent to det ( A ) � = 0 . Inverses from Algebra Why Cont. Fcns are Integrable
Left and Right Inverses From Math 2220 Class 30 Recall also for f : R n → R n and f ( x ) = Ax : V2c a Linear transformations take parallelograms to Schedule parallelograms. (Similarly in higher dim.) Left and Right Inverses b If e 1 , e 2 , . . . , e n are the standard basis vectors, Change of Coordinates then the i’th column of the n × n matrix A is Polar/Sph/Cyl given by Ae i . Problems c So, e.g. it is easy to map the unit square with Inverses from Algebra sides e 1 , e 2 in R 2 (and lower left corner at the Why Cont. origin) to any parallelogram with sides v 1 , v 2 (and Fcns are Integrable � � lower left corner at the origin. ( A = v 1 v 2 . ) d So A − 1 can take us from an arbitrary parallelogram with sides v 1 , v 2 to the unit square.
Left and Right Inverses From Math 2220 Class 30 V2c Recall also for f : R n → R n and f ( x ) = Ax : Schedule Left and Right e By composition of two of the above, we can map Inverses any parallelogram with sides w 1 , w 2 in R 2 (and Change of Coordinates lower left corner at the origin) to any Polar/Sph/Cyl Problems parallelogram with sides v 1 , v 2 (and lower left Inverses from corner at the origin. Algebra f Throwing in a translation lets us deal with Why Cont. Fcns are parallelograms for which the origin is not a vertex. Integrable
Change of Coordinates From Math ∆ A ∼ r ∆ r ∆ θ by geometry 2220 Class 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Change of Coordinates From Math The Polar Coordinate Transformation 2220 Class 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Change of Coordinates From Math � � � � x u x v 2220 Class 30 � � � � ∆ A ∼ � ∆ u ∆ v � � � � y u y v V2c � � � Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Change of Coordinates From Math 2220 Class 30 For a C 1 1:1 onto map F ( u , v ) = ( x ( u , v ) , y ( u , v ) taking V2c D ∗ ⊂ R 2 in the uv -plane to D ⊂ R 2 in the xy plane, Schedule Left and Right � � x u x v Inverses � � dA = dx dy = � dudv � � y u y v Change of � Coordinates Polar/Sph/Cyl lets us move between xy and uv integrals. Problems (Similarly for more than 2 variables.) Inverses from Algebra Why Cont. Fcns are Integrable
Change of Coordinates From Math 2220 Class 30 V2c Schedule Polar: dA = r dr d θ. Left and Right Inverses Cylindrical: dV = r dr d θ dz . Change of Spherical: dV = ρ 2 sin φ d ρ d θ d φ. Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Change of Coordinates From Math The Basic Spherical Coordinate Picture 2220 Class 30 V2c Schedule Left and Right Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Polar/Spherical/Cylindrical Problems From Math 2220 Class 30 V2c Schedule D ( x 2 + y 2 ) 2 dx dy for D the disk x 2 + y 2 ≤ 1 . 3 Left and Right �� Problem: Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Polar/Spherical/Cylindrical Problems From Math 2220 Class 30 V2c Schedule � ∞ −∞ e − x 2 dx . Left and Right Problem: Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Polar/Spherical/Cylindrical Problems From Math 2220 Class 30 V2c Schedule Problem: Volume of a right circular cone of height H and Left and Right Inverses largest radius R . Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Polar/Spherical/Cylindrical Problems From Math 2220 Class 30 V2c Problem: Volume of the portion of the Earth above latitude 45 ◦ . Schedule Ambiguous as written; could mean all points with colatitude φ Left and Right Inverses in spherical coordinates less than π 4 . Change of Or could refer to all points whose z value is above the z value Coordinates at this latitude. Polar/Sph/Cyl Problems Either integral could be set up . . . . Inverses from Algebra Why Cont. Fcns are Integrable
Polar/Spherical/Cylindrical Problems From Math 2220 Class 30 V2c Schedule Problem: Volume, using spherical coordinates, of a ball of Left and Right Inverses radius R with a hole of radius a (centered on a diameter) Change of drilled out of it. Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Inverses from Algebra From Math 2220 Class 30 V2c Schedule Left and Right Right inverses and existence of solutions Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Inverses from Algebra From Math 2220 Class 30 V2c Schedule Left and Right Left inverses and uniqueness of solutions Inverses Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Inverses from Algebra Consider the map F : D ∗ ⊂ R 2 → D ⊂ R 2 defined by From Math 2220 Class 30 V2c F ( x , y ) = ( x 2 − y 2 , x + y ) Schedule Left and Right Label the components of image points F ( x , y ) as ( u , v ); i.e. we Inverses think of the above transformation as Change of Coordinates x 2 − y 2 Polar/Sph/Cyl u = Problems v = x + y Inverses from Algebra Why Cont. Fcns are Integrable
Inverses from Algebra From Math 2220 Class 30 To study one-to-oneness and ontoness of F , consider the V2c algebra: Schedule Left and Right x 2 − y 2 u Inverses = = x − y Change of v x + y Coordinates = x + y v Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Inverses from Algebra From Math 2220 Class 30 V2c Adding and subtracting the above two equations: Schedule 1 v + u � � Left and Right x = Inverses 2 v 1 Change of v − u � � Coordinates y = 2 v Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
Inverses from Algebra You may take as given the fact that these formulae check From Math 2220 Class 30 showing that V2c G ( u , v ) = (1 v + u , 1 v − u Schedule � � � � ) 2 v 2 v Left and Right Inverses gives the inverse of F where everything is defined; i.e. Change of Coordinates Polar/Sph/Cyl F ( G ( u , v )) = ( u , v ) Problems G ( F ( x , y ) = ( x , y ) Inverses from Algebra Why Cont. Fcns are Integrable
Inverses from Algebra From Math 2220 Class 30 V2c Schedule What is the natural domain of the function G ? In other words, Left and Right Inverses describe the largest subset (call it U ) of the uv plane on which Change of G is defined. Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable
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