What is Line integral: Definition and 401 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}

View More On Wikipedia.org
  1. L

    Line integral of a vector field (Polar coordinate)

    Hi, I am not sure if I have solved task b correctly According to the task, ##\textbf{F}=f \vec{e}_{\rho}## which in Cartesian coordinates is ##\textbf{F}=f \vec{e}_{\rho}= \left(\begin{array}{c} \cos(\phi) \\ \sin(\phi) \end{array}\right)## since ##f \in \mathbb{R}_{\neq 0}## is constant...
  2. Graham87

    Line Integral Solution for Curve γ: Simplifying Substitutions

    Hello, How should I go about to solve this line integral along the line curve γ? I attempt to apply this relation but the substitutions get too messy. Thanks
  3. S

    Line Integral of circle in counterclockwise direction

    My attempt: Let ##x=a \cos \theta## and ##y=a \sin \theta## $$\int_{L} xy^2 dx-x^2ydy$$ $$=\int_{0}^{2\pi} \left( (a\cos \theta)(a\sin \theta)^2 (-a\sin \theta)-(a\cos \theta)^2 (a \sin \theta)(a\cos \theta)\right) d\theta$$ $$=-a^4 \int_{0}^{2\pi}\left( \sin^3 \theta \cos \theta+\cos^3 \theta...
  4. chiyu

    I Vector calculus: line element dr in cylindrical coordinates

    We were taught that in cylindrical coodrinates, the position vector can be expressed as And then we can write the line element by differentiating to get . We can then use this to do a line integral with a vector field along any path. And this seems to be what is done on all questions I've...
  5. WMDhamnekar

    I Applying Stokes' theorem

    Calculate surface integral ## \displaystyle\iint\limits_S curl F \cdot dS ## where S is the surface, oriented outward in below given figure and F = [ z,2xy,x+y]. How can we answer this question?
  6. 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...
  7. WMDhamnekar

    Evaluate the line integral

    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...
  8. M

    Gauss' law in line integral, Q=##ϵ_0 ∮E.n dl=-ϵ_0 ∮∂ϕ/∂n dl##

    I know the Gauss law for surface integral to calculate total charge by integrating the normal components of electric field around whole surface . but in above expression charge is calculated using line integration of normal components of electric field along line. i don't understand this...
  9. H

    Help with a line integral please

    ∫zds=∫acos(t)*( (acos(2t))^2+(2asin(t))^2+(-asin(t))^2 )^1/2 dt , (0≤t≤pi/2) Simplified : ∫a^2cos(t)*(cos^2(2t)+5sin^2(t) )^1/2 dt , (0≤t≤pi/2) However here i get stuck and i can´t find a way to rewrite it better or to integrate as it is. Can i please get some help in this?
  10. P

    Calculating eletric potential using line integral of electric field

    So, I am able to calculate the electric potential in another way but I know that this way is supposed to work as well, but I don't get the correct result. I calculated the electric field at P in the previous exercise and its absolute value is $$ E = \frac {k Q} {D^2-0.25*l^2} $$ This is...
  11. WMDhamnekar

    I Properties of Line Integral

    I don't have any idea to answer this question. So, any math help will be accepted. I know ##\nabla fg = f\nabla g + g\nabla f \rightarrow (1) ## But I don't understand to how to use (1) here?
  12. 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...
  13. WMDhamnekar

    MHB How to prove this corollary in Line Integral using Riemann integral

    . Let C be a smooth curve with arc length L, and suppose that f(x, y) = P(x, y)i +Q(x, y)j is a vector field such that $|| f|(x,y) || \leq M $ for all (x,y) on C. Show that $\left\vert\displaystyle\int_C f \cdot dr \right\vert \leq ML $ Hint: Recall that $\left\vert\displaystyle\int_a^b g(x)...
  14. greg_rack

    Calculating the Line Integral of F over C: Stokes' Theorem and Symmetry

    From Stokes we know that ##\iint_{\textbf{S}}^{}curl \textbf{F}\cdot d\textbf{S}=\int_{C}^{}\textbf{F}\cdot d\textbf{r}##. Now, we can calculate the surface integral of the curl of F by calculating the line integral of F over the curve C. The latter ends up being 0(I calculated it parametrizing...
  15. L

