Question

the divergence of a
Vector function f=30 x^2 i+5xy^2 j + xyz^3 k is​

Answer 1

Question

Compute the divergence of the vector function:

$\sf \bold{f} = 30x^2 \hat{\imath} + 5xy^2 \hat{\jmath} + xyz^3 \hat{k}$

Solution

Divergence, at its core idea, measures the outward flow of something. Consider the standard example of a vector function showing the velocity of the fluid.

The divergence measures the outward flow of the fluid.

Anyway, on to the math. Divergence of $\bold{f}$ is denoted as $\nabla . \bold{f}$

Divergence is a scalar quantity. The differentiation is straightforward, with only one rule being used:

$\dfrac{d x^n}{dx} = nx^{n-1}$

The divergence is then computed as follows:

$\displaystyle \nabla\ .\ \bold{f} = \frac{\partial f_x}{\partial x} + \frac{\partial f_y}{\partial y} + \frac{\partial f_z}{\partial z} \\\\\\ \implies \nabla\ .\ \bold{f} = \frac{\partial\ (30x^2)}{\partial x} + \frac{\partial\ (5xy^2)}{\partial y} + \frac{\partial\ (xyz^3)}{\partial z} \\\\\\ \implies \boxed{\nabla\ .\ \bold{f} = 60x + 10xy + 3xyz^2}$