I have task described below. I'm not sure if I started correctly and how to finish (include orientation to calculations). Could you check it and give me some explanations?
Let $S = \lbrace (x, y, z) \in \mathbb{R}^3: x^2 + y^2 +z^2 = 1, x+y+z=0, x\geq0 \rbrace$, find $p \in S$ such that vector $v = (1, -1, 0)$ is tangent at the point $p$.
Let $\omega = x~dy - y~dz$. Calculate $\int_S \omega$, where orientation is given by $v$ at the $p$.
$v$ have to be orthogonal to $\{x^2+y^2+z^2\}$ and $\{x+y+z=0\}$. It's not hard to show that $p = (\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{-2}{\sqrt{6}}) \in S$ is our point.
Then I tried to find parametrization. $z = - (x+y) \Rightarrow 2(x^2 + y^2 + xy) = 1$, now looking at $y$ as parameter:
$$ x^2 + yx + (y^2-\frac{1}{2}) = 0\\ \Delta = y^2-4(y^2-\frac{1}{2})=2-3y^2 \geq 0 \Leftrightarrow\\ \sqrt{\frac{2}{3}} \geq |y|\\ x = \frac{-y + \sqrt{\Delta}}{2} ~~~~ \because x \geq 0\\ y \leq \sqrt{\frac{1}{2}} ~~~~ \because x \geq 0\\ y \in \left[ -\sqrt{\frac{2}{3}}; \frac{1}{\sqrt{2}} \right] =A'\\ A = \text{int} A' $$ So I made parametrization (we removed only two extreme points - measure 0) $$ \psi(t) = \left( \frac{-y + \sqrt{2-3y^2}}{2}, y, - \frac{y + \sqrt{2-3y^2}}{2}\right) $$
And now forgetting about orientation, we can calculate $$ \int_S \omega = \int_A \frac{-y + \sqrt{2-3y^2}}{2} \cdot 1 - y \cdot \left(-\frac{1 - \frac{3y}{\sqrt{2-3y^2}}}{2}\right)~dy=\\ \frac{1}{2}\int_A (\sqrt{2-3y^2} - y) + y - \frac{3y^2}{\sqrt{2-3y^2}}~dy =\\ \frac{1}{2}\int_A \frac{2-3y^2}{\sqrt{2-3y^2}} - \frac{3y^2}{\sqrt{2-3y^2}} = \frac{1}{2} \int_A \frac{2-6y^2}{\sqrt{2-3y^2}}\\ = \frac{1}{4} $$
And how should I include information about orientation now, when given is vector $v$ at point $p$? I understand what orientation is (as one of two class of abstraction). When two parametrization has same orientation. Why I can check only at one point. Etc. I know some imagination methods like "if you are walking..." (I saw MIT lectures and I have to say that my imagination is very poor).
So, how can I check it algorithmically? Without using imagination, only with calculations. I will be very grateful for general answer for any manifold ($m$-dimentional in $\mathbb{R}^n$) and example on $S$ from given task.