The rate of increase, of a scalar field 𝑓(𝑥, 𝑦, 𝑧) = 𝑥𝑦𝑧, in the direction 𝒗 = (2, 1, 2) at a point (0, 2, 1) is 

Concept:

The gradient of a function f(x, y, z) is given by:

\(\rm ∇ f = \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j + \frac{\partial f}{\partial z} k\)

where i, j, and k are the unit vectors along the x, y, and z-axis respectively.

The gradient of any function f(x, y, z) represents the direction of the greatest rate of increase of any scalar field function.

Calculation:

Given, f(x, y, z) = xyz

\(∇ f = \frac{\partial (xyz)}{\partial x} ̂ i + \frac{\partial (xyz)}{\partial y} ̂ j + \frac{\partial (xyz)}{\partial z} ̂ k\)

∇ f = (yz)î + (xz)ĵ + (xy)k̂

Gradient of a function f(x, y, z) at P = (0, 2, 1) is:

∇ f = 2î 

The rate of increase, of a field in the direction of 𝒗 =\(∇ f.{\hat v}\)

 =\(\frac{ 2î .( 2î +{\hat j} +2{\hat k})}{\sqrt{2^2+1^2+2^2}}\)

=\(\frac{4}{3}\)