Gravitation: Curvature - An Introduction To General Relativity
Gravitation: CurvatureAn Introduction to General RelativityPablo LagunaCenter for Relativistic AstrophysicsSchool of PhysicsGeorgia Institute of TechnologyNotes based on textbook: Spacetime and Geometry by S.M. CarrollSpring 2013Pablo LagunaGravitation: Curvature
CurvatureCovariant DerivativesParallel Transport and GeodesicsThe Reimann Curvature TensorSymmetries and Killing VectorsMaximally Symmetric SpacetimesGeodesic DeviationPablo LagunaGravitation: Curvature
IntroductionThe metric defines the geometry in a manifold.Curvature in a manifold depends on the metric, but how.The form of the metric is strongly dependent on the coordinate system used.We need a formal definition of curvature.Curvature plays a central role in general relativity: a measure of locala measure of local spacetime curvaturematter energy densityPablo LagunaGravitation: Curvature
Gaussian CurvatureIn R3 , the Gaussian curvature is defined as the product of the principal curvatures; that is, K κ1 κ2 . Its value isintrinsic to the surface and does not depend on the embedding.Pablo LagunaGravitation: Curvature
Covariant DerivativesRecall: The partial derivative of a tensor is not in general a tensor.We need to have a generalization of equations such that ν T µν 0 to be tensorial, so they are invariantunder coordinate transformations.In flat space in inertial coordinates, µ is a map from (k, l) tensors to (k, l 1) that is linear and obey theLeibniz rule.Covariant Derivative : is a map from (k , l) tensor fields to (k , l 1) tensor fields (T S) T SLeibniz (product) rule: (T S) ( T ) S T ( S) .Pablo LagunaGravitation: Curvature
Covariant Derivative of VectorsConsider v(x α ) and v(x α dx α ) such that dx α t α with t α defining the direction of the covariantderivative.Parallel transport the vector v(x α t α ) back to the point x α and call it vk (x α )Covariant Derivative:vk (x α ) v(x α )α t v(x ) lim 0In a local inertial frame:( t v)αThus, β vβ t β vα β vααNotice: The above expression is not valid in curvilinear coordinates. In general,αδvk (x )αγδβ v α (x δ t δ ) Γβγ v (x )( t )component changesbasis vector changesTherefore, β vα β vPablo Lagunaαα Γβγ vγGravitation: Curvature
Transformation properties of ΓαβγSince the covariant derivative yields tensors, µ0 Vν0 x µ x ν 0 x µ0 µ V x νν.thus µ0 Vν0 µ0 Vν00and x µ x ν0 x ν x µ00 x µ x νν0 x λtherefore µ Vν0 x λVλλ0 Γµ0 λ0 V x µ x ν Γµ0 λ0ν0 µ Vν x µ x µ x ν0 x µ x µ0 x µVVν µ Vν0 x µ0 x ν x ν x µ x λ x ν00 x µ νΓµλ Vν0 x µ x λ x ν0 x µ0 x λ0 x ννΓµλ x µ x λ0 x µ0 x λνΓµλ V 2x ν0 x µ x λNotice: the connection coefficients are not the components of a tensor.Pablo Lagunaλ0 x νand finallyΓµ0 λ0 x λ0 x ν x µ x ν0 x µ0 x λν0 Γµ0 λ0 x µ x ν 0 λ x µ x νGravitation: Curvature.λVλ
Covariant differentiation of 1-formsA possibility is:λ µ ων µ ων eΓµν ωλTo find eΓλµν we need that commutes with contractions: µ (T λ λρ ) ( T )µ λ λρreduces to the partial derivative on scalars: µ φ µ φThenλ µ (ωλ V ) ( µ ωλ )V( µ ωλ )Vλλλ ωλ ( µ V )σλλλρ eΓµλ ωσ V ωλ ( µ V ) ωλ Γµρ VButλ µ (ωλ V )Thereforeλ µ (ωλ V ) ( µ ωλ )Vλλ ωλ ( µ V )λλσσ0 eΓµλ ωσ V Γµλ ωσ V .Since ωσ and V λ are completely arbitrary,σσeΓµλ Γµλ .Pablo LagunaGravitation: Curvature
