What is Line integrals: Definition and 131 Discussions

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as




{\displaystyle W=\mathbf {F} \cdot \mathbf {s} }
, have natural continuous analogues in terms of line integrals, in this case









{\displaystyle \textstyle W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} }
, which computes the work done on an object moving through an electric or gravitational field F along a path


{\displaystyle L}

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  1. Falgun

    I Equivalence of alternative definitions of conservative vector fields and line integrals in different metric spaces

    I have seen conservative vector fields being defined as satisfying either of the two following conditions: The line integral of the vector field around a closed loop is zero. The line integral of the vector field along a path is the function of the endpoints of the curve. It is apparent to me...
  2. chwala

    Evaluate the given integrals - line integrals

    My interest is on question ##37##. Highlighted in Red. For part (a) I have the following lines; ##\int_c A. dr = 4t(2t+3) +2t^5 + 3t^2(t^4-2t^2) dt ## ##\left[\dfrac {8t^3}{3}+ 6t^2+\dfrac{t^6}{3} + \dfrac{3t^7}{7} - \dfrac{6t^5}{5}\right]_0^1## ##=\dfrac{288}{35}## For part (b) for...
  3. Z

    Checking the Solution to -√2: Is It Right?

    The answer at the end of the book says ##-\sqrt{2}##. Is this correct or is my solution correct? Here is a depiction of the path where we are integrating
  4. WMDhamnekar

    Computing line integral using Stokes' theorem

    ##curl([x^2z, 3x , -y^3],[x,y,z]) =[-3y^2 ,x^2,3]## The unit normal vector to the surface ##z(x,y)=x^2+y^2## is ##n= \frac{-2xi -2yj +k}{\sqrt{1+4x^2 +4y^2}}## ##[-3y^2,x^2,3]\cdot n= \frac{-6x^2y +6xy^2}{\sqrt{1+4x^2 + 4y^2}}## Since ##\Sigma## can be parametrized as ##r(x,y) = xi + yj +(x^2...
  5. WMDhamnekar

    Is the Calculation of the Vector Line Integral Over a Square Correct?

    Author's answer: Recognizing that this integral is simply a vector line integral of the vector field ##F=(x^2−y^2)i+(x^2+y^2)j## over the closed, simple curve c given by the edge of the unit square, one sees that ##(x^2−y^2)dx+(x^2+y^2)dy=F\cdot ds## is just a differentiable 1-form. The...
  6. WMDhamnekar

    I Properties of Line Integrals question

    I don't have any idea to answer these questions. I am working on it by searching the reference books where similar questions have been solved by authors. Meanwhile, any member of Physics Forums may help me in answering these questions.
  7. WMDhamnekar

    A Solve Line Integral Question | Get Math Help from Physics Forums

    I don't have any idea about how to use the hint given by the author. Author has given the answer to this question i-e F(x,y) = axy + bx + cy +d. I don't understand how did the author compute this answer. Would any member of Physics Forums enlighten me in this regard? Any math help will be...
  8. F

    Mathematica Problem with line integrals in Mathematica

    Hello everyone. I am testing mathematica to work with some line integrals. I want to go from the point (0,0) to (2, 3) over a straight line. I do it with 3 different parametrizations. The problem is that each one offers me a different result. The original problem is a two dimensional gaussian...
  9. W

    Problem with line integrals for electric potential

    Homework Statement I have a problem understanding the equation $$\Delta V = -\int_{a}^{b} \vec{E} \cdot d \vec{l}$$ In the case of a parallel plate capacitor whereby the positive plate is placed at ##z=t## while the negative is at ##z = 0##, my integral looks like $$\Delta V = -\int_{0}^{t}...
  10. M

    Question about Finding a Force with line integrals

    Homework Statement [/B] F =< 2x, e^y + z cos y,sin y > (a) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along a straight path. (b) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along the curved path given by C : r(t)...
  11. M

    Question about Vector Fields and Line Integrals

    Homework Statement (a) Consider the line integral I = The integral of Fdr along the curve C i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C? ii) What is the value of I if the vector field F is is a unit vector...
  12. Marcin H

