Question

Let \(S\) be a closed surface and \(V\) the enclosed volume. Prove Green's first and second identities, respectively. a. \(\int_{S} \phi \nabla \psi \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d V\). b. \(\int_{S}[\phi \nabla \psi-\psi \nabla \phi] \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V\).

Short answer

Green's first identity is proven by the equivalence of the surface integral of the function \( \phi \nabla \psi \) and the volume integral of the function \( \phi \nabla^{2} \psi + \nabla \phi \cdot \nabla \psi \). Similarly, Green's second identity is shown by the equivalence of the surface integral of the function \( \phi \nabla \psi - \psi \nabla \phi \) and the volume integral of the function \( \phi \nabla^{2} \psi - \psi \nabla^{2} \phi \). Both identities exemplify the divergence theorem.

Step-by-step solution

Prove Green's First Identity

The left hand side of the equation can be written as a volume integral using the divergence theorem. The divergence theorem states that \(\int_{S} \mathbf{F} \cdot d\mathbf{S} = \int_{V} (\nabla \cdot \mathbf{F}) dV\). By selecting the vector field \(\mathbf{F} = \phi \nabla \psi\), the theorem yields \(\int_{V} \nabla \cdot (\phi \nabla \psi) dV\). Expanding the divergence yields two terms: \(\int_{V}\left(\phi \nabla^{2} \psi + \nabla \phi \cdot \nabla \psi\right) dV\), which matches the right-hand side of the equation.

Prove Green's Second Identity

Again, use the divergence theorem but now select the vector field \(\mathbf{F} = \phi \nabla \psi - \psi \nabla \phi\). The theorem yields \(\int_{V} \nabla \cdot (\phi \nabla \psi - \psi \nabla \phi) dV\). Expanding the divergence gives \(\int_{V}\left(\phi \nabla^{2} \psi + \nabla \phi \cdot \nabla \psi - \psi \nabla^{2} \phi - \nabla \psi \cdot \nabla \phi\right) dV\). The second and the fourth terms cancel out, leaving \(\int_{V}\left(\phi \nabla^{2} \psi - \psi \nabla^{2} \phi\right) dV\), which matches the right-hand side of the equation.

Key concepts

Mathematical Physics

Mathematical physics is a branch of physics that applies rigorous mathematical methods to problems in physics and develops new mathematical techniques suitable for such applications. A quintessential example of this symbiotic relationship is found in Green's identities, which are used in various areas such as electrodynamics, fluid dynamics, and quantum mechanics.

These identities are named after the British mathematician George Green and are derived from the application of the divergence theorem. They play a crucial role in simplifying complex physics problems by transforming surface integrals into volume integrals, thus reducing dimensions and often making the problem more tractable. Understanding Green's identities can provide deep insights into the behavior of physical fields and potential functions in space.

Divergence Theorem

The divergence theorem is a central result in vector calculus linking surface integrals and volume integrals. It states that the integral of a vector field's divergence over a volume is equal to the integral of the vector field itself over the surface enclosing the volume. Symbolically, \[ \int_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \int_{V} (abla \cdot \mathbf{F}) \, dV \], where \(\mathbf{F}\) represents a vector field, \(S\) is a closed surface, \(\mathbf{n}\) is the outward normal vector to the surface, and \(V\) is the volume enclosed by \(S\).

Applying the divergence theorem as demonstrated in Green's identities allows physicists and mathematicians to convert complex surface integrals over a closed surface into more manageable volume integrals over the region the surface encloses.

Vector Calculus

Vector calculus is a field of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space. It is essential for describing the physical world as many quantities of interest can be represented as vectors, such as force, velocity, and electric field.

Key operations in vector calculus include grad, div, and curl, which respectively represent the gradient, divergence, and curl of vector fields. These operations are used in Green's identities to transform products and gradients of scalar fields into forms that are easier to work with in physical and mathematical contexts.

Surface Integrals

Surface integrals are a form of integral where the function to be integrated is defined over a surface in space. They are used to calculate quantities like flux of a vector field through a surface, which is a common requirement in electromagnetism and fluid dynamics among other fields. The surface integral of a scalar field \(\phi\) over a surface \(S\) with an outward normal vector \(\mathbf{n}\) is represented by \[ \int_{S} \phi \mathbf{n} \, dS \].

In the context of Green's identities, surface integrals allow for the representations of physical phenomena at the boundaries of a spatial domain. Converting these to volume integrals via the divergence theorem often simplifies calculations and aids in solving partial differential equations.

Volume Integrals

Volume integrals extend the concept of integrating over a 2-dimensional surface to integrating over a 3-dimensional volume. They are used to calculate total quantities inside a volume, such as mass or electric charge density. Mathematically, a volume integral of a scalar function \(\phi\) over a volume \(V\) is denoted as \[ \int_{V} \phi \, dV \].

For Green's identities, volume integrals provide a way to express global properties of scalar fields within a bounded region. Being able to convert surface integrals to volume integrals using the divergence theorem simplifies the process of finding solutions to numerous physics and engineering problems, especially when dealing with complex geometries or boundary conditions.