Articles of line integrals

$C$ 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 not able to get the surface $S$ ; Please help . Thanks in advance

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 $$\int_C \vec{F} \cdot d\vec{r} = \pi \sqrt{3},$$ where $C$ is oriented in the counterclockwise direction when viewed from above […]

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 complex conjugate of $z$ equal this on the circle?

Residue Theorem for Gamma Function

I am kinda stuck and not sure what to do at this point of the calculation where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\,\sqrt{\, 2\,}\,\,\right)^{s}\Gamma\left(\,{s \over 2}\,\right)\,{\rm d}s $$ The Gamma Function produces multiple singularities and I am not sure if the Residue Theorem can be applicable here.

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 $\beta_1$, $\beta_2$, $\sigma$ and $x$ are real and $c>0$ Cauchy’s residue theorem is used and I am not sure how the integral can be simplified to apply the theorem

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: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$$ whereby $\beta > 0$, $\sigma > 0$, $c>0$ and $|x|<\beta$. $\beta$, $\sigma$, $c$ and $x$ are all real numbers

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

I am proving this integral: $$ \int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\beta^{1 \over 2}\,\right)^{s}\ \Gamma\left(\,s \over 2\,\right) \Gamma\left(\,{\beta +1 \over 2} – {s \over 2}\,\right)\,{\rm d}s = \Gamma\left(\,{\beta +1 \over 2}\,\right)\left(\,1+{1 \over \beta}\left(x \over \sigma\right)^2\,\right)^{-{\beta +1 \over 2}}$$ where $\beta>0$, $\sigma>0$ and $x$ is real number The clue I have is that Cauchy’s residue theorem […]

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 $$0<\rho<1-\cos\phi$$ Let a curve $C$ be the intersection of the boundary surface of the solid with the plane $y=0$ and equip $C$ with an anticlockwise orientation as seen from the positive $y-axis$. Find the circulation $$\int_{C}(z+e^x)\:dx+e^{x^3+z^3}\:dy+(\sin y-x)\:dz$$ What […]