    Flux Integral: How to find ds for line integrals in general

    Homework Statement Homework Equations flux = int(b (dot) ds) The Attempt at a Solution I just wanted clarification on finding ds. I understand why ds is in the positive yhat direction (just do rhr) but I don't understand where the dxdz come from. How do we find ds in general?
  13. R

    I Confusion regarding line integrals

    Sorry if this is the wrong place to post this, but I wasn't sure where exactly to put it. When we calculate the force on a closed loop of current-carrying wire in a uniform magnetic field, We calculate the line integral of the loop to be 0. However, when we evaluate the line integral for an...
  14. M

    A Time differentiation of fluid line integrals

    I am looking at a proof from a book in fluid dynamics on time differentiation of fluid line integrals - Basically I am looking at the second term on the RHS in this equation $$ d/dt \int_L dr.A = \int_L dr. \partial A / \partial t + d/dt \int_L dr.A$$ The author has a field vector A for a...
  15. Toby_phys

    Line Integrals around a Square on the x-y Plane

    Homework Statement Evaluate the following line integrals, showing your working. The path of integration in each case is anticlockwise around the four sides of the square OABC in the x−y plane whose edges are aligned with the coordinate axes. The length of each side of the square is a and one...
  16. toforfiltum

    Calculating work done using line integrals

    Homework Statement Sisyphus is pushing a boulder up a 100-ft tall spiral staircase surrounding a cylindrical castle tower. a) Suppose Sisyphus's path is described parametrically as $$x(t)=(5\cos3t, 5\sin3t, 10t)$$, $$\space 0\leq t\leq10$$. If he exerts a force with constant magnitude of 50 Ib...
  17. S

    A Line integrals of differential forms

    Consider a curve ##C:{\bf{x}}={\bf{F}}(t)##, for ##a\leq t \leq b##, in ##\mathbb{R}^{3}## (with ##x## any coordinates). oriented so that ##\displaystyle{\frac{d}{dt}}## defines the positive orientation in ##U=\mathbb{R}^{1}##. If ##\alpha^{1}=a_{1}dx^{1}+a_{2}dx^{2}+a_{3}dx^{3}## is a...
  18. superkraken

    Surface integrals and line integrals

    Homework Statement when we calculate the electric field due to a plane sheet or the magnetic field due to a wire,are we calculating it at a single point or the whole field due to the total wire? Homework EquationsThe Attempt at a Solution
  19. K

    I Green's theorem and Line Integrals

    (Sorry for my bad English.) I was reading about the Green's theorem and I notice that the book only shows for the case where the function is a vector function. When learning about line integrals, I saw that we can do line integrals using "ordinary" functions. For example, the line integral of...
  20. anhtu2907

    Proving a theorem in line integrals

    At the bottom of the picture, I couldn't understand why differentiating with respect to x gives the first integral at the right-hand side 0. Thanks for reading.
  21. A

    Does Gauss' Law use line integrals or surface integrals?

    In my physics textbook, I see Gauss' Law as https://upload.wikimedia.org/math/0/3/5/035b153014908c0431f00b5ddb60c999.png\ointE dA but in other places I see it as...
  22. kostoglotov

    Insight into determinants and certain line integrals

    I just did this following exercise in my text If C is the line segment connecting the point (x_1,y_1) to (x_2,y_2), show that \int_C xdy - ydx = x_1y_2 - x_2y_1 I did, and I also noticed that if we put those points into a matrix with the first column (x_1,y_1) and the second column (x_2,y_2)...
  23. kostoglotov

    A question about path orientation in Green's Theorem

    So if we have a non-simply-connected region, like this one to apply Green's Theorem we must orient the C curves so that the region D is always on the left of the curve as the curve is traversed. Why is this? I have seen some proofs of Green's Theorem for simply connected regions, and I...
  24. kostoglotov

    Line Integral Example - mistake or am I missing something?

    This is an example at the beginning of the section on the Fundamental Theorem for Line Integrals. 1. Homework Statement Find the work done by the gravitational field \vec{F}(\vec{x}) = -\frac{mMG}{|\vec{x}|^3}\vec{x} in moving a particle from the point (3,4,12) to (2,2,0) along a piece wise...
  25. kostoglotov

    Line Integral/Ampere's Law: is my logic valid?