Consequently:Covariant differentiation of Vectors β vα β vαα Γβγ vγCovariant differentiation of 1-formsλ µ ων µ ων Γµν ωλCovariant differentiation of general Tensorsµ µ ···µk σ T 1 2ν1 ν2 ···νl µ µ ···µk σ T 1 2ν1 ν2 ···νlµ1 T Γσλλµ2 ···µkν1 ν2 ···νlµ2µ λ···µk ΓσλT 1ν1 ν2 ···νl · · ·λµ1 µ2 ···µkλµ µ ···µk Γσν T 1 2λν2 ···νl Γσν2 Tν1 λ···νl · · ·1Pablo LagunaGravitation: Curvature
Properties of Γαµν :3Γαµν has n components; that is, 64 in 4-dimensions.Γαµν is called connection because it helps to transport tensors.bλSµν λ Γλµν Γµν is a tensor. Recallν0Γµ0 λ00νbΓµ0 λ0 x µ x λ x ν0 x µ x λ0 x µ x λ x ν0 x µ0 x λ0ν x νΓµλ 0 x ννbΓµλ x µ x λ 2x ν000 x µ x λ x µ x λ x µ x λ0 x µ0 x λ 2x ν0 x µ x λthenSµ0 λ0ν0 x µ x λ x ν0 x µ0 x λ0 x νSµλνλIf Γλµν is a connection, Γνµ is also a connection. Thus, the torsion tensor is defined byTµνλλλλ Γµν Γνµ 2Γ[µν] .λλA spacetime metric gµν induces a unique connection if Γαµν is (1) torsion-free: Γµν Γ(µν) and (2)metric compatible: ρ gµν 0.Metric compatibility implies that ρ gµν 0λ ρ (gµλ V ) ρ Vµgµλ ρ VPablo LagunaλGravitation: Curvature
Christoffel symbolsFrom metric compatibility:λλλλ ρ gµν µ gνρ ρ gµν Γρµ gλν Γρν gµλ 0λλ µ gνρ Γµν gλρ Γµρ gνλ 0 ν gρµ ν gρµ Γνρ gλµ Γνµ gρλ 0Subtract the second and third from the first,λ ρ gµν µ gνρ ν gρµ 2Γµν gλρ 0 .Multiply by g σρ to getChristoffel SymbolsσΓµν 1 σρg ( µ gνρ ν gρµ ρ gµν )2Pablo LagunaGravitation: Curvature
The Christoffel symbols vanish in flat space in Cartesian coordinatesThe Christoffel symbols do not vanish in flat space in curvilinear coordinates.For example, if ds2 dr 2 r 2 dθ 2 , it is not difficult to show that Γrθθ r and Γθθr 1/rAt any one point p in a spacetime (M, gµν ), it is possible to find a coordinate system for which Γσµν 0(recall local flatness)Very useful property: µ Vµ µ Vµµ Γµλ VλbutµΓµλ 1 µρg ( µ gλρ λ gρµ ρ gµλ )2q1 µρ11g λ gρµ λ ln g p λ g 22 g then µ Vµ µ Vµ Vµq1 µ g p g thus µ Vµq1µ p µ ( g V ) g Pablo LagunaGravitation: Curvature
Parallel TransportAs mentioned before, covariant differentiation involves computing how tensor change.However, tensor are maps from vectors and 1-forms to real numbers at a given point.Question: What is behind the changes computed by acting on tensors?Answer: gives the instantaneous rate of change of a tensor field in comparison to what the tensor wouldbe if it were parallel transported.The results of parallel transporting a tensor is path dependent.keep vectorconstantqpPablo LagunaGravitation: Curvature
Keeping the Tensor ConstantIn flat spacetime, the constancy of a tensor along a curve x µ (λ) can be states as ddλT µ µ ···µ1 2kν1 ν2 ···νl Define the directional derivative as:Ddλ dx σdλdx µdλµ µ ···µk σ T 1 2ν1 ν2 ···νl 0 µNotice that this derivative is a map from (k, l) tensors to (k, l) . Thus, the parallel transport condition or equation ofparallel transport is defined as: DdλT µ µ ···µ1 2kν1 ν2 ···νl dx σdλµ µ ···µk σ T 1 2ν1 ν2 ···νl 0For a vector, this equation takes the formddλVµµ Γσρdx σdλVρ 0If the connection Γµσρ is metric compatible, thenDdλgµν Pablo Lagunadx σdλ σ gµν 0Gravitation: Curvature
Theorem: Given to vectors V µ and W µ that are parallel-transported along a curve x α (λ), the inner productgµν V µ W ν is preserved along this curve.Proof:Ddλµν(gµν V W ) Ddλ gµνµV Wν gµνDdλVµ Wν gµν V0Lemma: The norm of vectors and orthogonality are preserved under parallel transportPablo LagunaGravitation: Curvatureµ DdλWν
