Oblique derivative

directional derivative

A derivative of a function $f$ defined in a neighbourhood of the points of some surface $S$, with respect to a direction $l$ different from the direction of the conormal of some elliptic operator at the points of $S$. Oblique derivatives may figure in the boundary conditions of boundary value problems for second-order elliptic equations. The problem is then called a problem with oblique derivative. See Differential equation, partial, oblique derivatives.

If the direction field $l$ on $S$ has the form $l=(l_1,\ldots,l_n)$, where $l_i$ are functions of the points $P\in S$ such that $\sum_{i=1}^n(l_i)^2=1$, then the oblique derivative of a function $f$ with respect to $l$ is

$$\frac{df}{dl}=\sum_{i=1}^nl_i(P)\frac{\partial f}{\partial x_i},\quad P=(x_1,\ldots,x_n),$$

where $x_1,\ldots,x_n$ are Cartesian coordinates in the Euclidean space $\mathbf R^n$.

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