    This is a problem from a section on Line Integrals in my Calculus Textbook, I haven't studied any physics relating to E&M yet, and the solutions manual only gives solutions for odd numbered problems. Sorry, if I'm posting in the wrong forum, I hope I'm not. 1. Homework Statement A steady...
  26. YogiBear

    Finding a parametric form and calculating line integrals.

    Homework Statement Let C be the straight line from the point r =^i to the point r = 2j - k Find a parametric form for C. And calculate the line integrals ∫cV*dr and ∫c*v x dr where v = xi-yk. and is a vector field Homework EquationsThe Attempt at a Solution For parametric form (1-t)i + (2*t)j...
  27. C

    The Fundamental Theorem for Line Integrals

    Homework Statement Determine whether or not f(x,y) is a conservative vector field. f(x,y) = <-3e^(-3x)sin(-3y),-3e^(-3x)cos(-3y) > If F is a conservative fector field find F = gradient of f Homework Equations N/A The Attempt at a Solution Fx = -3e^(-3x)(-3)cos(-3y) Fy =...
  28. PhysicsKid0123

    Confused about force and work in 3 Dimensions. Line integrals.

    So I am kind of confused about the role of force when calculating work. Specifically, when defining work using a line integral. There is a paragraph in my calculus book that is really throwing me off and its really bugging me so much I can't continue reading unless I fully understand what's...
  29. J

    Line integrals, gradient fields

    Homework Statement ##\nabla{F} = <2xyze^{x^2},ze^{x^2},ye^{x^2}## if f(0,0,0) = 5 find f(1,1,2)Homework Equations The Attempt at a Solution my book doesn't have a good example of a problem like this, am I looking for a potential? ##<\frac{\partial}{\partial x},\frac{\partial}{\partial...
  30. dwn

    Line Integrals and Finding Parametric Equations

    I am having a difficult time finding the parametric equations x = x(t) and y = y(t) for line integrals. I know how to find them when dealing with circles, but when it comes to finding them for anything else, I don't see the method...it all seems very random. I did fine with finding the...
  31. B

    Visualizing Non-Zero Closed-Loop Line Integrals Via Sheets?

    How do I visualize \dfrac{xdy-ydx}{x^2+y^2}? In other words, if I visualize a differential forms in terms of sheets: and am aware of the subtleties of this geometric interpretation as regards integrability (i.e. contact structures and the like): then since we can interpret a...
  32. A

    Does the orientation you evaluate line integrals matter?

    If instead of evaluating the above line integral in counter-clockwise direction, I evaluate it via the clockwise direction, would that change the answer? What if I evaluate ##C_1## and ##C_3## in the counter-clockwise direction, but I evaluate ##C_2## in the clockwise direction?
  33. T

    How to Determine Line and Surface Integrals with Rectangular Boundaries

    Homework Statement Consider a vector A = (2x-y)i + (yz^2)j + (y^2z)k. S is a flat surface area of a rectangle bounded by the lines x = +-1 and y = +-2 and C is its rectangular boundary in the x-y plane. Determine the line integral ∫A.dr and its surface integral ∫(∇xA).n dS Homework...
  34. T

    Volume, surface, and line integrals

    Homework Statement Consider a vector A = (x^2 - y^2)(i) + xyz(j) - (x + y + z)k and a cube bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0 and z = 1 Determine the volume integral ∫∇.A dV where V is the volume of the cube Determine the surface integral ∫A.n dS where s is the surface of...
  35. H

    Coordinate transformation for line integrals; quadrature rules

    Hi all, The context of this problem is as follows: I'm trying to implement a discontinuous finite element method and the formulation calls for the computation of line integrals over the edges of the mesh. Anyway, more generally, I need to evaluate \int_{e}f(x,y)ds, where e is a line segment...
  36. W

    ML-inequality, Estimation of Line Integrals

    Problem: Let ##\vec{F}## be a vector function defined on a curve C. Let ##|\vec{F}|## be bounded, say, ##|\vec{F}| ≤ M## on C, where ##M## is some positive number. Show that ##|\int\limits_C\ \vec{F} \cdot d\vec{r}| ≤ ML ## (L=Length of C).Attempt at a Solution: I honestly have no idea where...
  37. M

    Exploring the Relationship Between Line Integrals and Mass in Physics

    Wasn't sure which section to put this q in. Just reading now that f(x,y) can represent the density of a semicircular wire and so if you take a line integral of some curve C and f(x,y) you can find the mass of the wire... makes sense. What I don't get is that if I then move the wire around the...
  38. M

    What is the interpretation of a line integral with a 2D function?

