Questions tagged [line-integrals]

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.

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Residue Integration Without a Closed Contour

Can the Residue Integration method be used if the interval of integration does not result in a closed contour, or cannot be leveraged due to symmetry? For example: $$\int_{a}^{b}\frac{1}{k+cosx}\,dx \\...
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Line integral around the origin involving bivariate polynomials

I encountered a mathematical problem while studying mathematical analysis. The problem is as follows: Let $n\geq 1$ and consider a polynomial $R$ of degree $2n$ in two variables that is only equal to ...
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Must the orientation of a parameterized curve $\gamma:I\subseteq\mathbb{R}\to \mathbb{R}^n$ be in the direction of the increasing parameter?

In the context of evaluating the line integral of a vector field $\vec{F}:\vec{x}\in\mathbb{R}^n\mapsto \vec{F}(\vec{x})\in\mathbb{R}^n$ along a regular parameterized curve $\gamma:t\in{I}\subseteq\...
DarkLordOfPhysics's user avatar
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Consequences of the vanishing of the line integral of a 1-form on every closed piecewise polygonal with lines parallel to the axes

Let $A$ be a path-connected set of $\mathbb{R}^2$ and let $w\in\Omega^1(A)$ be a differential form of degree 1 $$ \omega = a(x,y)dx+b(x,y)dy. $$ Let's assume that $\tilde \gamma$ is any piecewise $C^...
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Caculate the line integral of a ellipse by a small circle.

Calculate the line integral $$ \int_\gamma \frac{y\,dx+(1-x)\,dy}{(x-1)^2+y^2} $$ where $\gamma$ is the ellipse $x^2 + 4y^2 = 4$ traversed two laps in positive direction. So I have been given a ...
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Directional derivative of a line integral along a straight line path

I'm currently working through a paper that uses quite a bit of multivariable calculus, which I am admittedly quite rusty at. At one point a line integral is taken over a straight path between two ...
Eqn6's user avatar
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Continuity of line integral given a sequence of curves

I have this question regarding the continuity of a line integral. Suppose we have a continuous function $f:\mathbb{R}^n \to \mathbb{R}$, and a curve $C_n$ such that $C_n \to C$ pointwise as $n\to \...
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Line Integral in Octave

I want to compute Line Integral in Octave. There is a sample code that works fine in Matlab, but it does not work in Octave. ...
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Finding a lower bound for $\lvert 1-e^{z} \rvert$ on a circle to compute a complex integral

I've been stuck on a specific part of a problem for the past few days. For $k \geq 1$ we consider the meromorphic function $f_k : z \rightarrow \frac{1}{z^{2k}(1-e^z)}$. Its poles are the $2 \pi m$ ...
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line integral with respect to angle

Suppose we have a parametric curve $x = x(s)$, $y = y(s)$, and $z = z(s)$. We define the unit tangential vector of the curve by $(\cos\theta_x, \cos\theta_y, \cos\theta_z)$, which we assume to be ...
Maozhu Peng's user avatar
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Conceptual Question: difference between Line Integral with respect to ds versus dx or dy

I am doing vector calculus and am having trouble gaining intuition/understanding about the difference between a line integral with respect to ds, and line integrals with respect to dx. $$\int_C f(x,y)\...
badcoder123's user avatar
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Using a line integral to finding a function for the distance between two points on a sphere

Per the title, suppose in R 3 we have a unit sphere. Two vectors intersect that sphere at two points. I know the distance between two points can be found as a function of the angle between the two ...
Peter Eremeev's user avatar
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Finding parameterization for exponential integrand

I'm currently working on a practice problem for my Calculus III class: "Evaluate the line integral along the negatively-oriented closed curve C, where C is the boundary of the triangle with ...
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Visual Representation of the Line Integral

I am trying to better understand the visual interpretation of a Line Integral. We are all told that a classic integral of a function represents the area under that function between two certain points -...
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Line integral over multiple lines by parameterizing the curves

I have solved this problem by setting a relationship between x and y for the 2 lines and got 11. However, when I try solving by the method of parameterization, I come up with 36. I tried setting $C_1 =...
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Minimizing a line integral in 2 dimensions : $\inf \int_{(a,b)}^{(c,d)} f(r(t)) |r'(t)| dt$

Let $x,y,z$ be real. Consider a scalar field $$z = f(x,y)$$ More specific; $f(x,y)$ is a (given) real polynomial in $x,y$ of degree at most $5$ such that For all $x,y$ $$f(x,y)> 0$$ For a given ...
mick's user avatar
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Mass of a curve

The curve $\gamma$ is the intersection between the cylinder $x^2+y^2=1$ and the plane $z=2-x$, so $$\gamma(t)=(\cos(t),\sin(t),2-\cos(t)),\quad 0\leq t\leq2\pi$$ I have to evaluate its mass, with ...
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Whats the difference between using line integral of the first kind and normal integral to find the mass of the given curve

I was working on a question that asked me to find the mass of a given curve using a specified density function. My initial approach was to use the line integral of the first kind, which intuitively ...
Raymonk Surya's user avatar
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What is the work done along a quarter circle in a vector field?

