Question

Explain the meaning of the surface integral in the Divergence Theorem.

Short answer

Answer: The surface integral in the Divergence Theorem represents the net flux of the vector field across a closed surface. It measures how much of the vector field is flowing into or out of the enclosed region.

Step-by-step solution

Understanding Vector Fields

A vector field is a function that assigns a vector to each point in space. It represents a quantity that has both magnitude and direction at each point. Examples of vector fields include fluid flow, electric fields, and gravitational fields.

Understanding Surface Integrals

A surface integral is a generalization of the line integral that is applied to the vector field over a surface, rather than just a curve. Instead of adding up the values of the vector field along a curve, a surface integral adds up the values of a scalar function across a surface. This is useful for calculating the total effect of a vector field on a surface.

Divergence Theorem

The Divergence Theorem, also known as Gauss's theorem, states that the surface integral of a vector field around a closed surface is equal to the volume integral of its divergence over the region enclosed by the surface. In mathematical terms, it is expressed as \(\iint_{S} \textbf{F} \cdot d\textbf{A} = \iiint_{V} \nabla \cdot \textbf{F} \,dV\), where \(\textbf{F}\) is a vector field, \(S\) is a closed surface, and \(V\) is the volume enclosed by the surface.

Meaning of Surface Integral in Divergence Theorem

In the Divergence Theorem, the surface integral, \(\iint_{S} \textbf{F} \cdot d\textbf{A}\), represents the net flux of the vector field across the closed surface. It measures how much of the vector field is flowing into or out of the enclosed region. The volume integral, \(\iiint_{V} \nabla \cdot \textbf{F} \,dV\), represents the total divergence (or sources minus sinks) of the vector field within the enclosed region. This is the total "expansion" or "contraction" of the field in the volume. By equating these two integrals, the Divergence Theorem states that the net flow of the vector field through a closed surface is equal to the total divergence of the field within the region enclosed by the surface. This provides an important connection between the global behavior of the vector field in the enclosed region and its local behavior at the boundaries. In summary, the surface integral in the Divergence Theorem represents the net flow of the vector field across a closed surface and the volume integral signifies the total divergence within the region enclosed by the surface. This theorem provides a deeper insight into the behavior of vector fields and has a wide range of applications in various fields like fluid dynamics, electromagnetism, and more.