Answer to Question #112254 in Field Theory for sahil

Question #112254
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^......
1
Expert's answer
2020-04-27T10:15:53-0400

The directional derivative of function f(x,y,z)f(x,y,z)at point (x0,y0,z0)(x_0, y_0, z_0) in the direction of v\bold v is f(x0,y0,z0),v\langle \nabla f(x_0, y_0, z_0), \bold v\rangle, where ,\langle \,, \rangle is the scalar product in R3\mathbb{R}^3.

The gradient of f(x,y,z)f(x,y,z) is f(x,y,z)=(6x,3y2z2,2zy3)\nabla f(x,y,z) = (6 x, -3 y^2 z^2, -2 z y^3) and is equal to (6,12,16)(6, -12, 16) at given point x0=(1,2,1)\bold x_0 = (1, 2, -1).

Hence, the directional derivative at x0\bold x_0 in the direction of (1,1,1)(1, 1, 1) is (6,12,16),(1,1,1)=10\langle (6, -12, 16), (1,1,1)\rangle = 10.


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