# The difference between a gradient vector to a surface and the normal vector to that s

• Nov 1st 2012, 04:31 PM
nicksbyman
The difference between a gradient vector to a surface and the normal vector to that s
...urface. Plz help I don't understand this difference. Thanks
• Nov 1st 2012, 05:48 PM
chiro
Re: The difference between a gradient vector to a surface and the normal vector to th
Hey nicksbyman.

If you have a definition, you should post it (if its from lectures notes or a book) but a gradient vector can refer to the tangential vector with respect to a particular variable (i.e. one based on the partial derivative of the surface at a particular point).

I think your problem is where you refer to the "gradient vector" of a surface. There is no such thing. Rather, the gradient vector is the gradient of a function. If we have a function f(x,y,z) then the "gradient of f", also written $\nabla f$, is the "vector" $\frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}+ \frac{\partial f}{\partial z}\vec{k}$. Given such a function, the equation f(x,y,z)= constant, could, theoretically, be "solved" for one of the variables in terms of the other two. Since we can then have z= g(x,y), say, that equation defines a surface. Given the equation f(x,y,z)= C, $\nabla f$, the gradient vector of the function is a normal vector to the surface at every point.