GeodesicsA geodesic generalizes the notion of a straight line in Euclidean space to curved space.Definition 1: A geodesic is the path of shortest distance.Definition 2: A geodesic is the path that parallel transports its own tangent vector.Definitions 1 and 2 are equivalent if the connection is the Christoffel connection.From Definition 2, let dx µ /dλ be the tangent vector to a path x µ (λ) The condition that dx µ /dλ be paralleltransported isD dx µ 0,dλ dλor alternativelyd2xµdλ2µ Γρσdx ρ dx σdλ dλ 0.This is called the geodesic equation.2 µ2Since in Euclidean space in Cartesian coordinates Γµρσ 0, the geodesic equation becomes d x /dλ 0,which is the equation for a straight line.Pablo LagunaGravitation: Curvature
From Definition 1, consider a time-like curve x α (λ) and its corresponding proper time functionalZτ dτ Z q gµν dx µ dx ν Z gµνdx µ dx ν!1/2dλ dλ dλZ p f dλµ dx ν. The extrema of this functional will give us thewhere we have introduced the following definition: f gµν dxdλ dλshortest-distance path. That is,Zδτ δpZ f dλ 12( f ) 1/2δf dλ 0Without loss of generality, we can selectR dx α /dλ to be the 4-velocity vector V α ; that is, λ τ and f 1.Therefore the stationary points of τ dτ are equivalent to the stationary points ofI 11Z2f dτ dx µ dx νZ2gµνdτdτdτConsider now changes in the proper time under infinitesimal variations of the path,µxgµν µµx δxσgµν δx σ gµν .thenδI 12Z σ gµνdx µ dx νdτPablo Lagunadτδxσ 2 gµνdx µ d(δx ν )dτdτGravitation: Curvature!dτ
Consider the last termZgµνdx µ d(δx ν )dτ!Zdτdτ gµνZ gµνd2xµdτ 2d2xµdτ 2 dgµν dx νdτ!δxdτ σ gµννdτ!dx σ dx νdτdτδxνdτThe δI becomes:Z "δI gµσd2xµdτ 2 12 σ gµν ν gµσ µ gνσ dx µ dx νdτdτ#which yieldsgµσd2xµdτ 2 12 σ gµν ν gµσ µ gνσ dx µ dx νdτdτord2xµdτ 2σ ΓµνPablo Lagunadx µ dx νdτdτ 0Gravitation: Curvatureσδx dτ 0 0
Properties of GeodesicsThe geodesic equation is a generalization of Newton’s law f ma for the case f 0.For the Lorentz force case, in general relativityd2xµdτ 2µ Γρσdx ρ dx σdτdτ qmµF νdx νdτThe transformation τ λ aτ b, for some constants a and b, leaves the geodesic equation invariant.λ is called an affine parameterNotice: The demand that the tangent vector be parallel-transported constrains the parametrization of thecurve.For a general parametrization,d2xµdα2µ Γρσwheref (α) Pablo Lagunadx ρ dx σdα dαd2αdλ2 f (α)! dαdx µdα, 2dλGravitation: Curvature
Properties of GeodesicsIn a spacetime with Lorentzian metric, the character (timelike/null/spacelike) of the geodesic never changes.For time-like curves with U α dx α /dτ the 4-velocity, the geodesic equation reads U λ λ U µ 0 .In terms of the 4-momentum pµ m U µ the geodesic equations reads pλ λ pµ 0For a null geodesic the proper time parameter τ vanishes. There is no preferred choice of affine parameterin that case. One can for instance pick the affine parameter λ such that pα dx α /dλThe energy of a particle (time-like or null) is the given by E pµ U µPablo LagunaGravitation: Curvature
