## Line integrals

Video in TIB AV-Portal: Line integrals

 Title Line integrals Title of Series Complex Analysis Part Number 3 Number of Parts 15 Author License CC Attribution - ShareAlike 3.0 Germany:You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. Identifiers 10.5446/34041 (DOI) Publisher Release Date 2014 Language English

 Subject Area Mathematics
Point (geometry) Group action Functional (mathematics) INTEGRAL Length Differential (mechanical device) Real number Direction (geometry) Multiplication sign Correspondence (mathematics) Sheaf (mathematics) Open set Permutation Power (physics) Plane (geometry) Complex number Theorem Nichtlineares Gleichungssystem Antiderivative Partial derivative Condition number Social class Area Multiplication Graph (mathematics) Theory of relativity Gradient Curve Calculus Price index Line (geometry) Limit (category theory) Time domain Vector potential Category of being Arithmetic mean Loop (music) Vector field Vector space Line integral Physicist Linearization Right angle Linear map Fundamental theorem of algebra Spacetime
Axiom of choice Differential (mechanical device) INTEGRAL Multiplication sign Modal logic 1 (number) Sheaf (mathematics) Parameter (computer programming) Dimensional analysis Subset Positional notation Different (Kate Ryan album) Scalar field Matrix (mathematics) Körper <Algebra> Partial derivative Partition (number theory) Social class Rotation Process (computing) Constraint (mathematics) Gradient Curve Physicalism Price index Flow separation Time domain Proof theory Dreiecksmatrix Symmetry (physics) Vector space Ring (mathematics) Chain Order (biology) Right angle Summierbarkeit Point (geometry) Functional (mathematics) Diagonal Connectivity (graph theory) Lemma (mathematics) Student's t-test Graph coloring Hypothesis Product (business) Element (mathematics) Term (mathematics) Modulform Theorem Stochastic kernel estimation Nichtlineares Gleichungssystem Maß <Mathematik> Condition number Standard deviation Mathematical analysis Mortality rate Line (geometry) Cartesian coordinate system Numerical analysis Vector potential Loop (music) Vector field Physicist Line integral
Rotation Point (geometry) LAN party INTEGRAL Length Differential (mechanical device) Curve Sheaf (mathematics) Time domain Tangent space Category of being Goodness of fit Positional notation Well-formed formula Right angle Summierbarkeit Set theory Social class
Beta function State of matter INTEGRAL Differential (mechanical device) Direction (geometry) Multiplication sign Correspondence (mathematics) Propositional formula Inverse element Parameter (computer programming) Mereology Subset Derivation (linguistics) Mathematics Sign (mathematics) Positional notation Many-sorted logic Different (Kate Ryan album) Oval Funktionalintegral Conservation law Körper <Algebra> Partition (number theory) Rotation Area Moment (mathematics) Curve Physicalism Price index Time domain Category of being Arithmetic mean Vector space Symmetry (physics) Mathematical singularity Summierbarkeit Right angle Parametrische Erregung Mathematician Sinc function Spacetime Point (geometry) Connectivity (graph theory) Mass Trigonometric functions 2 (number) Tangent space Prime ideal Latent heat Goodness of fit Term (mathematics) Analytic continuation Game theory Condition number Alpha (investment) Addition Dot product Standard deviation Forcing (mathematics) Lemma (mathematics) Physical law Independence (probability theory) Line (geometry) Vector field Line integral Physicist Maß <Mathematik>
Beta function Orientation (vector space) Multiplication sign Modal logic 1 (number) Mereology Derivation (linguistics) Sign (mathematics) Mathematics Many-sorted logic Physicist Different (Kate Ryan album) Scalar field Category of being Gradient Curve Entire function Substitute good Proof theory Vector space Order (biology) Chain Dew point Right angle Mathematician Directed graph Spacetime Point (geometry) Slide rule Functional (mathematics) Divisor Dual space Morphismus Goodness of fit Term (mathematics) Modulform Theorem Chain rule Set theory Alpha (investment) Complex analysis Dot product Shift operator Lemma (mathematics) Vector graphics Independence (probability theory) Line (geometry) Inclusion map Vector field Scalar field Line integral Physicist Calculation Coefficient
Axiom of choice Point (geometry) Functional (mathematics) Direction (geometry) Connectivity (graph theory) Maxima and minima Mathematical analysis Dual space Coordinate system Dimensional analysis Morphismus Group representation Positional notation Physicist Different (Kate Ryan album) Isomorphieklasse Scalar field Vector space Spacetime Dot product Differential (mechanical device) Military base Gradient Coordinate system Basis <Mathematik> Line (geometry) Product (business) Isomorphieklasse Vector space Scalar field Function (mathematics) Duality (mathematics) Linearization Linear map Spacetime
