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.

4
votes
1answer
264 views

Evaluate Complex Integral with $\frac{\Gamma(\frac{s}{2})} {\Gamma\big({\beta +1\over 2} - {s\over 2}\big)}$

I am proving this integral: $$ \int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\beta^{1 \over 2}\,\right)^{s}\ \Gamma\left(\,s \over 2\,\right) \Gamma\left(\,{\beta +1 \over 2} - {s \over 2}\...
1
vote
2answers
309 views

Finding the Circulation of a Curve in a Solid. (Vector Calculus)

A solid can in spherical coordinates \begin{equation} x=\rho\sin\phi\cos\theta\\ y=\rho\sin\phi\sin\theta\\ z=\rho\cos\phi \end{equation} be described by the following inequalities $$0&...
1
vote
1answer
129 views

Evaluate Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$

I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$$ whereby $\beta_1$, $\...
1
vote
1answer
122 views

Cauchy's Residue Theorem for Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$

This is a similar problem to the one I posted here. I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$$ ...
4
votes
3answers
701 views

Residue Theorem for Gamma Function

I am kinda stuck and not sure what to do at this point of the calculation where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\,\sqrt{\, 2\,}\,\,\right)^{s}\Gamma\left(\,{s \over 2}\,\...
3
votes
2answers
229 views

Motivation behind notation $\int_C P \, dx + Q \, dy$

I think the notation $$ \int_C P \, dx + Q \, dy $$ is a bit confusing. I understand fairly well the notation $\int_C \vec{F}\cdot d\vec{r}$ and I understand from my question here that they are the ...
2
votes
1answer
453 views

$C$ be curve of intersection of hemisphere $x^2+y^2+z^2=2ax$ and cylinder $x^2+y^2=2bx$ ; to evaluate $\int_C(y^2+z^2)dx+(x^2+z^2)dy+(x^2+y^2)dz$

$C$ be the curve of intersection of the hemisphere $x^2+y^2+z^2=2ax$ and the cylinder $x^2+y^2=2bx$ , where $0<b<a$ ; how to evaluate $\int_C(y^2+z^2)dx+(x^2+z^2)dy+(x^2+y^2)dz$ using Stoke's ...
0
votes
1answer
90 views

line integral (multivariable calculus)

Evaluate the line integral $$ \int_C (\ln y) e^{-x} \,dx - \dfrac{e^{-x}}{y}\,dy + z\,dz $$ where C is the curve parametrized by $r(t)=(t-1)i+e^{t^4}j+(t^2+1)k$ for $0\leq t\leq 1$
0
votes
1answer
54 views

Why does $x= \frac{1}{2}(z+\bar{z}) = \frac{1}{2}(z+\frac{r^{2}}{z})$ on the circle?

I some help computing $$\int_{|z|=r} x \, dz$$ by noting that $x= \frac{1}{2}(z+\bar{z}) = \frac{1}{2}(z+\frac{r^{2}}{z})$ on the circle but I don't understand why this is true. Why does the ...
3
votes
1answer
1k views

Find Surface Area Via a Line Integral (Stokes' Theorem)

I am trying to use Stokes' Theorem to calculate the surface area of a parametrized surface via a line integral. The surface is the part of $z= x^2+y^2$ below the plane $z=5$. I know this can be done ...
2
votes
2answers
119 views

Finding the Radius of a Circle in 3D Using Stokes Theorem

Let $$\vec{F} (x, y, z) = xy\hat{i} +(4x - yz)\hat{j} + (xy - z^{1/2}) \hat{k},$$ and let $C$ be a circle of radius $R$ lying in the plane $x + y + z = 5$. If $$\int_C \vec{F} \cdot d\vec{r} = \pi \...
4
votes
2answers
2k views

When is the line integral independent of parameterization?

Let $\alpha: [a,b] \rightarrow \mathbb{R}^2$ be a smooth path (i.e. $\alpha'$ is continuous on $[a,b]$), and let $f$ be a continuous vector field. The line integral of $f$ along $\alpha$ is defined as ...
2
votes
1answer
489 views

$C$ be the curve of intersection of sphere $x^2+y^2+z^2=a^2$ and plane $x+y+z=0$ ; to evaluate $\int_C ydx + z dy +x dz$ by Stoke's theorem?

Let $C$ be the curve of intersection of the sphere $x^2+y^2+z^2=a^2$ and the plane $x+y+z=0$ ; how to evaluate $\int_C ydx + z dy +x dz$ by Stoke's theorem ? $C$ is a great circle I think ; I am ...
0
votes
1answer
119 views

How do I extend RS-integral to bounded variation parameters?

