Question

Green's First Identity Prove Green's First Identity for twice differentiable scalar-valued functions \(u\) and \(v\) defined on a region \(D:\) $$\iiint_{D}\left(u \nabla^{2} v+\nabla u \cdot \nabla v\right) d V=\iint_{S} u \nabla v \cdot \mathbf{n} d S$$where \(\nabla^{2} v=\nabla \cdot \nabla v .\) You may apply Gauss' Formula in Exercise 48 to \(\mathbf{F}=\nabla v\) or apply the Divergence Theorem to \(\mathbf{F}=u \nabla v\)

Short answer

Question: Use Green's First Identity to prove that the flux of \(u \nabla v\) through the boundary surface \(S\) of a region \(D\) is equal to the volume integral of \(u \nabla^2 v + \nabla u \cdot \nabla v\) over the region \(D\).

Step-by-step solution

Calculate the divergence of u∇v

We need to find the divergence of \(\mathbf{F}= u \nabla v\). By definition, \(\nabla \cdot \mathbf{F} = \nabla \cdot (u \nabla v)\). We can use the product rule to expand it: $$\nabla \cdot (u \nabla v) = (\nabla u) \cdot (\nabla v) + u(\nabla \cdot \nabla v)$$

Use the Divergence theorem to write the volume integral

Applying the Divergence Theorem on \(\mathbf{F} = u \nabla v\), we get: $$\iint_{S} u \nabla v \cdot \mathbf{n} dS = \iiint_{D} \nabla \cdot (u \nabla v) dV$$ Replace \(\nabla \cdot (u \nabla v)\) with what we found in Step 1: $$ \iint_{S} u \nabla v \cdot \mathbf{n} dS = \iiint_{D} \left((\nabla u) \cdot (\nabla v) + u(\nabla \cdot \nabla v)\right) dV$$

Rearrange the equation

We now have the following equation: $$\iint_{S} u \nabla v \cdot \mathbf{n} dS = \iiint_{D} \left((\nabla u) \cdot (\nabla v) + u(\nabla \cdot \nabla v)\right) dV$$ Notice that \(\nabla \cdot \nabla v = \nabla^{2}v\), so we can rewrite it as: $$\iint_{S} u \nabla v \cdot \mathbf{n} dS = \iiint_{D} \left(u \nabla^{2} v + \nabla u \cdot \nabla v\right) dV$$ This is Green's First Identity.

Key concepts

Divergence Theorem

The Divergence Theorem is a powerful tool in vector calculus that relates the flow of a vector field through a surface to the behavior of the field inside the volume enclosed by that surface. In simple terms, it allows us to translate a challenging surface integral into a more manageable volume integral.

Mathematically, for a domain \(D\) in three-dimensional space with boundary surface \(S\), and a continuously differentiable vector field \(\textbf{F}\), the Divergence Theorem states:
$$\iiint_D (abla \cdot \textbf{F}) dV = \iint_S \textbf{F} \cdot \textbf{n} dS$$
Here, \(abla \cdot \textbf{F}\) represents the divergence of \(\textbf{F}\), a measure of how much the field spreads out from a point, and \(\textbf{n}\) is the unit outward normal to the surface \(S\). This theorem is crucial for solving problems in physics and engineering, such as calculating electric flux or fluid flow. In the context of Green's First Identity, the Divergence Theorem simplifies the assessment of the right-hand side of the identity by evaluating a volume integral instead of a more complex surface integral.

Vector Calculus

Vector calculus is an extension of classical calculus to vector fields. It provides a framework for dealing with physical quantities that have both magnitude and direction, such as velocity, force, and electric and magnetic fields. The core operations in vector calculus include the gradient, divergence, and curl.

The gradient of a scalar field measures the rate and direction of change in the maximum rate, the divergence mentioned in the Divergence Theorem indicates how much a vector field expands or contracts at a given point, and the curl shows the tendency of the field to rotate around a point. These operations are invaluable in formulating physical laws and equations, such as those found in electromagnetism and fluid dynamics. An understanding of vector calculus is crucial for correctly applying Green's First Identity, as it involves both the gradient and divergence of vector fields.

Triple Integral

A triple integral extends the concept of an integral to functions of three variables. It is a way to compute the volume under a surface in three-dimensional space, or more generally, to accumulate a quantity over a three-dimensional region. More formally, for a function \(f(x, y, z)\) over a region \(D\), the triple integral is expressed as:
$$\iiint_D f(x, y, z) dV$$
where \(dV\) is a differential volume element, often written in Cartesian, cylindrical, or spherical coordinates depending on the symmetry of the problem.

Triple integrals play a critical role in the application of Green's First Identity. The left-hand side of the identity involves a triple integral over a scalar-valued function, which quantifies the accumulation of the function within the region of interest. By performing the triple integral, we sum the contributions of the scalar fields throughout the volume, leading us to better understand physical scenarios being modelled.

Surface Integral

Surface integrals extend the idea of integrating functions of two variables to integrating over a curved surface in three-dimensional space. They are used to calculate quantities spread out over surfaces, like the flux of a vector field through a surface or the surface area itself.

For a vector field \(\textbf{F}\) over a surface \(S\), with \(\textbf{n}\) as the unit normal vector at each point on \(S\), the surface integral is:
$$\iint_S \textbf{F} \cdot \textbf{n} dS$$
In Green's First Identity, the right-hand side involves the surface integral of the vector field \(u abla v\). Here, \(abla v\) is the gradient of the function \(v\), and \(u\) is a weighting function. The dot product with the normal vector \(\textbf{n}\) determines the component of the vector field that is normal to the surface. This integral sums up the 'flow' of \(u abla v\) across the boundary surface \(S\), providing a boundary-condition perspective that complements the volumetric view given by the triple integral on the left-hand side of the identity.