DOCSLIB.ORG

The Mean Value Theorem of Line Complex Integral and Sturm Function 1 Introduction and Preliminary Notes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Mean Value Theorem of Line Complex Integral and Sturm Function 1 Introduction and Preliminary Notes Applied Mathematical Sciences, Vol. 8, 2014, no. 37, 1817 - 1827 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.4279 The Mean Value Theorem of Line Complex Integral and Sturm Function Jelena Vujakovi´c Faculty of Sciences and Mathematics, University of Priˇstina Lole Ribara 29, 38 220 Kosovska Mitrovica, Serbia Copyright c 2014 Jelena Vujakovi´c. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract For linear complex differential equation of second order w (z)+a (z)=0, where a (z) is analytical function, two solutions w1,w2 are obtained by itera- tions (according to results from [4] where are present successive line integrals on unclosed arc L of integration path. If L would be contour, then due to analytics and Cauchy theorem every first integral in order in these successive integrals would be equal to zero, then all iterations are equal to zero. In this way we obtain only trivial solution. That is way it has tried to find simple mean value formula, as in real integral. Mathematics Subject Classification: 34E05, 34K11 Keywords: The mean value theorem, Iteration, Sturm function, Complex differential equation, Oscillation 1 Introduction and Preliminary Notes It is common that term mean value of complex integral has been introduced for closed contours (for the further details, see [2]), according to which f (z)is analytical or at least continuous, while inside contour can be discontinuous and even non-analitical. This was suggested by Cauchy’s fundamental theorem 1818 J. Vujakovi´c f (z) dz =0. (1.1) L f(ς) Let z is point from the interior of contour L. Then function ς−z is not analytical all over inside L, so Cauchy’s fundamental integral formula is in effect 1 f (ς) f (z)= dς . (1.2) 2πi ς − z L If f (z) is analytical on L , then L can be replaced by the simplest closed contour, circle K of radius R with center at z. So standard formula (1.2) also ≡ f(ς) applied for L K. Under mean value of function we consider value ς−z dς K divided with thelenght of circle arc S =2Rπ. Having in (1.2) that ς = z + R eiϕ, 0 ≤ ϕ ≤ 2π, we obtain 2π 1 f (z)= f z + Reiϕ dϕ . (1.3) 2π 0 If f (z)=u (x, y)+iv (x, y) is analytical, then u (x, y) and v (x, y) are harmonic functions, so for them theorem on mean value also applies 2π 2π 1 1 u (z )= u z + reiϕ dϕ , v (z )= v z + reiϕ dϕ . (1.4) 0 2π 0 0 2π 0 0 0 However, in solving ordinary differential equations in real area, formulas from (1.1) to (1.4) are of small importance because closed contour has not bee used. So, certain method for easier calculus or estimation of complex line integral has been searched for, and as well for real integral. For real integral, as is known, of great importance is first theorem on mean value of certain integral. That is why, for unclosed pathL, we need approximate value of integrals a (z) dz, a (z) dz2, za(z) dz2, whereat we shall examine L L L L L approximation by module. Therefore, the following hypothesis applies. The mean value theorem... 1819 Formula a (z) dz = a (ς)(z1 − z0) (1.5) L where 1. a (z) is analytical function in area G , where is placed curve L on which integration is conducted 2. z0 and z1 are initial and final limits of integral 3. ς is certain point from G, doesn’t have to be L, but which satisfies certain standards of approximation, first according to module and then according to argument. 