To calculate the flux of the vector field $\mathbf{H} = x^2\mathbf{i} + y^2\mathbf{j} + z^2\mathbf{k}$ through the surface of the cube $S$, we use the Divergence Theorem. The Divergence Theorem states:

$\int_S \mathbf{H} \cdot \mathbf{dA} = \int_V (\nabla \cdot \mathbf{H}) \, dV$

where:

• $S$ is the closed surface of the cube,
• $V$ is the volume enclosed by $S$,
• $\nabla \cdot \mathbf{H}$ is the divergence of $\mathbf{H}$.

Step 1: Compute the divergence of $\mathbf{H}$

$\nabla \cdot \mathbf{H} = \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial y}(y^2) + \frac{\partial}{\partial z}(z^2)$

$\nabla \cdot \mathbf{H} = 2x + 2y + 2z$

Step 2: Set up the volume integral

The volume $V$ of the cube is defined by $0 \leq x, y, z \leq 1$. Substituting the divergence, the integral becomes:

$\int_V (\nabla \cdot \mathbf{H}) \, dV = \int_0^1 \int_0^1 \int_0^1 (2x + 2y + 2z) \, dx \, dy \, dz$

Step 3: Evaluate the triple integral

We can split the integral into three parts:

$\int_0^1 \int_0^1 \int_0^1 (2x + 2y + 2z) \, dx \, dy \, dz = 2 \int_0^1 \int_0^1 \int_0^1 x \, dx \, dy \, dz + 2 \int_0^1 \int_0^1 \int_0^1 y \, dx \, dy \, dz + 2 \int_0^1 \int_0^1 \int_0^1 z \, dx \, dy \, dz$

First term:

$2 \int_0^1 \int_0^1 \int_0^1 x \, dx \, dy \, dz = 2 \int_0^1 \int_0^1 \left[\frac{x^2}{2}\right]_0^1 dy \, dz = 2 \int_0^1 \int_0^1 \frac{1}{2} \, dy \, dz = 2 \int_0^1 \frac{1}{2} \, dz = 1$

Second term:

$2 \int_0^1 \int_0^1 \int_0^1 y \, dx \, dy \, dz = 2 \int_0^1 \int_0^1 y \int_0^1 dx \, dy \, dz = 2 \int_0^1 \int_0^1 y \, dy \, dz = 2 \int_0^1 \left[\frac{y^2}{2}\right]_0^1 dz = 2 \int_0^1 \frac{1}{2} \, dz = 1$

Third term:

$2 \int_0^1 \int_0^1 \int_0^1 z \, dx \, dy \, dz = 2 \int_0^1 \int_0^1 z \int_0^1 dx \, dy \, dz = 2 \int_0^1 \int_0^1 z \, dy \, dz = 2 \int_0^1 \left[\frac{z^2}{2}\right]_0^1 dz = 2 \int_0^1 \frac{1}{2} \, dz = 1$

Step 4: Add the results

Adding the contributions of all three terms:

$\int_V (\nabla \cdot \mathbf{H}) \, dV = 1 + 1 + 1 = 3$

Final Answer:

The flux of $\mathbf{H}$ through the surface of the cube is:

$\boxed{3}$

Would you like a detailed breakdown of any of these steps or have additional questions? Here are related problems for practice:

1. What is the flux of a constant vector field through a cube?
2. How does the Divergence Theorem apply to spherical surfaces?
3. Calculate the flux of $\mathbf{H} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ through a cube of side length 2.
4. How does the flux through a cube change if the vector field is $\mathbf{H} = x^3\mathbf{i} + y^3\mathbf{j} + z^3\mathbf{k}$?
5. What happens if the cube is rotated but retains the same side lengths?

Tip: Always check the boundaries of the volume carefully when applying the Divergence Theorem!