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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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Explanation of coefficient when evaluating contour around a branch for fractional version of Cauchy's Integral Formula

I am working on fractional derivatives which are defined by taking the Cauchy Integral formula and letting the order of the derivative be non-integer. Specifically, \begin{equation} f^{(\alpha)}(z)=\...
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mean of a field along the line

What is the proof of a mean field along any line? Or how can we define it? i.e \begin{equation} a=\int_0^1 b \, dx \end{equation} where $a$ is the mean of $b$ and $b(0)=b(1)=0$.
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Question about line integrals with various options

I am new to calculus and am looking for some feedback regarding the following question. Many thanks in advance! The following line integral is given: $\int_C(y + y cos(xy))dx + (x + x cos(xy))dy$. ...
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Calculating the mass of the ellipse curve

I want to calculate the mass of the ellipse curve $\vec\gamma(t)=(acos(t),bsin(t))$ for $a>0,b>0$ and for $t \in[0,2\pi]$, where line density $\lambda(x_1,x_2)=Cx_1x_2$ for $C>0$. After ...
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Triangle parameterisation

I get how to answer the qs below, the problem is actually finding path $2$ $ \left( 2, 0, 0 \right) $ to $ \left( 0, 1, 0 \right) $ I get $(2-t)i+tj$ yet the answer for path 2 is... $$ (2-t)i+(t/2)...
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A given perimeter length that is circular encloses the maximum area - which are the (analytic) proofs? [duplicate]

I'm guessing Newton, because of his integrals. But what proofs have been established, and which is the most mathematically intuitive one? I was looking for the tag "circumference", supplied the newer ...
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How to compute the line integral of $\int_{C^+}(x^2-y)dx+(y^2+x)dy$?

I need help with this problem: Compute the line integral$\int_{C^+}(x^2-y)dx+(y^2+x)dy$ where $C^+$ is the parabollic arc $y=x^2+1$, $0\leq x\leq 1$ oriented from $(0,1)$ to $(1,2)$. First I ...
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How to prove the integral formulae of the inverse path $\alpha^-$ and the product path $\alpha\beta$?

I need help with this problem: Let $f:S\subset\mathbb{R}^n\rightarrow\mathbb{R}$ be continuous on $S$, and let $\alpha:[a,b]\subset\mathbb{R}\rightarrow\mathbb{R}^n$ and $\beta:[c,d]\subset\mathbb{...
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How to evaluate the line integral (checking Stokes Theorem)

Consider the vector field: $$\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k$$ A closed curve $C$ lies in the plane $x + y + z = 3$, oriented counterclockwise. The parametric ...
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Work done by three-dimensional inverse square field

I'm confused on how to set up the problem. Any help would be great, thank you in advance!
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Define the path integral when scalar $f$ and curve $\mathbf{c}$ is in *curvilinear* coordinates

I'm doing a multivariable calculus course at the moment. I've seen path integrals in the cartesian coordinate system as the following definition: Definition. The path integral of $f(x,y,z)$ along the ...
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Using line integrals to find total surface area

I am given the equation for the height of a fence, which is h(x,y,z) = 4 + ((sinx)/2) + (y/3) +z. The line integral of this function will give the lateral surface area of one side of the fence. ...
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Integration problem with full derivative

I am trying to find out what type of integration is used to solve this problem and what are the rules behind it. I am sorry, if this question is quite basic, but I do not study Mathematics. The ...
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What kind of integration is this and what is the rule behind it?

I am trying to find out what type of integration is used to solve this problem and what are the rules behind it. I am sorry, if this question is quite basic, but I do not study Mathematics. The ...
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LHS and RHS of Stokes' theorem not equal

Question: I am trying to test my understanding of Stokes' theorem by calculating the left and right hand side of the theorems equality by way of example and seeing whether they equal each other. They ...
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Why calculate the line integral for a vector field? *Without using work/physics*

For a while now I've been trying to find motivation and a good intuition behind the line integral for a vector field. This is the first time I'm learning this topic and I'm not interested in too much ...
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line element as a 1 form

I'm studying differential forms and I know how to manipulate all the equations. On trying to find a pictorial understanding, I am a bit stuck on the following. A one-form is suppose to assign a ...
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How to show that two parametrizations are representing the same line in plane?

(Two parameterizations): $$\vec r_1(t) = (t^3, t + 1), t ∈ [0, 1] $$ $$\vec r_2(t) = (t^6, t^2 + 1), t ∈ [0, 1]$$ How can I show that these two parameterizations represent the same line in plane? ...
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How to convert an integration of a curl to the surface integral?