Slide rule Functional (mathematics) Dot product INTEGRAL Coordinate system Product (business) Power (physics) Goodness of fit Vector space Positional notation Linearization Modulform Maß <Mathematik> Set theory Spacetime
Statistical hypothesis testing Axiom of choice Group action Tensorfeld Differential (mechanical device) INTEGRAL Multiplication sign Sheaf (mathematics) Parameter (computer programming) Derivation (linguistics) Mechanism design Mathematics Many-sorted logic Positional notation Different (Kate Ryan album) Oval Scalar field Negative number Funktionalintegral Körper <Algebra> Series (mathematics) Area Euclidean vector Curve Physicalism Infinity Price index Substitute good Proof theory Isomorphieklasse Vector space Duality (mathematics) Chain Quantum mechanics Linearization Quantum Right angle Figurate number Mathematician Sinc function Resultant Spacetime Directed graph Point (geometry) Classical physics Slide rule Standard error Functional (mathematics) Vapor barrier Transformation (genetics) Real number Dual space Theory Product (business) Force Prime ideal Morphismus Goodness of fit Natural number Operator (mathematics) Modulform Chain rule Dot product Multiplication Standard deviation Military base Weight Forcing (mathematics) Mathematical analysis Diffuser (automotive) Algebraic structure Axialer Vektor Line (geometry) Loop (music) Invariant (mathematics) Vector field Line integral Physicist Calculation Gravitation Object (grammar) Valuation using multiples
and the today we'll deal with the line integral so what's continuous the section on line integrals and let me quickly what have you done so far 1st class we cover it complex multiplication and some of the stuff of complex numbers 2nd class we covered the costly Ramon equations and as a consequence of the uh C. linearity of the differential um and last class I started also with um line integrals so let me quickly review of the back I so so I'm an however so specific properties on a specific problem is to find the primitive call the find a primitive of the given function from the complex plane to complex numbers set of primitives f of gives little effort in the complex number k so a from C to C by integration of meaning that f prime equals f so this is what the the real numbers completed the um the a fundamental theorem of calculus to us and actually do this a little more generality so more general problem which i want to deal with here what is all the we want to find a given the vector field we want to find a potential so there's savers and find potential so s from uh mn while it's still this on the entire space uh a potential also a given vector fields Oscarsson vectors here has a aches from say are into our and all and an open subset tho and that means we want to find and a little F such that the gradient of f coincides with the X and I a nice and relation for this problem if you not a physicist that need this uh is uh think of s say from are 2 2 are and look at the graph what is the graph of efforts it's a hilly area right with the rest of for the case n equals tool and to so here what we look for is we look so all we have is a given vector field x and corresponding to the slope it's so here only of the domain we have where of say a vector field X and this is the gradient of f is equals x this means that the EC the given as the length corresponds to the slope so I want to find a a landscape such that such minutes uh for my given x and a such that x corresponds to the slope meaning to the direction and intensity of the essence the uh office really Netscape and as you know um so from this interpretation it's pretty obvious that this cannot be done in general why not well think of the think of following following you vector field the vector field always points to the direction of the S and so if you follow the if you follow the vector field on and suppose you have such an f you will always walk uphill right you will always go in the direction of the steepest S and so is nonzero so you will always move laughter so if X has has closed curves grossed integral curves here if there is a curve uh in the planes sir which is close to n to which x is tangential for this means on 1 hand you go uphill all the time on the other hand you must end up at the same point where you started that's impossible so effect closed if the vector field X has closed loops uh this uh this task here cannot be achieved OK so last time which I gave my stated the necessary condition for this problem here to the solution namely and necessary condition and so and low for a start to be solvable the is it is solvable I all of that is that some of the vector field X as the Jacobian a differential which is symmetric so it has some limitations onerous areas so that the x in symmetric or if you like you rather continued not a matrix but in a linear mapping is itself the giant which means that whatever the entries of the Jacobian well views of a partial derivatives of f of x so that means that the j also XII or this is my shorthand for d by dx j equals well well yeah the same thing with permuted indices and this must hold for each point x the cells for all x in OK say for AIX going from you in Rn the 2 power in making a more general so I requirements for all x in you for all points and for all indices hi j between 1 and between 1 and 10 the and let me call this
