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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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Proof of integral of complex function f(z)=z

Prove this equation by the definition of complex value function integral. Integral of z from z0 to z1 is equal to 0.5((z1)^2-(z0)^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)\...
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calculate this line integral

this is my curve : $r(t)=(\cos{t},\sin{t}-1,2\cos{\frac{t}{2}})$ , $t=[0,3\pi]$ $r'(t)=(-\sin{t},\cos{t},-\sin{t})$ $||r'(t)||=\sqrt{\sin^2{t}+1}$ I have to calculate: $\int{(y+1)}ds$ So I have : ...
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How to prove that the angle between two vectors on a closed plane curve is given by a line integral

i have to prove that for the angle between the two position vectors at the edge of a part of a closed plane curve is given by the following integral: $\theta$ $=$ $k$ *$\int_c {r\over r^2} \times dr $...
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Line Integral Work Done [closed]

I have a problem when comes to question 2, I don't know how to put this into parametric form. And I am not sure if this is a parabola. Thanks in advance.
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Line Integral Harmonization

Is there a connection between line integrals over scalar fields and line integrals over vector fields? For example, do the pair $f(x, y)$ and $F(x, y)$ which stand in a potential function and ...
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Explain why the area of $S$ is equal to $\int_C x\,d\sigma$ ( line integral )

Let $S$ be a surface in $\mathbb{R^3} $ with the parametrization $g(s, t) = (t, t^2 , st)$ where $g : [0, 1] × [0, 10] → \mathbb{R^3} $ . Explain why the area of $S$ is equal to $\int_C xdσ$ , ...
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Applying the residue theorem to calculate the Fourier transform of $\frac{1}{(x-\tau)^k}$

I'm trying to do this exercise from Daniel Bump's book, Automorphic Forms and Representations. For $f: \mathbb R \rightarrow \mathbb C$, the Fourier transform $\hat{f}$ is defined by $$\hat{f}(v) = \...
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Line integral in proof of Green's theorem

In wikipedia page about Green's theorem the following equality appears: $$ \int_{C_1} L(x,y)\, dx = \int_a^b L(x,g_1(x))\, dx $$ I do not understand it. Wikipedia page about line integral defines ...
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Find $\int F.dr$ where $F= \frac{y i - x j}{x^{2}+y^{2}}$ and $C$ is the circular path $x^{2} + y^{2} = 1$ in cartesion coordinate system.

Find $\int F.dr$ where $F= \frac{y i - x j}{x^{2}+y^{2}}$ and $C$ is the circular path $x^{2} + y^{2} = 1$ described in counter clockwise sense. The formula for flux is $\int {F_1 dy - F_2 dx}$ where ...
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Fundamental Theorem of Calculus when Non-Conservative

Let's say I have an integral of an objective function which is path-dependent (such as in an optimal control problem): $$ J = \int_{t_0}^{t_f} l\big( x(t) \big) dt$$ And want to compute the ...
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Line integrals with respect to x and y or differential form?

When taking line integrals with respect to x and y, is the result from the line integral the displacement in the x and y directions, or is it the area in the xz or yz plane that you're finding? For ...
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Verify Stokes' theorem on a hemisphere using a line integral

When solving this question, I wrote $r(t)$ as $\langle\cos t,0,\sin t\rangle$ then I differentiated it and got $dr=\langle-\sin t,0,\cos t\rangle\,dt$, then i substituted in the formula for line ...
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How to find the vector equation of the line segment.

Let $C$ be the line segment from $(0,0)$ to $(2,2)$, and let $f(x,y)=x^2+y$. Write down a vector equation $r(t)$ of the line segment, that is, find a parametrization of $C$. The answer given is $r(t)...
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Cauchy Integral Theorem with $f(z)=e^{z^2}$

I have $z(t)=t(1-t)e^t + \cos(2 \pi \cdot t^3)i$ with $0 \le t \le 1$ and need to evaluate the line integral of $e^{z^2}$. I know that the endpoints are $z(0)=z(1)=0+i$, so the line is a closed ...
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Line Integrals of Vector Fields, Homework Conundrum

I am a student and I have a conflict with a given answer in the textbook. The question is the following: Evaluate the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ for the given vector field $\...
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Line integral depending on a parameter is entire

Suppose you have a continuous function: $$\phi:[0,1]\rightarrow \mathbb{C}$$ define the complex function: $$f(z)=\int_0^1\phi(t)e^{itz}dt$$ prove that it is entire and calculate it's Taylor ...
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A line integral equals zero implies a real integral also is zero

I'm asked to check that the following line integral is zero: $$\int_{C(0,r)} \frac {\log(1+z)}z dz=0$$ (where $C(0,r)$ is the circle of radius $r$ centered at $0$) and then to conclude that for ...
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Finding the potential function of $F$

$F= \langle ye^{xy}+x^2,xe^{xy}+2y \rangle$. Find the potential function of $F$. My Try: $\varphi_x=f(x,y)=ye^{xy}+x^2 $ and $\varphi_y=g(x,y)=xe^{xy}+2y$ Now integrated the first equation with ...
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Are the bounds of t always [0, 1] for line integrals?

I was given the task to find the line integral $\int _C (x+y)ds$ where $C$ is the line segment from $(0,1,1)$ to $(3, 2, 2)$. I parameterised $C$ as $3t\vec{i}+(1+t)\vec{j}+(1+t)\vec{k}$, which means ...
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Stokes' Theorem - Vector Field

I am having problems trying to verify Stokes' theorem (below) as part of a question. $$\iint_{S} \text{curl} \vec F \cdot d\vec S=\oint_{c} \vec F \cdot d\vec r$$ The vector field in question is $\...
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Find the line integral of $\int\frac{e^{iz}}{z}\, dz$ [duplicate]

This is a homework problem, I need to calculate $\int\frac{e^{iz}}{z}\, dz$ over some curves, but I can find a useful parametrization of $z$ that make this calculations not so difficult. For example ...
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How to calculate $\int\frac{e^{iz}}{z}\, dz$ on the semi-circle given by $re^{i\theta}$ where $\theta:\,0\to\pi$

I was reading this post and in the comments someone said that the difficult in calculating the limit as $r$ goes to $0$ is a lot different than calculating the limit. I tried to calculate the integral ...
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Lineintegral of absolute value with path $\gamma: [0,1]\rightarrow\mathbb{C},t\mapsto i +\exp(i\pi t)$

Calculate: $\int_{\gamma} |z|dz$ with $\gamma: [0,1]\rightarrow\mathbb{C},t\mapsto i +\exp(i\pi t)$ I tried calculating it and actually made some progress, where one term vanished when splitting ...
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Work Done by a Force Field When Given Integral With dx and dy

This homework question is absolutely nothing like anything else my teacher taught us in our course. The problem is as follows: Find the work done by the force field $\vec{F} (x,y) = y \ \vec{i} - x \ ...
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Solving Line Integral When Given Vector and Bounds

I've been stuck on this problem for a while now. I have to evaluate the line integral below: $\int_{C}^{}(x+4 \sqrt{y}) ds \\ C: \vec{r}^{} (t)=t\vec{i}^{} + t\vec{j}^{} \\ 0\leq t \leq 1 $ What I ...