4. A (z) is primitive function for a (z), which is possible to find by quadrature can be adopted as some kind of mean value of integral, if integration path is open line L. Fundaments for this hypothesis are: 1. If z1 = z0, that is if L exists and closes down, then we obtain a (z) dz = 0, which is fundamental formula of Cauchy. L 2. For real z ≡ x we have section L =[x0,x1] and formula (1.5) becomes x1 a (x) dx = a (ς)(x1 − x0) ,x0 ≤ ς ≤ x, and that is formula for mean value x0 of integral in real area. 3. If a (ς) is discontinuous in given area G where path L is placed, that f(ς) is a (ς)= ς−z , then Cauchy’s fundamental theorem or Cauchy’s theorem on residues are valid. 4. Besides that, also are in effect formulas (1.3) and (1.4) on mean value of analytical function a (z), that is, on its harmonic parts u =Rea (ς) ,v= Im a (ς). So, basic knowledge about analytics fit into assumption (1.5), but its construc- tion demands existence of singular points from a (z) which don’t have to be on L. Thus, if L bends and closes and if the length of its arc is S =2Rπ, then L becomes a circle K of length S, so in order to have first characteristic valid, ς must be out of circle K. 2 Main Results These are the main results of the paper. Theorem 2.1. For integral of power function a (z)=zn,n≥ 0 on arbitrary open curve, from z0 =0to point z1, theorem on mean value is applied 1820 J. Vujakovi´c z1 a (z) dz = a (z) dz = a (ς) z1, (2.1) 0 where z ς = √ 1 . (2.2) n n +1 Integral of each Jordan curve inside the circle |z|≤|z1| = R can be replaced by integral on shortest path, that is, on direction Oz1 and (2.1), where the mid point on the path was given with (2.2) and the mean value of the function is zn √z1 1 a (ς)=a n n+1 = n+1 . Proof. Let’s first observe the case n = 0, that is, let a a (z)=c is complex constant. Since the integral of analytic function does not depend on path L but only on primitive function A (z)=z for given a (z)=c, and from starting and final point, implies a (z) dz = cdz = c (z1 − z0). Here L L is c (z1 − z0)=a (ς)(z1 − z0), and that is, actually, formula (1.5) in which a (ς)=c and ς any point on segment z0z1. For n = 1, that is for function a (z)=z, implies a (z) dz = zdz = L L z2−z2 1 z2 − z2 a ς z − z a ς ς ς 1 1 0 z1+z0 2 ( 1 0 )= ( )( 1 0). Hence, since ( )= we have = 2 z1−z0 = 2 . Therefore, ς is arithmetic mean of final points z0 and z1. Let’s further review the case n =2.Ifa (z)=z2, then a (z) dz = L 2 1 3 − 3 1 3 − 3 − 1 2 z dz = 3 (z1 z0 ), that is 3 (z1 z0 )=a (ς)(z1 z0)ora (ς)= 3 (z1 + z0z1 L 2 2 2 +z0 ) = ς . From here, due to many-valued characteristics of squarer root we z2 z z z2 ± 1 + 0 1+ 0 2 have two values for ς = 3 . Notice that ς is arithmetic mean of 2 2 three squares z1 ,z0z1,z0 , taken in the starting and final point. n We are continuing with the procedure. Let a (z)=z . Then a (ς)= n 1 n n n−1 n √z1 ς = n z1 + z1 z0 + ... + z0 .Forz0 = 0 implies ς = n n . Since √ +1 1 +1 n n +1 → 1 for n →∞the mean value ς → z1, for n →∞, on segment Oz1, and a (ς) tends to a (z1). So, we obtained sequence {ς n} of points, z1 √z1 √z1 √z1 , , 3 , ..., n , which on segment Oz tends to z . On the other hand, 2 3 4 n+1 1 1 with induction it easy to prove that The mean value theorem... 1821 z1 cdz = cdz = cz1 = a (ς) z1 =⇒ a (ς)=c = const L 0 z1 z2 z zdz = zdz = 1 = a (ς) z = ςz =⇒ ς = 1 2 1 1 2 L 0 z1 3 2 2 z1 2 z1 z dz = z dz = = a (ς) z1 = ς z1 =⇒ ς = √ 3 3 L 0 z1 4 3 3 z1 3 z1 z dz = z dz = = a (ς) z1 = ς z1 =⇒ ς = √ 4 3 4 L 0 . z1 n+1 n n z1 n z1 z dz = z dz = = a (ς) z1 = ς z1 =⇒ ς = √ . n +1 n n +1 L 0 Let a (z) is arbitrary analytical function, defined by arbitrary convergent +∞ n power series a (z)= anz . There is a question is it formula (2.1) valid or n=0 some other variance of this formula, as well as where mid point ς is placed and what is the amount of mean value of a (ς). n k Theorem 2.2. For arbitrary polynomial Pn (z)= akz applies for- k=0 mula (2.1), for mean value of integral, where ς is certain mean value of mod- ule lesser than |z1| = R1, ( |ς| <R1), and which depends on z1, that is on Pn (ak,z). Proof. For simple case, when P1 (z)=a0 + a1z is linear function we have z1 2 z (2.1) P (z) dz = (a + a z) dz = a z + a 1 = P (ς) z 1 0 1 0 1 1 2 1 1 L 0 z1 for P1 (ς)=a0 + a1ς.