    An iron cylinder inside a solenoid

    From ##\oint_{\Gamma}\vec{H}\cdot d\vec{l}=\sum I## by Ampere's Law which gives ##H \Delta l=\Delta N\cdot i\Leftrightarrow H=n i## where ##n=## number of turns per unit length so ##i=\frac{H}{n}=\frac{10^3 A / m}{\frac{200}{0.2m}}=1 A##. Since ##\vec{H}=\frac{\vec{B}-\mu_0\vec{M}}{\mu_0}## we...
  16. brotherbobby

    A line integral about a closed "unit triangle" around the origin

    Problem statement : As a part of the problem, the diagram shows the contour ##C##above on the left. The contour ##C## is divided into three parts, ##C_1, C_2, C_3## which make up the sides of the right triangle. Required to prove : ##\boxed{\oint_C x^2 y \mathrm{d} s = -\frac{\sqrt{2}}{12}}##...
  17. brotherbobby

    Line integral of a scalar function about a quadrant

    Problem : We are required to show that ##I = \int_C x^2y\;ds = \frac{1}{3}##. Attempt : Before I begin, let me post an image of the problem situation, on the right. I would like to do this problem in three ways, starting with the simplest way - using (plane) polar coordinates. (1) In (plane)...
  18. kmot

    I Why is this closed line integral zero?

    This problem comes from fluid dynamics where Kelvin circulation theorem states, that if density "rho" is a function of only pressure "p", then closed line integral of grad(p) / rho(p) equals zero. It seems so trivial, so that no one ever gives reason for this claim. When trying to solve it...
  19. S

    Line Integral to Verify the Magnetic Field B

    Hello folks, I'm working on a question as follows: I appreciate that there might be more sophisticated ways to do things, but I just want to check that my approach to the line integral is accurate. I will just give my working for the first side of the path. So I have set up the path as a...
  20. M

    Vector Line Integral Direction of Limits

    Hi, I apologise as I know I have made similar posts to this in the past and I thought I finally understood it. However, this solution seems to disagree on a technicality. I know the answer ends up as 0, but I still want to understand this from a conceptual point. Question: Evaluate the line...
  21. B

    Electric Potential inside an insulating sphere

    I used the potential at the surface of the sphere for my reference point for computing the potential at a point r < R in the sphere. The potential at the surface of the sphere is ## V(R) = k \frac {Q} {R} ##. To find the potential inside the sphere, I used the Electric field inside of an...
  22. K

    Curve integrals, del operator

    My attempt is below. Could somebody please check if everything is correct? Thanks in advance!
  23. V

    Find the potential using a line integral (Electromagnetism)

    Sorry - I wish I had some way of writing equations in this forum so the "relevant equations" section is easier to read. The answer to the first part is (a) so the rest follows from using the electric field given in B. If anyone is interested this question comes from Griffith's 3rd edition...
  24. D

    A complex line integral

    if ## \gamma (t):= i+3e^{2it } , t \in \left[0,4\pi \right] , then \int_0^{4\pi} \frac {dz} {z} ## in order to solve such integral i substitute z with ##\gamma(t)## and i multiply by ##\gamma'(t)## that is: ##\int_0^{4 \pi} \frac {6e^{2it}}{i+3e^{2it}}dt=\left.log(i+3e^{2it}) \right|_0^{4...
  25. H

    I Estimate the magnitude of a line integral exp(iz) over a semicircle

    Not homework, just trying to understand a statement in the book. On page 158 in Fisher, the following statement is made: In these applications of the Residue Theorem, we often need to estimate the magnitude of the line integral of e^{iz} over the semicircle = Re^{i\theta}, \; 0 \le \theta \le...
  26. T

    Multivariable Calculus, Line Integral

    The vector field F which is given by $$\mathbf{F} = \dfrac{(x, y)} {\sqrt {1-x^2-y^2}}$$ And the line integral $$ \int_{C} F \cdot dr $$C is the path of $$\dfrac{\ (\cos (t), \sin (t))}{ 1+ e^t}$$ , and $$0 ≤ t < \infty $$ How do I calculate this? Anyone got a tip/hint? many thanks
  27. S

