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
W
=
F
⋅
s
{\displaystyle W=\mathbf {F} \cdot \mathbf {s} }
, have natural continuous analogues in terms of line integrals, in this case
W
=
∫
L
F
(
s
)
⋅
d
s
{\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
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...
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...
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
##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...
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...
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.
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...
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...
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
$$\Delta V = -\int_{0}^{t}...
Homework Statement
[/B]
F =< 2x, e^y + z cos y,sin y >
(a) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along a straight path.
(b) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along the curved path given by C : r(t)...
Homework Statement
(a) Consider the line integral I = The integral of Fdr along the curve C
i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C?
ii) What is the value of I if the vector field F is is a unit vector...
Homework Statement
Homework Equations
flux = int(b (dot) ds)
The Attempt at a Solution
I just wanted clarification on finding ds. I understand why ds is in the positive yhat direction (just do rhr) but I don't understand where the dxdz come from. How do we find ds in general?
Sorry if this is the wrong place to post this, but I wasn't sure where exactly to put it.
When we calculate the force on a closed loop of current-carrying wire in a uniform magnetic field,
We calculate the line integral of the loop to be 0.
However, when we evaluate the line integral for an...
I am looking at a proof from a book in fluid dynamics on time differentiation of fluid line integrals -
Basically I am looking at the second term on the RHS in this equation
$$ d/dt \int_L dr.A = \int_L dr. \partial A / \partial t + d/dt \int_L dr.A$$
The author has a field vector A for a...
Homework Statement
Evaluate the following line integrals, showing your working. The path of integration in each case is anticlockwise around the four sides of the square OABC in the x−y plane whose edges are aligned with the coordinate axes. The length of each side of the square is a and one...
Homework Statement
Sisyphus is pushing a boulder up a 100-ft tall spiral staircase surrounding a cylindrical castle tower.
a) Suppose Sisyphus's path is described parametrically as $$x(t)=(5\cos3t, 5\sin3t, 10t)$$, $$\space 0\leq t\leq10$$.
If he exerts a force with constant magnitude of 50 Ib...
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...
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
(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...
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.
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...
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)...
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...
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...
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...
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...
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 =...
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...
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...
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...
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...
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?
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...
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...
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...
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...
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...
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...
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...
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"...
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...
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...
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...
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...
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...
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...
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...
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}...