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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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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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Integrating $ \int_\gamma (z-4)dz $, where $\gamma$ is a quarter of the unit circle in the first quadrant, and a segment from $i$ to $-3$ [on hold]

I need to integrate the next function, but I don't know how. $$ \int_\gamma (z-4)dz $$ $\gamma$ is a quarter of circumference of radius 1, centered in the origin in the first quadrant and a ...
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Line Integral Work Done [on hold]

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 ...
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How to calculate the line integral with respect to the circle in counterclockwise direction

Consider the vector field $F=<y,-x>$. Compute the line integral $\int_cF\cdot dr$ where $C$ is the circle of radius $3$ centered at the origin counterclockwise. My Try: The circle is $x^2+y^2=...
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Computing the line integral of the oriented curve

Computing the line integral given that vector field $F=<-y,x,z>$ and the oriented curve $r(t)=<2\cos t,2\sin t ,\dfrac{t}{2\pi}>,0\le t\le2\pi$. Find $\int_cF.T\ ds$ My try: First I ...
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Line Integral of Clockwise Circle

Considering the circle $x^2+y^2=9$ going in the clockwise direction, I am evaluating the line integral $\int_{C}$ $Fdr$ from $(3,0)$ to $(0,3$). I have parametrization $x=3cost$ and $y=3sint$ and I ...
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Difference between $\int_c F \cdot dr$ and $\int_c F\times dr$?

$\boxed{Q . 1}$ If $\phi=2xyz^2$, $\vec F=xy \hat i-z \hat j +x^2 \hat k$ and C is the curve $x=t^2,y=2t,z=t^3$ from $t=0$ to $t=1$ then evaluate $\int_c F\cdot dr$ and $\int_c F\times dr$ I don't ...
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Line integral of a vector field along a curve C with two segments

Vector field $ \vec F = (3x^2y^3+8x)\vec i + 3x^3y^2\vec j$, along a curve C consisting of two segments C$_1$ and C$_2$. Line segment C$_1$ given by $y = 0$ and $0 ≤ x ≤ x_0$ and the line segment C$...
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Solve the line integral without Green's formula

I'm starting on double and line integrals, and I'm stuck at this question. It asks of me to calculate the following integral without using Green's theorem. Usually with Green's theorem I use polar ...
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Lineintegral $\int_{\gamma}|z|^2dz$ over ellipse

Let $a,b\in\mathbb{R}_{>0}$ and $\gamma: [0,2\pi]\rightarrow\mathbb{C},t\mapsto a\cos(t)+ib\sin(t)$ calculate the line integral $\int_{\gamma}|z|^2dz$ My calculation turns out to be really ugly. ...
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Complex line integrals

Suppose we have an analytic function then Why complex integral of that function does not depend on the path of integration?
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2 types of Line integrals in scalar field

Consider the line integral, $\int _ c$f(x,y)$\vec dr$ , where f(x,y) is a scalar field, and it is evaluvated on a curve c . After integration we get a vector let it be $\vec I$ . $\int _ c$f(x,y)$\...
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Compute the integral $\int_{|z|=\rho}|z-a|^{-4}|dz|$ with $|a|\neq \rho$

I need help in computing the integral indicated above. What I've tried so far: Parametrize the curve indicated by $|z|=\rho$ with $\gamma = z(t) = \rho \cos t + i\sin t$. Then by definition $$ \int_\...
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Picking an Ellipse in Vector Field Integral

I have difficulty understanding the part of the solutions on identifying some ellipse bounded by $L$, and the purpose which it serves. Overall I am unclear of the general direction of the solution and ...
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intersection of 2 surfaces for line integral

I am asked to find the curve of intersection of the following: $x+y=2$ and $x^2+y^2+z^2=2(x+y)$. I suppose (1) is a cylinder and unable to find what (2) is, however I think their intersection will ...
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Show that $\int_C y\,dz=-S$

Let $C$ be a simple closed contour bounding an area $S$. Prove that $$\displaystyle{\int_{C}y\,dz}=-S.$$ Let $u(x,y)=0,\,v(x,y)=y$, which leads to $v_y=1$ and $u_x=u_y=v_x=0$, so that $$\...
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Line Integral Symmetry Trick

(I was wondering how to use the given hint in this question.) Evaluate $$ \int_L \frac{1}{x^2+y^2}\, ds $$ where $L$ is the straight line $Ax+By=C$, $C\ne 0$. Hint: use the symmetry of the ...
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How to solve a line integral when one of the points is not in the domain?