Load more

Recommended publications
  • The Directional Derivative the Derivative of a Real Valued Function (Scalar field) with Respect to a Vector
    Math 253 The Directional Derivative The derivative of a real valued function (scalar field) with respect to a vector. f(x + h) f(x) What is the vector space analog to the usual derivative in one variable? f 0(x) = lim − ? h 0 h ! x -2 0 2 5 Suppose f is a real valued function (a mapping f : Rn R). ! (e.g. f(x; y) = x2 y2). − 0 Unlike the case with plane figures, functions grow in a variety of z ways at each point on a surface (or n{dimensional structure). We'll be interested in examining the way (f) changes as we move -5 from a point X~ to a nearby point in a particular direction. -2 0 y 2 Figure 1: f(x; y) = x2 y2 − The Directional Derivative x -2 0 2 5 We'll be interested in examining the way (f) changes as we move ~ in a particular direction 0 from a point X to a nearby point . z -5 -2 0 y 2 Figure 2: f(x; y) = x2 y2 − y If we give the direction by using a second vector, say ~u, then for →u X→ + h any real number h, the vector X~ + h~u represents a change in X→ position from X~ along a line through X~ parallel to ~u. →u →u h x Figure 3: Change in position along line parallel to ~u The Directional Derivative x -2 0 2 5 We'll be interested in examining the way (f) changes as we move ~ in a particular direction 0 from a point X to a nearby point .
    [Show full text]
  • Lecture 11 Line Integrals of Vector-Valued Functions (Cont'd)
    Lecture 11 Line integrals of vector-valued functions (cont’d) In the previous lecture, we considered the following physical situation: A force, F(x), which is not necessarily constant in space, is acting on a mass m, as the mass moves along a curve C from point P to point Q as shown in the diagram below. z Q P m F(x(t)) x(t) C y x The goal is to compute the total amount of work W done by the force. Clearly the constant force/straight line displacement formula, W = F · d , (1) where F is force and d is the displacement vector, does not apply here. But the fundamental idea, in the “Spirit of Calculus,” is to break up the motion into tiny pieces over which we can use Eq. (1 as an approximation over small pieces of the curve. We then “sum up,” i.e., integrate, over all contributions to obtain W . The total work is the line integral of the vector field F over the curve C: W = F · dx . (2) ZC Here, we summarize the main steps involved in the computation of this line integral: 1. Step 1: Parametrize the curve C We assume that the curve C can be parametrized, i.e., x(t)= g(t) = (x(t),y(t), z(t)), t ∈ [a, b], (3) so that g(a) is point P and g(b) is point Q. From this parametrization we can compute the velocity vector, ′ ′ ′ ′ v(t)= g (t) = (x (t),y (t), z (t)) . (4) 1 2. Step 2: Compute field vector F(g(t)) over curve C F(g(t)) = F(x(t),y(t), z(t)) (5) = (F1(x(t),y(t), z(t), F2(x(t),y(t), z(t), F3(x(t),y(t), z(t))) , t ∈ [a, b] .