    I Help with a 3D Line Integral Problem (segmented line)

    Hi all, I'm finding it difficult to start this line integral problem. I have watched a lot of videos regarding line integrals but none have 3 line segments in 3D. If someone can please point me in the right direction, it would help a lot. I've put down the following in my workings: C1...
  28. G

    Line integral where a vector field is given in cylindrical coordinates

    What I've done so far: From the problem we know that the curve c is a half-circle with radius 1 with its center at (x,y) = (0, 1). We can rewrite x = r cos t and y = 1 + r sin t, where r = 1 and 0<t<pi. z stays the same, so z=z. We can then write l(t) = [x(t), y(t), z ] and solve for dl/dt...
  29. R

    Line integral around a circle centered at the origin

    Hi everyone, I am confused in this question. First I solved it by noticing that the gradient of the function will be zero (without substitution the hit) I got that it's a conservative field so the integral should be zero since it's closed path. Then I solved it by the hit and convert it as any...
  30. bottle_shadow

    I Line integral for work done by gravity

    Dear Physics Forums people, My problem lies in understanding how the following line integral, which represents work done by the gravitational force, was calculated Specifically, in the integral after the 2nd = sign, they implicitly used \hat{r}\cdot d\vec{s} = dr I wish to understand what...
  31. M

    Electrostatics: Understanding this "Work Done" Line Integral Question

    I have a quick question about the work done concept here, especially the line integral part of it. So I understand the fact that the work done from getting from point A to B is: \int_{a}^{b} \vec F \cdot d\vec r . However, within the context of electric fields, when we define electrostatic...
  32. JD_PM

    Particle moving in conservative force field

    I don't get why ##F \cdot dr = \frac{mv^2}{2}## I know this has to be really easy but don't see it. Thanks.
  33. JD_PM

    Why the magnetic field doesn't have to describe a circle?

    Homework Statement Imagine an infinite straight wire pointing at you (thus, the magnetic field curls counterclockwise from your perspective). Such a magnetic field equals to: $$B = \frac{\mu I}{2 \pi s} \hat{\phi}$$ I want to calculate the line integral of ##B## around the circular path of...
  34. A

    Ampere's Law: Double Negative Line Integral Help

    Hi all, I’m having some trouble finding a minus sign in a standard calculation I have been doing. I am trying to show that if there is no enclosed current around the example loop in the enclosed jpeg, the four piecewise paths add up to zero (for the line integral part of Amp’s law). For this...
  35. Burhan Uddin

    Calculating the work done using a line integral

    Homework Statement a) A point charge + q is placed at the origin. By explicitly calculating the relevant line integral, determine how much external work must be done to bring another point charge + q from infinity to the point r2= aŷ ? Consider the difference between external work and work...
  36. Q

    Cylindrical Coordinates: Line Integral Of Electrostatic Field

    Homework Statement An electrostatic field ## \mathbf{E}## in a particular region is expressed in cylindrical coordinates ## ( r, \theta, z)## as $$ \mathbf{E} = \frac{\sin{\theta}}{r^{2}} \mathbf{e}_{r} - \frac{\cos{\theta}}{r^{2}} \mathbf{e}_{\theta} $$ Where ##\mathbf{e}_{r}##...
  37. L

    B Line Integral, Dot Product Confusion

    From my interpretation of this problem (image attached), the force applied to the point charge is equal and opposite to the repulsive Coulomb force that that point charge is experiencing due to the presence of the other point charge so that the point charge may be moved at a constant velocity. I...
  38. C

    I Force fields in curvilinear coordinate systems

    I am trying to solve problems where I calculate work do to force along paths in cylindrical and spherical coordinates. I can do almost by rote the problems in Cartesian: given a force ##\vec{F} = f(x,y,z)\hat{x} + g(x,y,z)\hat{y}+ h(x,y,z)\hat{z}## I can parametricize my some curve ##\gamma...
  39. gibberingmouther