What is the work done along a quarter circle in a particular vector field? To better understand different types of line integrals, I set up the problem below, and request verification: Let $F$ be a ...
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Line integrals and invariance of parameterization

The line integral of a scalar function $f(x,y)$ along a curve $\vec{r}(t)$ for $a \leq t \leq b$ is defined to be $$ \int\limits_{\vec{r}(t)} f(\vec{r}(t)) \, ds = \int_a^b f(\vec{r}(t)) \, ||\vec{r}\,...
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Calculate a line integral of the first kind

Calculate the curvilinear integral of the first kind $\int_{C}ydl$, where $C: x^2 + y^2 = ax$ Solvement $$x^2 - ax + \frac{a^2}{4} - \frac{a^2}{4} + y^2 = 0$$ $$(x - \frac{a}{2})^2 + y^2 = \frac{a^2}{...
Nick Schemov's user avatar
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Solve parameterized line integral

In my physics problem I've modelled the following integral. \begin{equation} p(X,Y,x_0,y_0) = \int_{0}^{2\pi} \frac{1}{\sqrt{(X-R\cos(\Psi))^2+(Y-R\sin(\Psi))^2}} \delta(g) d\Psi \end{equation} Where $...
Matthew James's user avatar
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Using Line Integral to Find Work done by a Force along a Curve

[Find the work done by the force $ \boldsymbol{F}=-10y\boldsymbol{i}+4x\boldsymbol{j}$ along one loop of the curve $r=\sin(9\theta)$] I have some trouble formatting/understanding the question. I have ...
Learningmath's user avatar
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How to transform area integral $\int_{D} \omega^2 \ dx \ dy$ into boundary integral $\oint_{C} \square \ ds$?

Let $\omega$ be a function that satisfies the Laplace's equation $$\nabla^2 \omega = 0$$ The values $\omega$ and $\dfrac{\partial \omega}{\partial n}$ are known in the boundary, but not in the ...
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Proof of Exactness of Closed Forms Using Leibniz' Rule Given Parametrized Curve in R^2

I have run into trouble with a foundational proof regarding the exactness of closed forms which are $C^1$-differentiable. I have seen other proofs utilizing integration factors and I see why the ...
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Line integral and attraction of a material point by a material curve in $\mathbb{R}^3$

According to Newton's law of universal gravitation, a material point $P$ with mass $m$ attracts a material point $P_0$ with mass $m_0$ with a force directed from $P_0$ towards $P$, of size $k\cfrac{...
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Line integral and attraction of a material point by a material curve in R^3

According to Newton's law of universal gravitation, a material point $P$ with mass $m$ attracts a material point $P_0$ with mass $m_0$ with a force directed from $P_0$ towards $P$, of size $k\cfrac{...
Paull's user avatar
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From where does this differential come from?

In one my physics classes I was finding the charge of a half sphere with constant radius $R$ (and I got this expression (just for context purposes): $$Q=\int_S \rho dS$$ and now my teacher did this $...
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Line integral exercise of a real vector field

I'm trying to solve the following exercise of a vector field over line integral: $$\int\limits_C\frac{-y}{4x^2+9y^2}dx+\frac{x}{4x^2+9y^2}dy,$$ where $C$ is the closed curve formed by the equations $y=...
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Line Integral along a Ellipse over a Scalar field

Given is the Line Integral: $$\int_C = \sqrt{\frac{a^2y^2}{b^2} + \frac{b^2x^2}{a^2}}ds$$ the Path $C$ is along the border of the ellipse with: $$\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$$ and moves ...
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Vector Line Integral For Biot Savart Law

How would one go about computing the vector line integral presented in the Biot-Savart law: $$\vec{B}=\int_c\frac{\mu_0I}{4\pi} \frac{d\vec{l}\times\hat{r}}{r^2}$$ I know how to compute vector line ...
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Find the attraction of an infinite homogeneous line of a unit mass located at a distance h from the line.

We know that \begin{equation} X=m_0\int_{\sigma} \frac{f(P)\cos{\alpha}}{r^2} ds, \hspace{1cm} Y=m_0\int_{\sigma} \frac{f(P)\sin{\alpha}}{r^2} ds; \end{equation} where $r$ is the length of the ...
Paull's user avatar
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Line integral and attraction of a material point by a material curve

According to Newton's law of universal gravitation, a material point $P$ with mass $m$ attracts a material point $P_0$ with mass $m_0$ with a force directed from $P_0$ towards $P$, of size $k\cfrac{...
Paull's user avatar
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Functional derivative of line integral

If we can interpret the line integral of a function over a path as a functional. What would be its functional derivative? For instance, in this example, let $\gamma$ be a path in $\mathbb{R}^3$. And ...
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Upper bound of a complex line integral

