Question

If φ = 2xz4 – x2y, find ∇φ and ‖∇φ‖ at the point (2, -2, -1).

Ans. 10 𝒂̂𝑥 - 4 𝒂̂𝑦 - 16 𝒂̂𝑧 , 2√93

Answer 1

The gradient of scalar field is given by

$$\nabla\varphi=\frac{\partial \varphi}{\partial x}{\hat i}+\frac{\partial \varphi}{\partial y}{\hat j}+\frac{\partial \varphi}{\partial z}{\hat k}$$

We get

$$\nabla\varphi=\left(\frac{\partial }{\partial x}{\hat i}+\frac{\partial }{\partial y}{\hat j}+\frac{\partial }{\partial z}{\hat k}\right)(2xz^4-x^2y)$$

$$\nabla\varphi=(2z^4-2xy){\hat i}-x^2{\hat j}+8xz^3{\hat k}$$

$$=(2*(-1)^4-2*2*(-2)){\hat i}-2^2{\hat j}\\+8*2*(-1)^3{\hat k} =10{\hat i}-4{\hat j}-16{\hat k}$$

$$|\nabla \varphi|=\sqrt{10^2+(-4)^2+(-16)^2}=2\sqrt{93}$$