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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Complex integral with fractional singularities
If one consider the complex value function
$$
f(z)=\frac{1}{\sqrt{z-1}\sqrt{z-2}}
$$
with branch cut chosen to be between $z=1$ and $z=2$. Could someone please explain why
$$
2\int_1^2 f(x)dx=\oint f(...
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Integral of Complex logarithm makes sense?
For example, I know that the principal branch of logarithm is not defined over negative real axis.
I think the integral of this logarithm along a circle doesn’t make sense.
Moreover, I know that the ...
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Question about the use of Green's theorem in Do Carmo's proof of the local Gauss-Bonnet theorem.
This is Do Carmo's statement and proof of the local Gauss-Bonnet theorem in "Differential Geometry of Curves and Surfaces":
My question is: how can he assert that
$$\sum_{i=0}^k \int_{s_i}^{...
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What's the definition of a line integral on a possibly disconnected curve?
I'm trying to understand this paper, and I see this integral (page 2):
$$
\int_{B\ \cap\ \mathcal{C}} (1 - y)dy,
$$
where $\mathcal{C}$ is the curve given by $x = ye^{1-y}$ and $B$ can be any ...
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Scalar line integrals in the plane
I understand that $\mathbf{r}=\left[\begin{array}{c} x\\ y \end{array}\right]= \left[\begin{array}{c} \cos{s} \\ \sin{s} \end{array}\right]$, whenever $0\le s \le 2\pi$.
However, I don't understand ...
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How has this integral been written as a line integral?
I am right now self-learning Green's functions for partial differential equations and I am stuck at the very last step of deriving the adjoint operator.
To begin, the PDE of interest is $\textbf L u = ...
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Evaluation of the given line integral
Question: Evaluate $\int_{C}$B.dr along the curve $x^{2}$+$y^{2}$=1,$z$= 1 in the positive direction from (0,1,2) to (1,0,2);given
B= (xz²+y)i+(z-y)j+(xy-z)k
The question itself is easy,but I don't ...
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Gaussian line integral over a polygon
Problem definition
Let $y=z+v$, where $z$ is uniformly distributed over the contour of a polygon with vertices $V_1,\dots,V_n \in \mathbb{R}^2$ and $v\sim\mathcal{N}(0,R)$. Let $\ell$ be the set of ...
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Let C be the path in $\Bbb R^2$ from from $(0,0)$ to $(2,1)$ defined by the equation $x^4-6xy^3-4y^2=0$. Find $\int_C (10x^4-2xy^3)dx -3x^2y^2dy.$ [duplicate]
Let C be the path in $\Bbb R^2$ from from $(0,0)$ to $(2,1)$ defined by the equation $x^4-6xy^3-4y^2=0$. Find $\int_C (10x^4-2xy^3 ) dx -3x^2y^2dy$.
I have no idea how to do this line integral. In our ...
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Is there oriented scalar field line integral, or non-oriented vector field line integral?
I'm studying complex (integral) analysis and struggling with it. BTW, this is my first math stackexchange question, so please forgive any mistakes on my part.
We have definition of complex line ...
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Do line integrals of closed curves depend on the orientation of the curve or of the vector field?
I am trying to understand how circulation works as a line integral with the curl in green's theorem. I know that a line integral describes the relationship between a vector field and a path, i.e. how ...
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Evaluate the line integral of $\textbf{F}(x,y,z) = (f(x),g(y),h(z))$ along the curve $x^{2/3} + y^{2/3} = 1$
Consider the vector field $\vec{F}(x,y,z)=f(x) \vec{i} + g(y) \vec{j}+ h(z) \vec{k}$ where $f,g,h$ are continuous functions. Evaluate
$$
\int_{\mathcal{C}}\vec{F} \cdot \vec{dr},
$$
where $\mathcal{C}$...
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Line integral of an exponential
Let $V_1, V_2\in \mathbb{R}^2$ be the vertices of a segment $s$, that is
\begin{equation}
s\triangleq\{z(\alpha) = (V_2-V_1)\,\alpha + V_1 \}_{\alpha\in[0,1]}
\end{equation}
Now let $y\in\mathbb{R}^2$ ...
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Line Integral with discontinuous functions
The line integral in polar coordinates with scalar field is defined as:
\begin{equation}
\oint f(r,\theta) \sqrt{r^2+(\frac{dr}{d\theta})^2} d\theta
\end{equation}
Imagine that we have for example ...
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Confusion about line integral? two different ways derive two different result
Let $C$ be a counterclockwise path with circle which is centered at origin and radius is $2$ and $y\geq 1$.
Evaluate
$$\int_C \frac{y}{x^2 + y^2 + 1}dx + \frac{x}{x^2 +y^2 + 1}dy$$
I tried two ...
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