The image shows a mathematical expression related to surface integrals. It involves a vector field $\mathbf{F} = ax \hat{i} + by \hat{j} + cz \hat{k}$, where $a$, $b$, and $c$ are constants. The task is to evaluate the surface integral of this vector field $\mathbf{F}$ over the surface of a unit sphere.

This type of question involves applying the Divergence Theorem to simplify the calculation.

Approach:

1. The Divergence Theorem states that the surface integral of a vector field over a closed surface is equal to the volume integral of the divergence of the vector field inside the region bounded by the surface.

   Mathematically: $\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) dV$ Where $S$ is the closed surface (in this case, the surface of a sphere), and $V$ is the volume inside the sphere.

2. Compute the Divergence of $\mathbf{F}$: $\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(ax) + \frac{\partial}{\partial y}(by) + \frac{\partial}{\partial z}(cz)$ Since $a$, $b$, and $c$ are constants: $\nabla \cdot \mathbf{F} = a + b + c$

3. Volume Integral: The volume integral becomes: $\iiint_V (a + b + c) \, dV$ Since $a + b + c$ is a constant, the integral is: $(a + b + c) \cdot \text{Volume of the sphere}$ The volume of a unit sphere is $\frac{4}{3} \pi$.

   Thus, the surface integral is: $\iint_S \mathbf{F} \cdot d\mathbf{S} = (a + b + c) \cdot \frac{4}{3} \pi$

4. Result: Therefore, the final result of the surface integral is: $\iint_S \mathbf{F} \cdot d\mathbf{S} = 4\pi (a + b + c)$

Summary:

The integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$, where $\mathbf{F} = ax \hat{i} + by \hat{j} + cz \hat{k}$, evaluates to $4\pi(a + b + c)$, with $S$ being the surface of the unit sphere.

Would you like further details or clarifications on any part of the solution? Here are some follow-up questions:

1. How is the Divergence Theorem derived?
2. How do we compute the surface integral for non-constant vector fields?
3. Can the Divergence Theorem be used for non-spherical surfaces?
4. How is the divergence of a vector field computed in general?
5. What is the relationship between surface integrals and flux?

Tip: When dealing with vector fields and surface integrals, always consider using the Divergence Theorem to simplify the calculations if the surface is closed and the region is easily defined.