Directional derivatives

Point P is located at a distance rho from $P_0$ along the direction vector l. Its coordinates in the xy-plane are given by ($x_0$ + rho*cos(alpha), $y_0$ + rho*sin(alpha)), where alpha is the angle of the direction vector.

Conditions: $P_0$ has coordinates ($x_0$, $y_0$).; The direction vector l makes an angle alpha with the positive x-axis.; rho is the scalar distance from $P_0$ to P in the xy-plane.

The directional derivative f'_l($x_0$, $y_0$) is defined as the limit as rho approaches 0 of the ratio of the change in function value to the displacement rho along the direction l.

Conditions: The function f is defined at ($x_0$, $y_0$) and nearby points.; The direction l is specified by angle alpha.; rho represents the displacement along the direction l in the xy-plane.; The limit is taken as rho -> 0.

The directional derivative represents the slope of the tangent line to the curve formed by intersecting the surface $z=f(x,y)$ with a vertical plane passing through $P_0$ and parallel to the direction vector l.

Conditions: The function $z=f(x,y)$ defines a surface in 3D space.; A specific point $P_0(x_0, y_0)$ is selected on the surface.; A direction vector l is defined by an angle alpha in the xy-plane.; A vertical plane passes through $P_0$ along direction l.

The vertical plane passing through $P_0$ and parallel to the direction vector l intersects the surface $z=f(x,y)$ to form a space curve. The directional derivative is the slope of the tangent to this specific curve at $P_0$.

Conditions: The plane is vertical (parallel to the z-axis).; The plane contains the point $P_0$.; The plane is parallel to the direction vector l in the xy-plane.

As the distance rho between the two points on the curve decreases, the secant line connecting them rotates and approaches a limiting position. This limiting line is defined as the tangent line to the curve at the point $P_0$.

Conditions: The two points lie on a smooth curve (the section curve).; One point is fixed at $P_0$.; The other point moves along the curve towards $P_0$.; rho represents the distance between the two points.

A secant line is constructed by connecting the initial point $P_0$ on the surface to a second point P located at a distance rho along the direction vector l. The slope of this secant line approximates the directional derivative before taking the limit.

Conditions: Point $P_0(x_0, y_0)$ is fixed on the surface $z=f(x,y)$.; A direction vector l is defined by angle alpha.; A second point P is chosen at distance rho from $P_0$ along l.; The coordinates of P are ($x_0$ + rho*cos(alpha), $y_0$ + rho*sin(alpha)).

The direction vector l is represented parametrically by the angle alpha it makes with the positive x-axis. The displacement components are given by rho*cos(alpha) in the x-direction and rho*sin(alpha) in the y-direction.

Conditions: The direction is defined in the xy-plane.; alpha is the angle with the positive x-axis.; rho is the magnitude of the displacement.