    [Show full text]
  • Cauchy Integral Theorem
    Cauchy’s Theorems I Ang M.S. Augustin-Louis Cauchy October 26, 2012 1789 – 1857 References Murray R. Spiegel Complex V ariables with introduction to conformal mapping and its applications Dennis G. Zill , P. D. Shanahan A F irst Course in Complex Analysis with Applications J. H. Mathews , R. W. Howell Complex Analysis F or Mathematics and Engineerng 1 Summary • Cauchy Integral Theorem Let f be analytic in a simply connected domain D. If C is a simple closed contour that lies in D , and there is no singular point inside the contour, then ˆ f (z) dz = 0 C • Cauchy Integral Formula (For simple pole) If there is a singular point z0 inside the contour, then ˛ f(z) dz = 2πj f(z0) z − z0 • Generalized Cauchy Integral Formula (For pole with any order) ˛ f(z) 2πj (n−1) n dz = f (z0) (z − z0) (n − 1)! • Cauchy Inequality M · n! f (n)(z ) ≤ 0 rn • Gauss Mean Value Theorem ˆ 2π ( ) 1 jθ f(z0) = f z0 + re dθ 2π 0 1 2 Related Mathematics Review 2.1 Stoke’s Theorem ¨ ˛ r × F¯ · dS = F¯ · dr Σ @Σ (The proof is skipped) ¯ Consider F = (FX ;FY ;FZ ) 0 1 x^ y^ z^ B @ @ @ C r × F¯ = det B C @ @x @y @z A FX FY FZ Let FZ = 0 0 1 x^ y^ z^ B @ @ @ C r × F¯ = det B C @ @x @y @z A FX FY 0 dS =ndS ^ , for dS = dxdy , n^ =z ^ . By z^ · z^ = 1 , consider z^ component only for r × F¯ 0 1 x^ y^ z^ ( ) B @ @ @ C @F @F r × F¯ = det B C = Y − X z^ @ @x @y @z A @x @y FX FY 0 i.e.
    [Show full text]
  • 1 Mean Value Theorem 1 1.1 Applications of the Mean Value Theorem
    Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary In Section 1, we go over the Mean Value Theorem and its applications. In Section 2, we will recap what we have covered so far this term. Topics Page 1 Mean Value Theorem 1 1.1 Applications of the Mean Value Theorem . .1 2 Midterm Review 5 2.1 Proof Techniques . .5 2.2 Sequences . .6 2.3 Series . .7 2.4 Continuity and Differentiability of Functions . .9 1 Mean Value Theorem The Mean Value Theorem is the following result: Theorem 1.1 (Mean Value Theorem). Let f be a continuous function on [a; b], which is differentiable on (a; b). Then, there exists some value c 2 (a; b) such that f(b) − f(a) f 0(c) = b − a Intuitively, the Mean Value Theorem is quite trivial. Say we want to drive to San Francisco, which is 380 miles from Caltech according to Google Map. If we start driving at 8am and arrive at 12pm, we know that we were driving over the speed limit at least once during the drive. This is exactly what the Mean Value Theorem tells us. Since the distance travelled is a continuous function of time, we know that there is a point in time when our speed was ≥ 380=4 >>> speed limit. As we can see from this example, the Mean Value Theorem is usually not a tough theorem to understand. The tricky thing is realizing when you should try to use it. Roughly speaking, we use the Mean Value Theorem when we want to turn the information about a function into information about its derivative, or vice-versa.
    [Show full text]
  • A Brief Tour of Vector Calculus
    A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0.
    [Show full text]
  • MATH 25B Professor: Bena Tshishiku Michele Tienni Lowell House
    MATH 25B Professor: Bena Tshishiku Michele Tienni Lowell House Cambridge, MA 02138 [email protected] Please note that these notes are not official and they are not to be considered as a sub- stitute for your notes. Specifically, you should not cite these notes in any of the work submitted for the class (problem sets, exams). The author does not guarantee the accu- racy of the content of this document. If you find a typo (which will probably happen) feel free to email me. Contents 1. January 225 1.1. Functions5 1.2. Limits6 2. January 247 2.1. Theorems about limits7 2.2. Continuity8 3. January 26 10 3.1. Continuity theorems 10 4. January 29 – Notes by Kim 12 4.1. Least Upper Bound property (the secret sauce of R) 12 4.2. Proof of the intermediate value theorem 13 4.3. Proof of the boundedness theorem 13 5. January 31 – Notes by Natalia 14 5.1. Generalizing the boundedness theorem 14 5.2. Subsets of Rn 14 5.3. Compactness 15 6. February 2 16 6.1. Onion ring theorem 16 6.2. Proof of Theorem 6.5 17 6.3. Compactness of closed rectangles 17 6.4. Further applications of compactness 17 7. February 5 19 7.1. Derivatives 19 7.2. Computing derivatives 20 8. February 7 22 8.1. Chain rule 22 Date: April 25, 2018. 1 8.2. Meaning of f 0 23 9. February 9 25 9.1. Polynomial approximation 25 9.2. Derivative magic wands 26 9.3. Taylor’s Theorem 27 9.4.
