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
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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
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?
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...
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}}##...
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)...
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...
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...
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
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...
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}##...
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...
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...
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...
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...
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'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...
In our section on path independence, we were asked to find the potential function given a vector field. Our teacher says to use only line integrals to find the potential function, and not any other method. Like if we have ##F=\left< M,N,P \right> ## The first step is to determine if the domain...
Hello. I'm new to physics, and the problem I have seems so basic, mathematically speaking. I'm just failing to grasp exactly what is being asked. If I can find that, I believe I can find the answer. Here it is:
1. Homework Statement
Let A = x2ˆx + y2ˆy + z2ˆz
Consider the parabolic path y2 =...
Homework Statement
I need to find the work done by the force field:
$$\vec{F}=(5x-8y\sqrt{x^2+y^2})\vec{i}+(4x+10y\sqrt{x^2+y^2})\vec{j}+z\vec{k}$$
moving a particle from a to b along a path given by:
$$\vec{r}=\frac{1}{2}\cos(t)\vec{i}+\frac{1}{2}\sin(t)\vec{j}+4\arctan(t)\vec{k}$$
The Attempt...
This is part of a larger question, but this is the part I am having difficulty with. I have had an attempt, but am not sure where I am making a mistake. Any help would be very, very appreciated.
1. Homework Statement
Let C2 be the part of an ellipse with centre at (4,0), horizontal semi-axis...
Homework Statement
Evaluate line integral(x+sqrt(y)) over y=x^2 from (0,0) to (1,1) and y=x from (1,1) to (0,0)
Homework Equations
n/a
The Attempt at a Solution
I set up integral from 0 to 1 of 2t(sqrt(1+4t^2))dt for the parabola part and then added integral from 0 to 1 of...
I'm used to parameterizing however I'm not sure how to solve these types of problems, any help would be much appreciated.
1) Calculate the line integral ∫v⋅dr along the curve y=x3 in the xy-plane when -1≤x≤2 and v=xyi+x2j
2) a) Find the work that the force F = (y2+5)i+(2xy-8)j carries...
We just started going over line integrals in calculus, and have been told that the integral over any closed surface is 0. What I don't get is then why can we do the line integral of a circle to get 2##\pi##r? Since a circle is a closed surface, shouldn't the line integral then be 0?
Homework Statement
F[/B]=(y + yz- z, 5x+zx, 2y+xy )
use stokes on the line C that intersects: x^2 + y^2 + z^2 = 1 and y=1-x
C is in the direction so that the positive direction in the point (1,0,0) is given by a vector (0,0,1)
2. The attempt at a solution
I was thinking that I could decide...
Homework Statement
A closed curve C is described by the following equations in a Cartesian coordinate system:
where the parameter t runs monotonically from 0 to 2π, thus defining the direction of C. Calculate the area vector of the planar region enclosed by C, using the formula:
2. The...
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)...
The question states: The calculation of the magnetic moment of a current loop leads to the line integral
∮ r x dr
I am puzzled - shouldn't this be ∮ r x dl where r is the radius of the loop and dl is the small change along the loop?
(I think dr would be in the same direction as r, so no cross...
Homework Statement
First, let's take a look at the complex line integral.
What is the geometry of the complex line integral?
If we look at the real line integral GIF:
[2]: http://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif
The real line integral is a path, but then you...
Homework Statement
A force F has components F sub x = axy-by^2, F sub y= -axy+bx^2 where a= 2N/m^2 & b=4N/m^2 Calculate the work done on an object of mass 4kg if it is moved on a closed path from (x,y) values of (0,1) to (4,1), to (4,3) to (0,3) and back to (0,1). all coordinates are in metres...
Homework Statement
Hello
I was given the vector field: \vec A (\vec r) =(−y(x^2+y^2),x(x^2+y^2),xyz) and had to calculate the line integral \oint \vec A \cdot d \vec r over a circle centered at the origin in the xy-plane, with radius R , by using the theorem of Stokes.
Another thing, when...