See [[Del in cylindrical and spherical coordinates]].
==Further properties and applications==
===Level sets===
{{see also|Level set#Level sets versus the gradient}}
If the partial derivatives of ''f'' are continuous, then the [[dot product]] <math>(\nabla f)_x\cdot v</math> of the gradient at a point ''x'' with a vector ''v'' gives the [[directional derivative]] of ''f'' at ''x'' in the direction ''v''. It follows that in this case the gradient of ''f'' is [[orthogonal]] to the [[level set]]s of ''f''. For example, a level surface in three-dimensional space is defined by an equation of the form ''F''(''x'', ''y'', ''z'') = ''c''. The gradient of ''F'' is then normal to the surface.
More generally, any [[embedded submanifold|embedded]] [[hypersurface]] in a Riemannian manifold can be cut out by an equation of the form ''F''(''P'') = 0 such that ''dF'' is nowhere zero. The gradient of ''F'' is then normal to the hypersurface.
Let us consider a function ''f'' at a point P. If we draw a surface through this point P and the function has the same value at all points on this surface,then this surface is called a 'level surface'.
===Conservative vector fields===
The gradient of a function is called a gradient field. A (continuous) gradient field is always a [[conservative vector field]]: its [[line integral]] along any path depends only on the endpoints of the path, and can be evaluated by the [[gradient theorem]] (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function.