    [Show full text]
  • The Mean Value Theorem) – Be Able to Precisely State the Mean Value Theorem and Rolle’S Theorem
    Math 220 { Test 3 Information The test will be given during your lecture period on Wednesday (April 27, 2016). No books, notes, scratch paper, calculators or other electronic devices are allowed. Bring a Student ID. It may be helpful to look at • http://www.math.illinois.edu/~murphyrf/teaching/M220-S2016/ { Volumes worksheet, quizzes 8, 9, 10, 11 and 12, and Daily Assignments for a summary of each lecture • https://compass2g.illinois.edu/ { Homework solutions • http://www.math.illinois.edu/~murphyrf/teaching/M220/ { Tests and quizzes in my previous courses • Section 3.10 (Linear Approximation and Differentials) { Be able to use a tangent line (or differentials) in order to approximate the value of a function near the point of tangency. • Section 4.2 (The Mean Value Theorem) { Be able to precisely state The Mean Value Theorem and Rolle's Theorem. { Be able to decide when functions satisfy the conditions of these theorems. If a function does satisfy the conditions, then be able to find the value of c guaranteed by the theorems. { Be able to use The Mean Value Theorem, Rolle's Theorem, or earlier important the- orems such as The Intermediate Value Theorem to prove some other fact. In the homework these often involved roots, solutions, x-intercepts, or intersection points. • Section 4.8 (Newton's Method) { Understand the graphical basis for Newton's Method (that is, use the point where the tangent line crosses the x-axis as your next estimate for a root of a function). { Be able to apply Newton's Method to approximate roots, solutions, x-intercepts, or intersection points.
    [Show full text]
  • Appendix a Short Course in Taylor Series
    Appendix A Short Course in Taylor Series The Taylor series is mainly used for approximating functions when one can identify a small parameter. Expansion techniques are useful for many applications in physics, sometimes in unexpected ways. A.1 Taylor Series Expansions and Approximations In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It is named after the English mathematician Brook Taylor. If the series is centered at zero, the series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin. It is common practice to use a finite number of terms of the series to approximate a function. The Taylor series may be regarded as the limit of the Taylor polynomials. A.2 Definition A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function f(x) about a point x ¼ a is given by; f 00ðÞa f 3ðÞa fxðÞ¼faðÞþf 0ðÞa ðÞþx À a ðÞx À a 2 þ ðÞx À a 3 þÁÁÁ 2! 3! f ðÞn ðÞa þ ðÞx À a n þÁÁÁ ðA:1Þ n! © Springer International Publishing Switzerland 2016 415 B. Zohuri, Directed Energy Weapons, DOI 10.1007/978-3-319-31289-7 416 Appendix A: Short Course in Taylor Series If a ¼ 0, the expansion is known as a Maclaurin Series. Equation A.1 can be written in the more compact sigma notation as follows: X1 f ðÞn ðÞa ðÞx À a n ðA:2Þ n! n¼0 where n ! is mathematical notation for factorial n and f(n)(a) denotes the n th derivation of function f evaluated at the point a.
