does this end up having the same effect as a partial derivative?
keenan
@mkmm The term "partial derivative" is usually used when a function is parameterized in terms of a particular set of variables, e.g., for a function $f(a,b,c)$ one might talk about the partial derivatives $\partial f/\partial a$ with respect to $a$. Functions over Euclidean space are naturally parameterized in terms of some orthonormal basis $x_1, \ldots, x_n$, so it's common to think about the partial derivatives $\partial f/\partial x_i$.
The directional derivative is in some sense a more geometric and "coordinate free" idea. When I think about the directional derivative, I don't think about what coordinate system I'm using, or how my function is parameterized, but simply ask for the change $D_{\mathbf{u}} f$ in the function $f$ along some given direction $\mathbf{u}$.
The two are of course related:
$$\frac{\partial f}{\partial x_i} = D_{x_i} f,$$
i.e., the directional derivative along one of the coordinate directions gives the corresponding partial derivative. (Maybe that's all you were really asking! :-))
does this end up having the same effect as a partial derivative?
@mkmm The term "partial derivative" is usually used when a function is parameterized in terms of a particular set of variables, e.g., for a function $f(a,b,c)$ one might talk about the partial derivatives $\partial f/\partial a$ with respect to $a$. Functions over Euclidean space are naturally parameterized in terms of some orthonormal basis $x_1, \ldots, x_n$, so it's common to think about the partial derivatives $\partial f/\partial x_i$.
The directional derivative is in some sense a more geometric and "coordinate free" idea. When I think about the directional derivative, I don't think about what coordinate system I'm using, or how my function is parameterized, but simply ask for the change $D_{\mathbf{u}} f$ in the function $f$ along some given direction $\mathbf{u}$.
The two are of course related:
$$\frac{\partial f}{\partial x_i} = D_{x_i} f,$$
i.e., the directional derivative along one of the coordinate directions gives the corresponding partial derivative. (Maybe that's all you were really asking! :-))