Let $\overrightarrow{\mathrm{E}}(\mathrm{x},\mathrm{y},\mathrm{z})=2{\mathrm{x}}^2\hat{\mathrm{i}}+5\mathrm{y}\hat{\mathrm{j}}+3\mathrm{z}\hat{\mathrm{k}}$ the value of ∭v $(\overrightarrow{\mathrm{\nabla }}.\overrightarrow{E})dV$, where v is the volume enclosed by the unit cube defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and 0 ≤ z ≤ 1 is

The correct answer is 10.

To find the value of the triple integral:

${\iiint }_V(\overrightarrow{\nabla }·\overrightarrow{E})dV$

we need to first compute the divergence of the given vector field, $\overrightarrow{E}(x,y,z)$.

The vector field is given by:

$\overrightarrow{E}(x,y,z)=2x^2\hat{i}+5y\hat{j}+3z\hat{k}$

Step 1: Calculate the divergence of $\overrightarrow{E}$
The divergence of a vector field is the sum of its partial derivatives with respect to each coordinate variable:

$\overrightarrow{\nabla }·\overrightarrow{E}=\frac{\partial }{\partial x}\left(2x^2\right)+\frac{\partial }{\partial y}\left(5y\right)+\frac{\partial }{\partial z}\left(3z\right)$

Evaluating the partial derivatives:
• $\frac{\partial }{\partial x}\left(2x^2\right)=4x$
• $\frac{\partial }{\partial y}\left(5y\right)=5$
• $\frac{\partial }{\partial z}\left(3z\right)=3$

Adding these together gives the divergence:

$\overrightarrow{\nabla }·\overrightarrow{E}=4x+5+3=4x+8$

Step 2: Set up and evaluate the triple integral
The volume $V$ is a unit cube bounded by:

$0\le x\le 1,0\le y\le 1,0\le z\le 1$

Therefore, the volume integral is defined as:

${\int }_0^1{\int }_0^1{\int }_0^1(4x+8)dxdydz$

Since the integrand depends only on $x$, we can separate the integrals:

$\left({\int }_0^1\right(4x+8\left)dx\right)·\left({\int }_0^{1}dy\right)·\left({\int }_0^{1}dz\right)$

The integrals with respect to $y$ and $z$ are both equal to 1:

${\int }_0^{1}dy={\left[y\right]}_0^1=1$

${\int }_0^{1}dz={\left[z\right]}_0^1=1$

Now, evaluate the integral with respect to $x$:

${\int }_0^1(4x+8)dx={[2x^2+8x]}_0^1$

$=(2{\left(1\right)}^2+8(1\left)\right)-(2{\left(0\right)}^2+8(0\left)\right)$

$=2+8=10$

Thus, the value of the triple integral is 10.