    [Show full text]
  • How to Compute Line Integrals Scalar Line Integral Vector Line Integral
    How to Compute Line Integrals Northwestern, Spring 2013 Computing line integrals can be a tricky business. First, there's two types of line integrals: scalar line integrals and vector line integrals. Although these are related, we compute them in different ways. Also, there's two theorems flying around, Green's Theorem and the Fundamental Theorem of Line Integrals (as I've called it), which give various other ways of computing line integrals. Knowing what to use when and where and why can be confusing, so hopefully this will help to keep things straight. I won't say much about what line integrals mean geometrically or otherwise. Hopefully this is something you have some idea about from class or the book, but I'm always happy to talk about this more in office hours. Here the focus is on computation. Key Point: When you are trying to apply one of these techniques, make sure the technique you are using is actually applicable! (This was a major source of confusion on Quiz 5.) Scalar Line Integral • Arises when integrating a function f over a curve C. • Notationally, is always denoted by something like Z f ds; C where the s in ds is just a normal s, as opposed to ds or d~s. • To compute, find parametric equations x(t); a ≤ t ≤ b for C and use Z Z b f ds = f(x(t)) kx0(t)k dt: C a • In the special case where f is the constant function 1, Z ds = arclength of C: C • For scalar line integrals, the orientation (i.e.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Line Integrals
    Line Integrals Kenneth H. Carpenter Department of Electrical and Computer Engineering Kansas State University September 12, 2008 Line integrals arise often in calculations in electromagnetics. The evaluation of such integrals can be done without error if certain simple rules are followed. This rules will be illustrated in the following. 1 Definitions 1.1 Definite integrals In calculus, for functions of one dimension, the definite integral is defined as the limit of a sum (the “Riemann” integral – other kinds are possible – see an advanced calculus text if you are curious about this :). Z b X f(x)dx = lim f(ξi)∆xi (1) ∆xi!0 a i where the ξi is a value of x within the incremental ∆xi, and all the incremental ∆xi add up to the length along the x axis from a to b. This is often pictured graphically as the area between the curve drawn as f(x) and the x axis. 1 EECE557 Line integrals – supplement to text - Fall 2008 2 1.2 Evaluation of definite integrals in terms of indefinite inte- grals The fundamental theorem of integral calculus relates the indefinite integral to the definite integral (with certain restrictions as to when these exist) as Z b dF f(x)dx = F (b) − F (a); when = f(x): (2) a dx The proof of eq.(2) does not depend on having a < b. In fact, R b f(x)dx = R a a − b f(x)dx follows from the values of ∆xi being signed so that ∆xi equals the value of x on the side of the increment next to b minus the value of x on the side of the increment next to a.
    [Show full text]
  • Calculus Formulas and Theorems
    Formulas and Theorems for Reference I. Tbigonometric Formulas l. sin2d+c,cis2d:1 sec2d l*cot20:<:sc:20 +.I sin(-d) : -sitt0 t,rs(-//) = t r1sl/ : -tallH 7. sin(A* B) :sitrAcosB*silBcosA 8. : siri A cos B - siu B <:os,;l 9. cos(A+ B) - cos,4cos B - siuA siriB 10. cos(A- B) : cosA cosB + silrA sirrB 11. 2 sirrd t:osd 12. <'os20- coS2(i - siu20 : 2<'os2o - I - 1 - 2sin20 I 13. tan d : <.rft0 (:ost/ I 14. <:ol0 : sirrd tattH 1 15. (:OS I/ 1 16. cscd - ri" 6i /F tl r(. cos[I ^ -el : sitt d \l 18. -01 : COSA 215 216 Formulas and Theorems II. Differentiation Formulas !(r") - trr:"-1 Q,:I' ]tra-fg'+gf' gJ'-,f g' - * (i) ,l' ,I - (tt(.r))9'(.,') ,i;.[tyt.rt) l'' d, \ (sttt rrJ .* ('oqI' .7, tJ, \ . ./ stll lr dr. l('os J { 1a,,,t,:r) - .,' o.t "11'2 1(<,ot.r') - (,.(,2.r' Q:T rl , (sc'c:.r'J: sPl'.r tall 11 ,7, d, - (<:s<t.r,; - (ls(].]'(rot;.r fr("'),t -.'' ,1 - fr(u") o,'ltrc ,l ,, 1 ' tlll ri - (l.t' .f d,^ --: I -iAl'CSllLl'l t!.r' J1 - rz 1(Arcsi' r) : oT Il12 Formulas and Theorems 2I7 III. Integration Formulas 1. ,f "or:artC 2. [\0,-trrlrl *(' .t "r 3. [,' ,t.,: r^x| (' ,I 4. In' a,,: lL , ,' .l 111Q 5. In., a.r: .rhr.r' .r r (' ,l f 6. sirr.r d.r' - ( os.r'-t C ./ 7. /.,,.r' dr : sitr.i'| (' .t 8. tl:r:hr sec,rl+ C or ln Jccrsrl+ C ,f'r^rr f 9.
    [Show full text]