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in a Riemannian manifold can be cut out by an equation of the form such that is nowhere zero. The gradient of is then normal to the hypersurface.

Similarly, an affine algebraic hypersurface may be defined by an equation , where is a polynomial. The gradient of is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector.

Conservative vector fields and the gradient theorem

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