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I need to find the normal vector(in a point (a,b)) for a surface F(x,y)=0, that we can't write as y=f(x) and F(x,y) doesn't satisfies the conditions of the implicit function theorem. For example: the surface $x^2 + y^2=1$ (b can be 0 so we can't use the implicit theorem). I found that: the normal vector for the cirlces is $grad(x^2+y^2)|_{(a,b)}=(2a,2b)$ and I don't understand why(I mean that I don't understand why gradient solve this, but the solution (a,b) is clear because of geometry.

Can you also recommend how to find the tangent for these kind of surfaces?

Michael
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1 Answers1

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If you are looking at a hypersurface which is defined by an equation of the form $F(x)=0$, then you are looking at the set of point along which $F$ does not change. The gradient of $F$ points into the direction in which $F$ changes most, so it is normal. In formulas: if $c=c(t)$ is a curve in $F=0$, then $F\circ c(t)=0$, hence also it's derivative: $$0=\frac{d}{dt}F\circ c(t)= \langle (\nabla F)\circ c(t), c^\prime(t)$$ This shows that the gradient of $F$ is orthogonal to any curve tangent to the surface.

(The tangent vectors are just the derivatives of curves such that $F\circ c = 0$. For this you need in fact that $\nabla F\neq 0$).

Then, if it is not $=0$, it is a normal vector. In you example it does not matter if $b= 0$, as long as the vector $2(a,b)^T\neq 0$.

Thomas
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