In vector calculus, the surface gradient is a vector differential operator that is similar to the conventional gradient.
, the surface gradient is defined and notated as where
[1] Examining the definition shows that the surface gradient is the (conventional) gradient with the component normal to the surface removed (subtracted), hence this gradient is tangent to the surface.
The surface gradient arises whenever the gradient of a quantity over a surface is important.
In the study of capillary surfaces for example, the gradient of spatially varying surface tension doesn't make much sense, however the surface gradient does and serves certain purposes.