    Line Integral for Electromagnetic Force

    http://web.mit.edu/sahughes/www/8.022/lec01.pdf So I'm trying to understand how to get from F = ∫[(Q*λ)*dL*r]/(r^2) to F=∫q*λ*[(xx+ay)/(a^2+x^2)^(3/2)]*dx Like I don't understand why the x and y components of r are negative, or why "The horizontal r component is obviously zero: for every...
  40. SSGD

    I Area between two closed curves

    I have been trying to find an answer to this problem for some time. So I was hoping the community might be able to point me in the right direct. https://drive.google.com/open?id=1Y2hYkRG94whLroK20Zmce07jslyRL3zZ I couldn't get the image to load. So above is a link to an image of the problem...
  41. K

    Calculating Line Integral in xy-Plane

    Homework Statement Calculate the line integral ° v ⋅ dr along the curve y = x3 in the xy-plane when -1 ≤ x ≤ 2 and v = xy i + x2 j. Note: Sorry the integral sign doesn't seem to work it just makes a weird dot, looks like a degree sign, ∫.2. The attempt at a solution I have to write something...
  42. karush

    MHB 16.1.9 Line Integral over space curves

    Evaluate $\displaystyle \int_C(x+y)ds$ where C is the straight-line segment $x=t, y=(1-t), z=0, $ from (0,1,0) to (1,0,0) ok this is due tuesday but i missed the lecture on it so kinda clueless. i am sure it is a easy one.
  43. K

    MHB Multivariable calculus line integral work

    calculate the work done by the force field $F(x,y)=(ye^{xy})i+(1+xe^{xy})j$ by moving a particle along the curve C described by gamma (γ):[0,1] in $R^2$, where gamma (γ)=(2t-1, t²-t)
  44. Pushoam

    Line integral of a curve

    Homework Statement Homework EquationsThe Attempt at a Solution Line integral of a curve ## I = \int_{ }^{ } yz dx + \int_{ }^{ } zx dy + \int_{ }^{ } xy dz ## with proper limits. ## I = \int_{\frac { \pi }4}^{ \frac { 3 \pi} 4} abc ( \cos^2 t - \sin^2 t ) dt = -abc ## |I| = abc...
  45. yecko

    Line integral of a vector field

    Homework Statement [/B] I would like to ask for Q5b function G & H Homework Equations answer: G: -2pi H: 0 by drawing the vector field The Attempt at a Solution the solution is like: by drawing the vector field, vector field of function G is always tangential to the circle in clockwise...
  46. M

    Using Green's Theorem for a quadrilateral

    Homework Statement Evaluate the line integral of (sin x + y) dx + (3x + y) dy on the path connecting A(0, 0) to B(2, 2) to C(2, 4) to D(0, 6). A sketch will be useful. Homework Equations Sketching the points, I have created a parallelogram shape. I also know that green's theorem formula, given...

    I Evaluation of a line integral

    calculate the line integral for a vector field F= -xy⋅j over a circle which is c(t)=costi+sintj, so I used x=cost y=sint and ∫(0 to 2pi) -(sintcost)(cost)dt=(cos^3(2pi)-cos^3(o))/3=0 now here is the problem, if this enclosed line integral is zero then why is the vector field not conservative?
  48. E

    Line Integral Notation wrt Scalar Value function

    I'm getting a bit confused by the specific notation in the question and am unsure what exactly it is asking here/how to proceed. Homework Statement Given a scalar function ##f## find (a) ##∫f \vec {dl}## and (b) ##∫fdl## along a straight line from ##(0, 0, 0)## to ##(1, 1, 0)##.Homework...
  49. R

    Line integral of vector field from Apostol calculus

    Homework Statement Here are the three problems that i couldn't solve from the book Calculus volume 2 by apostol 10.9 Exercise 2. Find the amount of work done by the force f(x,y)=(x^2-y^2)i+2xyj in moving a particle (in a counter clockwise direction) once around the square bounded by the...
  50. P

    MHB Luca's question via email about a line integral....

    I am assuming that this line integral is along the straight line from $\displaystyle \begin{align*} (0,0,0) \end{align*}$ to $\displaystyle \begin{align*} \left( 5, \frac{1}{2}, \frac{\pi}{2} \right) \end{align*}$, which has equation $\displaystyle \begin{align*} \left( x, y, z \right) = t\left(...