Definition of Riemann-Stieltjes integration Let $\alpha:[a,b]\rightarrow \mathbb{R}$ be a monotonically increasing function and $f:[a,b]\rightarrow \mathbb{R}$ be a bounded function. Then, $...
0
votes
2answers
190 views

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/...
3
votes
1answer
72 views

Complex substitution allowed but changes result

It is well known that $$ I := \int_L \frac{1}{z} ~\text{d}z = 2 \pi i $$ where $L$ is the complex unit circle, parametrized by $\gamma(t) = e^{it}, 0 \leq t \leq 2 \pi$. However, using complex ...
2
votes
1answer
240 views

Using Green's Theorem to Calculate the Counter-Clockwise Circulation for the Field $\mathbf{F}$ and Curve $C$.

I have this problem Use Green’s Theorem to find the counter-clockwise circulation for the field $\mathbf{F}$ and curve $C$. with this image Green's Theorem says that the counter-clockwise ...
2
votes
1answer
319 views

Cauchy's Residue Theorem with Multiple Gamma Functions

I previously posted a similar problem here and here. This time however I am dealing with multiple gamma functions. This is the problem I am dealing with right now: $$ \int_{c\ -\ j\infty}^{c\ +\ j\...
2
votes
1answer
173 views

Integrating over a specific vector field

I am trying to show that the solution of the following integral is as follows: Define the stopping time: $C(a) = \inf(u \ge 0 : H(\pi(0) |\mu)-H(\pi(u) | \mu) > a)$ Where $H(\pi(t)|\mu(t))=\sum^{...
1
vote
2answers
556 views

How to show that a line integral is independent of the path of integration

Show that the following line integral is path-independent $$ \int_C (\ln y) e^{-x} dx - \dfrac{e^{-x}}{y}dy + zdz $$
1
vote
1answer
146 views

An integral with the $\Gamma$ function: $\int_{c- i\infty}^{c+i\infty} u^{s}\:\Gamma(\beta +s-1) \:ds$

I previously posted a similar problem here and I have solved many of the problems from the answers given with explanations. This time however I am at this point of integration where: $$\int_{c\ -\ i\...
1
vote
2answers
124 views

Compute the following line integral along a path of your choice (Finding potential)

Consider the following vector field: $$\vec A(x,y,z)=(yz)\hat i+(xz)\hat j+(xy)\hat k$$ Compute the line integral of $A$ along a path of your choice connecting $(0,0,0)$ to $(1,1,1).$ I recognise ...
1
vote
1answer
85 views

Is $\int_C P dx + Q dy = \int_C \vec{F}\cdot d\vec{r}$?

I am a bit confused about some notation. For a vector field $\vec{F}$, I understand the notation $$ \int_C \vec{F}\cdot d\vec{r}. $$ But I have also seen the notation $$ \int_C P dx + Q dy $$ If $\vec{...
1
vote
0answers
353 views

Integrals over a Surface Using Stokes Theorem, C not parallel to coordinate axes

F(x, y, z) is a vector field, specifically, F (x, y, z) = 2zi + xyj + 4yk C is the ellipse obtained by intersecting the plane y + z = 4 with the cylinder x^2 + y^2 =4 oriented in a counterclockwise ...
0
votes
1answer
491 views

Prove conservative implies zero line integral over closed without path independence

Based on Stewart - Calculus Is this proof correct? If $F$ is conservative then $\exists f$ s.t. $F = \nabla f$. If $C$ is closed, then r(b)=r(a) hence RHS of the fundamental theorem of line ...
0
votes
1answer
135 views

Evaluate the complex integral $\oint_{|z|=1} z^3\cos z~\mathrm dz$

$$\oint_{|z|=1} z^3\cos z~\mathrm dz$$ I tried using $z=r\mathrm e^{i\theta}$, but the integral gets very complicated before evaluating at the bounds. Was this not the right approach? Also, what ...
0
votes
1answer
50 views

Evaluate the following line integral.

I have to find the line integral of the following. $$\int_Cxe^ydx+x^2ydy, C: 0 \leq x \leq 2, y = 3$$ I am trying to understand the concept of line integrals, but in this case, I am confused as to ...
0
votes
2answers
622 views

To evaluate a line integral along the curve of intersection of the cylinder $x^2+y^2=a^2$ and the plane $x/a+z/b=1$

Let $C$ be the curve of intersection of the cylinder $x^2+y^2=a^2$ and the plane $\frac{x}{a}+\frac{z}{b}=1$; how to evaluate $$\int_C (y-z) dx+(z-x)dy+(x-y)dz?$$ I know that the parametrization for $...
0
votes
2answers
111 views

Integral of 1-form $\omega=\dfrac{-y \,dx + x \,dy}{x^2 +y^2}$ over a triangle. [duplicate]

I'm trying to evaluate the integral of the $1$-form $$\omega=\dfrac{-y \,dx +x\,dy}{x^2 +y^2}$$ through the corners of a triangle with the vertices $A= (-5,-2)$, $B=(5,-2)$, $C=(0,3)$. I've ...