I have the following problem, to estimate an upper bound of the modulus of $$\int\limits_C\frac{z+4}{z^3-1},$$ where $C$ is the circumference arc that passes from (2,0) to (0,2i) in the first quadrant,...
Fernando's user avatar
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Help for evaluating the line integrals with Green's Theorem

Earlier, when I scrolled the Instagram posts I found a mathematical problem uploaded by The Vegan Math Guy like the following because this problem looks interesting to me to be solved. The ...
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Generalization of rectifiable curves and integrals over them to higher dimensions

Length of a curve can be defined for an arbitrary rectifiable curve(even in an arbitrary metric space). As is shown in this answer we can define line integral over any such curve(even in an arbitrary ...
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line integral for lapacian of green function

I was proving that $$-\Delta_x G(x-y)=\delta_y \text { on } D^{\prime}\left(\mathbb{R}^3\right) $$ but during the calculation there is $$\frac{1}{4 \pi \varepsilon^2} \int_{\partial B_{\varepsilon}(y)}...
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Cauchy Theorem Integration Issue

I need to compute the integral $\int_{0}^{2\pi}\frac{1}{1 + \sin^2(x)}dx = 4.4429$ utilizing the Cauchy theorem. However, I'm encountering a discrepancy where my result is consistently half of the ...
MyLight's user avatar
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Evaluating an Integral Using Cauchy's Theorem and Root-Finding

I am tasked with evaluating the following integral utilizing the Cauchy theorem: $$\int_{0}^{2\pi} \frac{1}{(2 - \cos t)^{2}}dt$$ I have initiated my calculation as follows: $$\begin{align*} \int_{0}^{...
MyLight's user avatar
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Calculation of an Integral Using the Cauchy Formula

I need assistance with calculating the following integral using the Cauchy formula. I have been encountering incorrect results and would greatly appreciate your help in identifying the mistake in my ...
MyLight's user avatar
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Help using Green's theorem to find volume of function inside a polygon

For my work, I am trying to find the volume under a two dimensional function $f(x,y)$ bounded by a polygon of $n$ vertices. My dim memory of undergrad is that Green's theorem is the way to go for this,...
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How can I interpret this object? Is it a 1-form?

THE PROBLEM I am dealing with the following object: $$I = \displaystyle\int_C f(x,y)\,dxdy,$$ where $C$ is a curve on the Cartesian plane and $f: \mathbb{R}^2\rightarrow \mathbb R$ is any Riemann ...
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Every line integral starting and ending at the boundary vanishes implies that the vector field is $0$?

Let $V\subset\mathbb{R}^3$ be compact and possess a piecewise smooth boundary $\partial V$. Let $\mathbf F$ be a continuously differentiable vector field defined on a neighborhood of $V$. Suppose that ...
P.S. Dester's user avatar
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Understanding theorem in Apostol that specifies implicit solution of differential equation $P(x,y)dx+Q(x,y)dy=0$ when the latter is exact.

There is a specific theorem that I would like to make sure I understand correctly. Here is the statement of the theorem as it appears in Apostol's Calculus, Volume II, section 10.19 (on application of ...
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Problem doing a line integral $\int_C P(x,y)dx+Q(x,y)dy$

Evaluate $$\int_C P(x,y)dx+Q(x,y)dy$$ where $P(x,y) = y^2 $, $Q(x,y) = x$, and $C$ is the part of the graph $x = y^3$ from $(-1,-1)$ to $(1,1)$. I was trying the parametrization: $$x = t $$ $$y = \...
SirMrpirateroberts's user avatar
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Explanation for arc length in parametrized curve

let $\Delta s_i$ be a piece of arc length hence: $$\Delta s_i = \int_{i-1}^i \sqrt{[x'(t)]^2+[y'(t)]^2+[z'(t)]^2}dt$$ Why is that the length of $\Delta s_i$ in 2d I know that the as $\Delta x$ -> ...
SirMrpirateroberts's user avatar
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Help With Fundamental Theorem of Line Integrals

I am achieving the wrong answer, but I am unsure where I went wrong about it. I am given a conservative vector field and the line $C$: $$F = \langle \frac{4x}{y^2+1}, -\frac{4y(x^2+1)}{(y^2+1)^2} \...
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Why $A=\int^1_0xdx-\int^1_0x^2dx$ is not the same as measuring the line integral of $\int_\mathcal{C}f(x)ds=\int^1_0x\sqrt{1+4x^2}$?

Today I started learning about line integrals and I have a doubt about the concept itself of the line integral. As I understood, measuring the line integral was the sum of the $ds$ elements multiplied ...
PedroRotondo's user avatar
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Line integral - why is $ds= \dfrac{ds}{dt}dt$?

I’ve been reading on my own about this because it’s on the syllabus of the next semester and I’m very confused on something about the line integral. I have the curve C with parametrization $C(t)=(x(t),...
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