The Riemann Curvature TensorRecall that in flat space:Parallel-transport around a closed loop leaves a vector unchanged.Covariant derivatives of tensors commute.Initially parallel geodesics remain parallel.How do these properties get modified by the presence of curvature and how can we quantify those changes?Recall also that:Parallel-transport of a vector around a closed loop in a curved space will lead to a transformation of thevector.The resulting transformation depends on the total curvature enclosed by the loop.Goal: to have a local description of the curvature at each point. Such description is provided by the Riemanncurvature tensor.Pablo LagunaGravitation: Curvature
Consider the following situation: The parallel-transport of a vector V µ around the loop in the figure below.(b a, bb)µA(0, bb)BiBi(ba, 0)µA(0, 0)The change δV µ experience by V µ as it is parallel-transported and returned to the starting point must beproportional to V µdepend on Aµ and B µanti-symmetric in Aµ and B µ to indicate the direction followed in the loopthusδVρνµρ (δa)(δb)A B R σµν Vσwhere R ρ σµν is a (1, 3) tensor known as the Riemann or curvature tensor. Notice:ρρR σµν R σνµPablo LagunaGravitation: Curvature
Commutator of two covariant derivatives: it measures the difference between parallel transporting the tensor firstone way and then the other, versus the opposite ordering.6µ6i6iµ6That is:[ µ , ν ]Vρρ ν µ Vρλρ µ ν V µ ( ν V ) Γµν λ V µ ν Vρ ( µ Γνσ )Vρρσ Γµσ ν Vρσρρρ Γµσ ν Vρ Γνσ µ Vσ Γµσ Γνλ Vρρλσ (µ ν)λ Γµν λ Vλ( µ Γνσ ν Γµσ Γµλ Γνσ Γνλ Γµσ )Vρσλρ Tµν λ V σwhereRiemann TensorρρρρλρλR σµν µ Γνσ ν Γµσ Γµλ Γνσ Γνλ ΓµσPablo Lagunaλρ Γµν Γλσ VGravitation: Curvatureλ 2Γ[µν] λ VR σµν V ρ (µ ν)ρλσρσ
Important to notice:The antisymmetry of R ρ σµν in µν is obvious.The derivation depends only connection (no mention of the metric was made).Thus, the definition is true for any connection, whether or not it is metric compatible or torsion free.The action of [ ρ , σ ] can be generalized to a tensor of arbitrary rank:µ ···µk[ ρ , σ ]X 1ν1 ···νlλµ ···µk Tρσ λ X 1ν1 ···νl µλµ2 ···µkµµ λ···µk R 1 λρσ Xν1 ···νl R 2 λρσ X 1ν1 ···νl · · ·µ ···µkµ1 ···µkλλ R ν1 ρσ X 1λν ···ν R ν2 ρσ Xν λ···ν · · ·21lOne can the view the torsion tensor and the curvature tensors asT (X , Y ) X Y Y X [X , Y ]R(X , Y )Z X Y Z Y X Z [X ,Y ] Z X λ (Y η Z ) Y λ (X η Z )ληληρ(X λ Y Y λ X ) η Zwhere X X µ µ . That isρµνR σµν X Y Zσ Pablo LagunaληρλGravitation: Curvatureηρl
Theorem:If the components of the metric are constant in some coordinate system, the Riemann tensor will vanish.If the Riemann tensor vanishes we can always construct a coordinate system in which the metriccomponents are constant.Proof:ρρPart 1: If in some coordinate system σ gµν 0, then Γρµν 0 and σ Γµν 0; thus R σµν 0.Part 2: see textbookPablo LagunaGravitation: Curvature
Properties of the Riemann TensorHow many independent components does the Riemann tensor have?In principle, it has n4 independent components in n-dimensions.The anti-symmetry in the last two indices implies there are only n(n 1)/2 independent values these lasttwo indices can take on, there are n3 (n 1)/2 independent components.Consider Rρσµν gρλ R λ σµν in Riemann:Rρσµν λλgρλ ( µ Γνσ ν Γµσ )1λτgρλ g( µ ν gστ µ σ gτ ν µ τ gνσ ν µ gστ ν σ gτ µ ν τ gµσ )21( µ σ gρν µ ρ gνσ ν σ gρµ ν ρ gµσ )2thenRρσµν RσρµνRρσµν RµνρσRρ[σµν] 0Since these are tensorial equations, they are also true in any coordinate system.There are 1/12n2 (n2 1)independent components in the Riemann tensor. In 4-dimensions, there are 20independent components.Pablo LagunaGravitation: Curvature