$\displaystyle\int \left[ \nabla \times \dfrac {M(r')}{r} \right] d\tau =\oint \dfrac{1}{r} \left[ M(r')×da' \right]$ I came across this integral from the David Griffiths introduction to ...
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Evaluating $\oint_C \vec{F} \cdot d\vec{r}$

After trying a couple of times, but failing to find a way to solve these problems, I decided I should perhaps ask the people on this forum for help. Problem 1 Let $C$ be the curve $(x-1)^2+y^2=16$, ...
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path independence of a scalar field line integral

I had seen mathematical proofs of why it's true. But I couldn't wrap around my head with the intuition behind it. For a single variable function, $\begin{equation} \int_{a}^{b}f(x) \,dx = -\int_{b}^{...
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Finding the maximum value of $\oint_C \vec{F} \cdot d\vec{r}$ of a vector field

I'm stuck with trying to find the maximum value for $\oint_C \vec{F} \cdot d\vec{r}$ of the vector field $\vec{F}=\langle5z, x, y \rangle$ where $C$ is a simple closed curve in the plane $2x+3y+z=7$. ...
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Integrating Factors Via Minimization

I am trying to algorithmize an idea to find integrating factors for inexact differential equations via optimization. Currently, I'm sticking to simple cases that can already be solved by other means. ...
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For Green's theorem, why is the region of integration of the line integral a weird partial derivative character?

Why the weird $\partial{Q}$ notation for the integral region for Green's Theorem? $$\int_{\partial{Q}} W \cdot ds = \iint_Q \frac{\partial{g}}{\partial{x}} - \frac{\partial{f}}{\partial{y}} dx\ dy$$ ...
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Evaluate the line integral of a vector field around a square

I am asking this question because I believe the answer sheet I was given has an incorrect solution. The task is to evaluate (by hand!) the line integral of the vector field $\mathbf{F}(x,y) = x^2y^2 \...
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Shortcuts for $\int_{\kappa}F dx$

Let $F(x,y,z):=\begin{pmatrix} x^{2}+5y+3yz \\ 5x +3xz -2 \\ 3xy -4z \end{pmatrix}$ and $\kappa: [0, 2\pi] \to \mathbb R^{3}, t\mapsto\begin{pmatrix}\sin t\\ \cos t \\ t \end{pmatrix}$ I was asked ...
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Finding the value of $\int{F \cdot dr}$

Find the value of $\int{F \cdot \mathrm{d}r}$, where $$F(x,y) = \langle 5e^y+ye^x,e^x+5xe^y \rangle$$ and $$C: r(t) = \left\langle\sin\left(\frac{\pi t}{2}\right),\ln(t)\right\rangle; 1\le t\le2$$ So ...
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Line integral for surface area

Use a line integral to find the area of the surface that extends upward from the semicircle $y=\sqrt{4-x^2}$ in the $xy$-plane to the surface $z=3x^4y$. I know how to compute line integrals but I'm ...
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Help in evaluating this line integral

Let $\gamma (t)=1+e^{it}, 0\le t\le 2\pi$. I have to find the line integral of $\frac{1}{(z^2-1)}$ with respect to $\gamma$. My attempt : $\gamma$ is the unit circle in the complex plane centered at ...
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Parametrization of a spherical triangle?

I want to calculate an integral in the curve defined by a spherical triangle with vertices $\left(a,0,0\right),\left(0,a,0\right),\left(0,0,a\right)$ over the sphere of center $\left(0,0,0\right)$, ...
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Question about the continuity of the length of a continous parametrized curve

Any hint or demo to prove that the length of a continous parametrized curve defines a continous function in a normed space?
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Line integral path independence proof check

Find the work done by the force $F(x, y, z) = (x^4y^5, x^3)$ along the curve C given by the part of the graph of $y$ = $(x^3)$ from $(0, 0)$ to $(-1, -1)$. I first checked for independence, which ...
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Closed form solution to $\int_0^{\pi/4}\frac{e^{ic\sqrt{1+ br^2(\theta)}}}{\sqrt{1+ br^2(\theta)}}\,d\theta$

Is there closed form solution to this integral $$\int_0^{\pi/4}\frac{e^{ic\sqrt{1+ br^2(\theta)}}}{\sqrt{1+ br^2(\theta)}}\,d\theta$$ $r(\theta)=\frac{a}{\cos(\theta)}$ is radius vector from the ...
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Line integral, independence of path when to use it

Let $F(x,y) = (3x^2,4y^3)$. Determine the value of $\int_c F(x,y)\cdot \mathrm dr$, where $c$ is the path from $(0,1)$ to $(\pi,-1)$ along graph of $y=\cos x$. Is it good to always check for path ...
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Calculating the area of a surface given by a set $S$