    When I think line integral - I understand when I'm taking a line integral for a function f(x,y) which is in 3D space above a curve that the integral is this curtain type space, just like if you had a 2D function and you find the area under the curve, except now it's turned on its side and it's...
  39. W

    Line Integral Homework: Integrate (xe^y)ds

    Homework Statement Integrate some area C of (xe^y)ds where C is the arc of the curve x=e^y Homework Equations What is the indeffinite integral and why is it that? Answer is (1/3)e^3y + C The Attempt at a Solution Integral of (xe^y)((e^y)^2 + 1)^(1/2) = Integral of (e^2y)(e^2y...
  40. C

    Help understanding closed line integrals

    Hi I'm currently studying Electromagnetism, and we keep coming across this symbol: \oint A closed line integral, something I have never really been able to understand. If a normal integral works like this: http://imageshack.us/a/img109/3732/standardintegral.png where f(x) is the "height"...
  41. M

    Curl and its relation to line integrals

    hey all i know and understand the component of curl/line integral relation as: curlF\cdot u=\lim_{A(C)\to0}\frac{1}{A(C)} \oint_C F\cdot dr where we have vector field F, A(C) is the area of a closed boundary, u is an arbitrary unit vector, dr is an infinitely small piece of curve C my...
  42. Vorde

    Work-Energy Theorem with Line Integrals

    Homework Statement The problem is to prove the work-energy theorem: Work is change in kinetic energy.Homework Equations Line integral stuff, basic physics stuff. The Attempt at a Solution I'm given the normal definitions for acceleration, velocity and I'm given Newton's second law. I'm...
  43. U

    Integrate curve f ds Line Integrals

    Homework Statement Compute ∫f ds for f(x,y)= √(1+9xy), y=x^3 for 0≤x≤1 Homework Equations ∫f ds= ∫f(c(t))||c'(t)|| ||c'(t)|| is the magnitude of ∇c'(t) The Attempt at a Solution So, with this equation y=x^3 ... I got the that c(t)= <t,t^3> c'(t)=<1,3t^2> I know that from the equation...
  44. S

    Line Integrals for trajectories

    So I was wondering if I defined a vector field F, and a Trajectory of a particle x=t y=.5at^2+vit+si and I can find the work done by the field on a particle moving on a path with a line integral ∫F.dr, so what would this equate to for a projectile does it apply to this?, could you give me a real...
  45. D

    Finding the amount of work done (line integrals)

    Homework Statement Find the amount of work (ω) done by moving a point from (2;0) to (1;3) along the curve y=4-(x^2), in the effect of force F=(x-y;x). Homework Equations The Attempt at a Solution ω = ∫((x-y)dx + xdy) ω = ∫(x-4+x^2)dx + ∫√(4-y) dy In the end, I get this...
  46. D

    Fundamental theorem for line integrals

    Hi, I have a question. In my calculus book, I always see the fundamental theorem for line integrals used for line integrals of vector fields, where f=M(x,y)i + N(x,y)j is a vector field.The fundamental theorem tells me that if a vector field f is a gradient field for some function F, then f is...
  47. B

    Solving Line Integrals with Vector Cross Products

    I posted an actual problem in advanced physics but no answer so i will try to get an math part answer from it. Suppose I have to solve this integral: I=\int {\vec{dl} × \vec A } Where \vec A = -\frac {1}{x} \vec a_{z} So it has only a z component and I have to find the vector cross of the...
  48. B

    Force on triangular current loop: line integrals

    Homework Statement http://pokit.org/get/img/65e8ba92c1d00bf7fc8be2b178757ed8.jpg If a=5b, and I1 and I2 are known, find the force on the triangular loop. Homework Equations The Attempt at a Solution For start, the field from the infinitely long wire is : \vec B=\large -\frac{\mu _{0}...