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Question from my engineering mathematics course:

Let $C$ be the vertical circle given by $y = a \sin t$ and $z = a \cos t$ with $x = 0$. Use Stoke's Theorem to evaluate $\displaystyle \int_C(x^2,y,0)\cdot \mathrm d \mathbf r$. Check your answer by calculating the integral directly.

Answer from solutions:

It is clear that the unit normal vector to $D$ is $\mathbf n = \pm \mathbf i$. By inspecting the diagram we see that the $\mathbf n$ must be $-\mathbf i$ to ensure that $S$ stays on the left as we walk around $C$ in the direction of increasing $t$. We also have $\mathrm dA = \mathrm dy \, \mathrm dz = s \,\mathrm ds \, \mathrm dt$, so

$$\operatorname{curl}(\mathbf u) \cdot \mathrm d\mathbf A=(-1,0,-x^2)\cdot (-\mathbf{i}) s \, \mathrm ds \, \mathrm dt = s \, \mathrm ds \, \mathrm dt$$

$$\int_S \operatorname{curl}(\mathbf u) \cdot \mathrm d\mathbf A=\int_{t = 0}^{2\pi}\int_{s=0}^a s \, \mathrm ds \, \mathrm dt = \int_{t=0}^{2\pi}\frac 1 2 a^2 \, \mathrm dt = \pi a^2$$

... Full Answer

I don't understand how the normal ends up pointing in the -'ve x axis direction.

My understanding says that using the right hand rule, you point your first finger in the direction of the first vector (so in this example, the y axis), second finger points in the direction of the 2nd vector (so the z axis) and then thumb points towards direction of normal (which gives POSITIVE x).

Am I understanding this incorrectly?

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    The path starts from the positive z axis and goes to the positive y axis with this parameterization, to use the right hand rule it's easier to wrap your hand around the path and your thumb will point in the negative direction. To use the right hand rule like you want to, you have to use the tangent and normal vector to this path – 2017-01-22

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The right-hand-rule convention for this kind of integral is done a little differently than the cross product of vectors $\mathbf x \cdot \mathbf y$ that you're thinking of. For the curl, you use your fingers to be curved in the orientation of the integration (this depends on how you parameterize the curve), "holding" the normal vector in your right hand, like if you reached to grab a long wire and wrapped your fingers around it. Then the normal vector is in the direction of your thumb if you make a thumbs-up.