Show that the right-hand side of \((84.16)\) can be written in the form $$ \int_{V} \nabla \cdot \mathbf{T} d v $$ where \(\mathrm{T}\) is the Maxwell stress tensor $$ \mathbf{T}={ }_{0} \mathbf{E} \mathbf{E} \mathbf{E}+\frac{\mathbf{1}}{\mu_{0}} \mathbf{B B}-\mathbf{1}\left(\frac{1}{2} \epsilon_{0} E^{2}+\frac{B^{2}}{2 \mu_{0}}\right) $$ Then apply Gauss' theorem to convert the integral to the surface-integral form $$ \int_{S} \mathbf{T} \cdot d \mathbf{S} . $$

Understanding Gauss' theorem, also known as Gauss' divergence theorem, is foundational for solving various problems in vector calculus and electromagnetism.

Gauss' theorem connects the flow of a vector field through a closed surface to the behavior of the vector field inside the volume. In more intuitive terms, it tells us that the 'net outflow' of a vector field through a surface is equal to the 'source strength' within the surface. Mathematically, it states that the volume integral of the divergence of a vector field over a volume, V, is equivalent to the surface integral of that vector field through the boundary surface, S. This can be written as:\[\int_{V} abla \cdot \mathbf{F} \, dV = \int_{S} \mathbf{F} \cdot d\mathbf{S}\] where \(\mathbf{F}\) is a vector field. In our exercise, we see this principle applied to the Maxwell stress tensor, allowing us to express complex electromagnetic interactions with simpler surface integrals.

The electric field, denoted by \(\mathbf{E}\), describes the electric force experienced by a stationary point charge at any location in space. It is a vector field, which means it has both magnitude and direction.

Electric fields are created by electric charges, or by changing magnetic fields, and they exert forces on other electric charges within the field. The strength of the electric field at a point in space is directly proportional to the force experienced by a unit positive charge placed at that point. In equations, the Maxwell stress tensor incorporates the electric field to calculate the force per unit area on a hypothetical surface within the field, making it a powerful tool in analyzing electromechanical stress.

The magnetic field, represented by \(\mathbf{B}\), is the magnetic force per unit charge and velocity that a moving charge would experience in a given location. Similar to the electric field, the magnetic field is also a vector field.

Magnetic fields arise from moving electric charges (currents) and are fundamental to the operation of electrical devices and the existence of magnetic materials. They play a crucial role in Maxwell's equations, which describe the fundamentals of electromagnetism. In the context of this exercise, the magnetic field interacts with the electric field within the Maxwell stress tensor to determine the electromagnetic force distribution.

A volume integral represents the total quantity of some scalar valued function throughout a three-dimensional region. It is a type of multiple integral where integration is extended over a three-dimensional volume.

In electromagnetism, volume integrals are used to calculate properties that are distributed throughout a space, such as charge, mass, or energy density. When you perform a volume integral of a vector field's divergence, like when applying the Maxwell stress tensor, you're essentially summing up the 'source' or 'sink' contributions of that vector field throughout the volume of interest.The Maxwell stress tensor involves the volume integral, which ultimately transforms to a surface integral. This step simplifies the calculations and is a practical application of Gauss' theorem.

A surface integral can be thought of as the two-dimensional analog of a line integral, extended over a surface. It deals with integrating a scalar function or the normal component of a vector field over a surface.

Surface integrals are a critical tool in physics and engineering, especially when calculating flux across a surface. In the context of the Maxwell stress tensor and electromagnetism, performing a surface integral of the stress tensor over a closed surface gives you a measure of the total force acting on that surface due to the electric and magnetic fields. This force calculation is essential for understanding and designing systems that contain or generate electromagnetic fields.