condition yeah let me call this condition practiced seminar for symmetric him so this is a necessary condition and sensor process so easy let me let me recall prove once again it's just if x if x is the gradient of f rich it means that the ice component the ice component of X is the i th component of a gradient of that is the eyes partial of so that means we need to verify so if f at this so that if it exists then then we have DJ and i now D I S and then I can OK I should actually now that a right of the proof I remember that hypotheses I should actually require that X is a C 2 vector fields since I want to apply the lemma of products of to say that this is the same as DID J. F what this the i X that this was a huge 6 OK so this is true of the lemma of the shots lemma holds for c 2 functions I I guess I should better write this out although I've formulated this is a theorem that list is like C 2 of you 2 and where you is in our who sorry OK so this clearly is a necessary condition actually we did this in the uh and analysis to course last year and also in physics it's common to write this the notation for this is uh use or rotation and so physics the standard way of writing this out is that the the uh how well let me stick here to the German notation that the color uh vanishes for all for all arguments x in you and what is what is the color What is the rotation and well it depends on dimension where for n equals 2 as you know uh we also covered his last term for rotation of X goes from form is a scalar function namely it's simply the difference of these 2 will be only significant difference the 1 which around is as I keep forgetting this is the 1 this is correct so that like out right Costa x equals uh t 1 x 2 minus the 2 X 1 end for any constraint uh the rotation goes from you'll which is now in uh so this is our tool and now it's in of 3 to and now the target is also vector hassles are free and now or products then after 1 to write out all components it's d 2 extreme minus Steve freely connects to now we have all differences with uh indices uh which do not coincide so and D 1 x 2 minus 2 gates what so OK end of so this is called are in english that's kernel and uh it's actually usually abbreviated call of X or if you physicists then you write the and not uh times X so the investors and because of this is the uh this is the rotation rotation and if you like it's rotation rate of the vector field and and the sum so this also shows you what the rotation and several dimensions should be doesn't come up and physics really that cell so tell me how many components do I need enough for now to write out the same things I 1 to give this condition for all indices i j uh uh as scalar equations how many equations to a need and an for n equals 4 PC then how many choices of nonequal indices are there for n equals 4 show 6 yeah that's correct know so it's it's actually for general and it would be what is it it's like the lower diagonal of the matrix is n times n minus 1 over 2 conditions it and I don't know all the ones you like to call these a not too bad this is what you need a the all k and l and the Hamming by chance for any that's the uh these 2 numbers uh coincide OK so this this um this is the kernel and uh and um it's up to you if you want the index notation or you can actually also view it as as the symmetry of the Jacobian matrix and then this these this means that the lower that sorry that the lower diagonal of the matrix coincides with the upper diagonal and there's no uh Abbott triangular matrix and there's no condition on the diagonal here so these these are the same that's exactly the rotation condition the OK then just gives a simple example which I'm sure you know uh which is uh if I have 3 standard rotation vector fields in our tool uh which is minus Y X so what is this near 0 0 head on the X. AIX's it points up or down and the y axis same thing to the left to and here I like this so it actually it's looks like a rotation and in fact if you compute uh efforts commutator D 1 X minus the 1 x 2 let's just do it so I D 1 so the kernel x is the 1 of x minus the 2 of the uh um of minus y which is the 1st component so this is this is 1 minus minus 1 equals 2 non-zero is expected OK so this for instance is a field uh with uh nonzero rotation and adverse