$S=\{(x,y,z):x^2+y^2+z^2=4, (x-1)^2+y^2 \leq 1 \}$. $x^2+y^2+z^2=4 \iff \frac{x^2}{4}+\frac{y^2}{4} +\frac{z^2}{4}=1$ I'm not exactly sure what to parametrize the set $S$ by I thought of using ...
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define work $W=\int_C \mathbf{F}\cdot d\mathbf{r}$

We define work as follows : $$W=\int_C \mathbf{F}\cdot d\mathbf{r}$$ and we know that $\mathbf{F}(\mathbf{r}(t),\mathbf{v}(t),t)$ now can we write ? : $$W=\int_C \mathbf{F}\cdot d\mathbf{r}\overset{?}{...
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Line Integral Over a Vector Field of a set of points where a sphere intersects $2z$ $+$ $x$ $=$ $0$

One of my practice problems asks me to compute $\int_C zdx+xdy$ where $C$ is the set of points satisfying $$x^2+y^2+z^2=4 \quad\text{and}\quad 2z+x=0$$ where $C$ is oriented counterclockwise. If I ...
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Calculate line integral in a vector field

I have the following problem and I'm not able to solve it. Given $F(x,y,z) = (1-2z, 0, 2y)$, calculate the line integral of C, where C is the contour of the surface. $S=\{(x,y,z) / x\geq 0, y \geq 0, ...
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Confusion regarding vector line integrals

Hi so I read online that the vector line integral can be represented this way. But I also read that we can use when derivatives appear inside the integral. So why is the last line still in terms of ...
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Prove that $\omega$ is closed if $\int_{c}\omega \in \mathbb{Q}$.

Let $\omega$ be a differentiable 1-from defined on an open subset $U \subset \mathbb{R}^{n}$. Suppose that for each closed differential curve $c$ in $U$, $\int_{c}\omega \in \mathbb{Q}$. Prove that $\...
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Decomposition of the absolute value of a complex line integral

I just would like someone to help me understand how $$\Bigg\vert{\int^{b}_{a} g(t) dt} \Bigg\vert = e^{-i \theta} \int_{a}^{b} h(t) dt$$
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Two Tests for Exactness

Suppose we have a differential equation of the form $M(x,\ y)\ dx + N(x,\ y)\ dy = 0$, where $\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \neq 0$. It is thus inexact, and absent ...
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Evaluating line integral over an ellipse

How can I evaluate $$\int\limits_C(1+x^2y)\ ds$$ where $C$ is the first quarter of the ellipse $\frac{x^2}9+\frac{y^2}4=1$ I tried parameterizing the curve but I couldn't get rid of the square root $...
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Evaluation of a few line integrals

1. Evaluate $\displaystyle\oint_C\vec{F}.d\vec{r}$, where $\vec{F}=(x^2-3y^2)\hat{i}+(y^2-2x^2)\hat{j}$ and the closed curve $C$ is given by $x=3\cos{t}, y=2\sin{t}$, where $0 \leq t<2 \pi$ in the $...
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Find wrapping angle of helix on a torus

I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus. The wrapping angle (or the angle measured around and/...
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What is really an exact differential and how does it relate to conservative fields

If there is a differential form $ A(x,y,z) dx + B(x,y,z) dy + C(x,y,z) dz$ where there exists some function $\psi(x,y,z)$ Let $ \psi = \psi (x,y,z)$ Then the total differential is $ d \psi = \left(\...
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Line integral method calculate work done by a particle

I'm having trouble knowing how to go about solving this question: Q: The force on a particle at a point with position vector $r = xi + yj + zk$ exerted by a charge at the origin is $F(r)=\left(\frac{...
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Sketch and find the volume of the solid in the first octant bounded by the coordinate planes, plane x+y=4 and surface z=root(4-x)

I understand that this can be done with triple integrals, but my class has yet to be taught those and we will be assessed on our ability to perform a question similar to this one with the principles ...
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The Jacobian and Particular Solutions to an Underdetermined Equation

I was wondering if the factor $\sqrt{(x')^2 + (y')^2}$ in the line integral formula $$\int_a^b\ f(x,\ y)\ \sqrt{(x')^2 + (y')^2}\ dt$$ can also be thought of as a Jacobian determinant, due to the ...
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Piecewise Line Integral of a Vector Field

I'm currently brushing up on some vector calc for a class on electromagnetics, and this particular practice question is giving me a bit of a headache: If: $$\mathbf H=(x - y)\mathbf a_x + (x^2 + zy)\...