calculate normal vector in differential geometry I am reading differential geometry from do carmo book.here he gave the formula to calculate normal vector as $$N(q)=\frac{Xu \wedge Xv}{|Xu \wedge Xv|}$$.
But I am not sure how to calculate for Cartesian.
As example for plane $$ax+by+cz+d=0$$ or sphere $$x^2+y^2+z^2=1$$ or cylinder how should I calculate normal vector using the formula.
Can someone elaborate the procedure please.
 A: I'm a little unsure about your notation, but typically you start with a parametrization $\mathbf{x}(u, v): \mathbb{R}^2 \rightarrow \mathbb{R}^3$.
For example, a plane can be parametrized as $\mathbf{x}(u, v) = u\mathbf{a} + v\mathbf{b} + \mathbf{c}$ where $\mathbf{a, b}$ are vectors parallel to the plane and $\mathbf{c}$ is a point that lies inside the plane.  
Likewise, you can find a parametrization for a sphere here.
Once you have a parametrization, then you compute $\displaystyle \frac{ \partial \mathbf{x} }{ \partial u}$ and $\displaystyle \frac{ \partial \mathbf{x} }{ \partial v}$, which are what $X_u$ and $X_v$ are here.  Finally, your normal vector will be given by $\displaystyle \frac{ \partial \mathbf{x} }{ \partial u} \times \frac{ \partial \mathbf{x} }{ \partial v}$.
As per convention, we finish up by normalizing it, so you'll want to divide by $\displaystyle \left\lvert \frac{ \partial \mathbf{x} }{ \partial u} \times \frac{ \partial \mathbf{x} }{ \partial v} \right\rvert$.
A: The formula you cited from Do Carmo's book applies in the case where you have parametric equations $(u,v) \mapsto \mathbf{x}(u,v)$ that give a surface point $\mathbf{x}(u,v)$ corresponding to any given parameter values $(u,v)$.
If the surface is given in "implicit" form $f(x,y,z)=0$, then there's a different formula for calculating surface normal:
$$
\mathbf{N} = \operatorname{grad} f = \nabla f =\left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, 
\frac{\partial f}{\partial z} \right)
$$
If you want a unit vector, divide this vector by its length, of course.
So, you have two choices: (1) invent some parametric equations for your surface, and use your formula, or (2) use the formula I gave above. For your plane and sphere examples, the second approach is easier, I think. If you prefer to use the first approach, then @Kaj showed you how.
More info here.
