Answer to Question #112254 in Field Theory for sahil

Question

Obtain the directional derivative for a scaler field (x,y,z) = 3x^2-y^3z^2 at the point ( 1, -2,-1)in the direction of i^+ j^+ k^......

Answer 1

The directional derivative of function $f(x,y,z)$at point $(x_0, y_0, z_0)$ in the direction of $\bold v$ is $\langle \nabla f(x_0, y_0, z_0), \bold v\rangle$, where $\langle \,, \rangle$ is the scalar product in $\mathbb{R}^3$.

The gradient of $f(x,y,z)$ is $\nabla f(x,y,z) = (6 x, -3 y^2 z^2, -2 z y^3)$ and is equal to $(6, -12, 16)$ at given point $\bold x_0 = (1, 2, -1)$.

Hence, the directional derivative at $\bold x_0$ in the direction of $(1, 1, 1)$ is $\langle (6, -12, 16), (1,1,1)\rangle = 10$.