The Bianchi IdentityConsider the covariant derivative of the Riemann tensor, evaluated in Riemann normal coordinates: λ Rρσµν λ Rρσµν1 λ ( µ σ gρν µ ρ gνσ ν σ gρµ ν ρ gµσ ) .2and consider λ Rρσµν ρ Rσλµν σ Rλρµν1 ( λ µ σ gρν λ µ ρ gνσ λ ν σ gρµ λ ν ρ gµσ2 ρ µ λ gσν ρ µ σ gνλ ρ ν λ gσµ ρ ν σ gµλ σ µ ρ gλν σ µ λ gνρ σ ν ρ gλµ σ ν λ gµρ ) 0.Since Rρσµν Rσρµν thenBianchi identity [λ Rρσ]µν 0 .Pablo LagunaGravitation: Curvature
The Ricci TensorRicci TensorλRµν R µλν .Because of Rρσµν RµνρσRµν Rνµ ,AlsoRicci scalarµR R µ gPablo LagunaµνRµν .Gravitation: Curvature
The Weyl TensorWeyl TensorCρσµν Rρσµν 2 (n 2)gρ[µ Rν]σ gσ[µ Rν]ρ 2(n 1)(n 2)Rgρ[µ gν]σ .Notice:The Ricci tensor and the Ricci scalar contain information about “traces” of the Riemann tensor. The Weyltensor is the Riemann tensor with “all of its contractions removed. ”All possible contractions of Cρσµν vanish, while it retains the symmetries of the Riemann tensor:CρσµνCρσµνCρ[σµν] C[ρσ][µν] ,Cµνρσ ,0.The Weyl tensor is only defined in three or more dimensions, and in three dimensions it vanishes identically.For n 4 it satisfies a version of the Bianchi identity,ρ Cρσµν 2(n 3)(n 2) [µ Rν]σ 12(n 1)!gσ[ν µ] R.The Weyl tensor is that it is invariant under conformal transformations. That is, Cρσµν for some metric gµνand C̃ρσµν for a metric g̃µν Ω2 (x)gµν are the same. So the Weyl tensor is a.k.a. the conformaltensor.Pablo LagunaGravitation: Curvature
The Einstein TensorContract twice the Bianchi identity λ Rρσµν ρ Rσλµν σ Rλρµν 0to get0 νσ µλg g( λ Rρσµν ρ Rσλµν σ Rλρµν )µν Rρµ ρ R Rρν ,orµ Rρµ 12 ρ R 0DefineEinstein TensorGµν Rµν 12R gµν ,to getContracted Bianchi Identityµ Gµν 0Pablo LagunaGravitation: Curvature
ExampleConsider the two-sphere, with metric22222ds a (dθ sin θ dφ ) ,where a the radius of the sphere. Then coefficients areφΓθφΓφφθ sin θ cos θφ cot θ . ΓφθThe only non vanishing component of the Reimann tensor isθR φθφθθθλθλ θ Γφφ φ Γθφ Γθλ Γφφ Γφλ Γθφ (sin θ cos θ) (0) (0) ( sin θ cos θ)(cot θ) sin θ .222ThusRθφθφFrom Rµν gαβλ gθλ R φθφθgθθ R φθφ a sin θ . 22Rαµβν one getsφφRθθRθφ gRφθφθ 1Rφθ 0Rφφ gθθ2Rθφθφ sin θ .and the Ricci scalarR gθθRθθ gPablo LagunaφφRφφ 2a2.Gravitation: Curvature
Example cont.The Ricci scalar for a two-dimensional manifold completely characterizes the curvatureFor this example it is a constant over the two-sphere.The manifold is maximally symmetricIn any number of dimensions, the curvature of a maximally symmetric space satisfies (for some constant a)Rρσµν a 2Pablo Laguna(gρµ gσν gρν gσµ ) ,Gravitation: Curvature
Symmetries and Killing VectorsA manifold M possesses a symmetry if the geometry is invariant under a certain transformation that maps M intoitself.Isometry: a symmetry of the metric.Example: 4-dimensional Minkowski space-time. The metric2ds ηµν dxµdxνhas the symmetriesxµ xxµ ΛNotice: σ gµν 0 µ aµνxµνtranslationsLorentz transformationσσσx x a Pablo Lagunais a symmetryGravitation: Curvature
Recall the geodesic equation0ν p ν pµ p ν pµ Γµν p pδ m m νδdpµνdx νdτδν ν pµ Γµν p pδ1 δλ νg µ gνλ ν gµλ λ gµν p pδ21 ν λm µ gνλ ν gµλ λ gµν p pdτ2dpµ1ν λm µ gνλ p pdτ2dτdpµ therefore4-Momentum Conservation σ gµν 0along a geodesicPablo Laguna dpσ dτGravitation: Curvature 0