Consider the vector field $\mathbb{R}^2\setminus\{0\}$, $F = \frac{2x}{\sqrt{(x^2 + y^2)}}\hat{i} + \frac{2y}{\sqrt{(x^2 + y^2)}}\hat{j}$ Compute the integral: $\int_{C}^{}F\cdot{dr}$ for the ...
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Stuck on limit of line integral

I am trying to show that the limit of $\int_{\partial B_\rho} u_x dy -u_y dx =0$ as $\rho \rightarrow 0$, where $\partial B_\rho$ is the boundary of a circle of radius $\rho$ centred at the origin, ...
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Line Integral in second quadrant of Unit Circle

If I am asked to compute $$\int_c F . dr$$ Where $$F(x,y) = <d/dx f(x, y), d/dy f(x,y)>$$ and $$f(x,y) =\sin(x^3 + y^3)$$ and C is the portion of the unit circle in the second quadrant, ...
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How to evaluate $ \int_c(2x+y)\,ds$ where c is defined by $x^2+y^2=25$ from the point $(3,4)$ to $(4,3)$ why it gives me $-15$?

If I parametrize the curve with $\begin{cases}x=5\cos t \\ y=5\sin t\end{cases}$ it gives me $-15$. Why if I parametrize the curve with $\begin{cases}x=t \\ y=\sqrt {25-t^2}\end{cases}$ the correct ...
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Find $\int_C F.dr $ where C is the smooth curve joining $(1,0,0) $ to $(0,0,2)$.

Suppose the vector field is the gradient of the function $f(x,y,z)=-\frac{1}{x^2+y^2+z^2}.$ Find $\int_C F\cdot dr $ where $C$ is the smooth curve joining $(1,0,0) $ to $(0,0,2)$. I found $\...
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Equality of complex line integrals along the closed contours $C$ and $C'$ where $C'$ is inside $C$ when we join them with a straight line

In the book of The theory of Functions by Titchmarsh, at page 78, after Cauchy's integral theorem, it is given that Suppose again that $C$ is a simple closed contour, and $C'$ another simple ...
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Need help trying to interpret line integral of a vector field as area.

So I was watching some videos on Line-Integrals on Khan Acadamy and when Sal was discussing about Line Integrals of Scalar Functions, Sal mentioned that one can think of a line integral as the area of ...
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Reference contour integral

I don't feel safe about Contour Integral for complex analysis. Espacially because of the hypothesis (regularity, Jordan's curve, ...). I think my complex lesson avoid it ! I would like to study the ...
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Conditions to tell that the total flux in a vector field across a closed surface is zero

When is the total flux across a closed surface zero. I am trying to find a set of values for an equation to prove that it has a total flux of 0 across a closed surface. I am not given any specific ...
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Complex line integral of $\frac{e^z}z$

I have an integral $$\oint_{|z|=1}\frac{e^z}{z}\,\mathrm dz.$$ I defined $g: [0,2\pi]$, $g(t)=e^{it}$. The integral then becomes. $$\int_0^{2\pi}e^{e^{it}}\mathrm dt.$$ I used the property that ...
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Complex integration along a circle not centered at the origin

I need to evaluate $∮_C\overline{z}^2dz$ around |$z-1|=1$. I understand that this is a circle with radius $1$ centered at $(1,0)$. I know how to do this if the circle is centered at the origin. $\...
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Parameterizing to evaluate a line integral with complex numbers

I'm trying to evaluate $\int_C(z^2+3z)dz$ along the circle $|z|=2$ from (2, 0) to (0, 2) going counterclockwise. I have an answer, but I was told it was wrong. It apparently should be $\frac{-44}{3}-...