Consider the following coordinate transformation on the 4-dimensional Minkowski metric: z ẑx. Then2ds ηµν dxµdxν222222 dt (1 z )dx dy x d ẑ 2x ẑdx d ẑWhere did the translational symmetries go? We need a method to define symmetries invariant under coordinatetransformations.Define K σ such that σ gµν 0.µThis is equivalent to K µ ( σ )µ δσ. If K is the generator of an isometry, then pσ K ν pν Kν pν is invariant along the K -directionThat isdpσ dτµ 0νp µ (Kν p ) 0Expanding the last equation0µν p µ (Kν p ) p Kν µ p µµ ν p p µ Kνµ νµ νp p µ Kν p p (µ Kν)Therefore (µ Kν) 0ν µνp µ (Kν p ) 0Killing’s Equation (µ Kν) 0Pablo LagunaGravitation: Curvature
Theorem: If a vector K is a Killing vector, it is always possible to find a coordinate system in which K σ . Ifthere are more than one Killing vector. In general it is not possible to find coordinates for which all vectors are of theform K σ .Notice: For a Killing vector, the Riemann equation [µ ν] Ktakes the form µ ν Kρρthus µ ν Kρ R σµν Kρ R νµσ Kµ Rνσ KσσσFrom the contracted Bianchi identity0 νµ Rµν 1 ν R K 2 K Rµν K µ R νµµν2µµν2 (K Rµν ) 2 Rµν Kµλ(µ2 ( λ µ K ) 2 Rµν µλ 2 µ Kλ 2 R K µ Rµνµ K µ RKν)µ K µ Rµ (µ Kν) K µ RµPablo LagunaGravitation: Curvature
Killing vectors in flat spaceTranslations:Xµ (1, 0, 0)Yµ (0, 1, 0)Zµ (0, 0, 1)Rotations:Rµ ( y , x, 0)Sµ (z, 1, x)Tµ (0, z, y)Pablo LagunaGravitation: Curvature
Geodesic Deviation EquationThe notion of parallel does not extend naturally from flat to curved spaces.Instead, construct a one-parameter family of non-intersecting geodesics, γs (t); namely, for each s Rthere is a geodesic γs with affine parameter t.The collection γs (t) defines a smooth two-dimensional surface.Chose s and t to be the coordinates in this surface.The entire surface is the set of points x µ (s, t) M.There are two natural vector fields:TµSµ x µtangent vectors t x µdeviation vectors sa s ( t)TµtSµsPablo LagunaGravitation: Curvature
DefineVµ ( T S)aµ ( T V )µ T ρ Sρµrelative velocity of geodesicsµ T ρ Vρµrelative acceleration of geodesicsSince S and T are basis vectors adapted to a coordinate system, [S, T ] 0Since we are considering vanishing torsion, [S, T ] 0 impies thatρS ρ Tµρ T ρ Sµ 0Thusaµ ρσµT ρ (T σ S )ρσµT ρ (S σ T )ρσµρ σµ(T ρ S )( σ T ) T S ρ σ Tµµνρσµρ σ(S ρ T )( σ T ) T S ( σ ρ T R νρσ T )ρσµσρµσρµµν ρ σ(S ρ T )( σ T ) S σ (T ρ T ) (S σ T ) ρ T R νρσ T T Sµν ρ σR νρσ T T S .ThereforeGeodesic deviation equationaµ D2dt 2Sµµνρ σ R νρσ T T SPablo Laguna,Gravitation: Curvature
Gravitation:Curvature An Introduction to General Relativity Pablo Laguna Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.
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dius of curvature R c 2.5 cm, exerts on a straight stent inserted into a curved vessel. A measure of curvature of a bent stent was calculated as a reciprocal of the radius of curvature for a middle curve of each stent (a small radius of curvature means a large curvature). Interpreting the Models In the relevant figures, blue/cyan denotes .
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Engineering Mathematics – I Dr. V. Lokesha 10 MAT11 8 2011 Leibnitz’s Theorem : It provides a useful formula for computing the nth derivative of a product of two functions. Statement : If u and v are any two functions of x with u n and v n as their nth derivative. Then the nth derivative of uv is (uv)n u0vn nC
