Question

Green's Second Identity Prove Green's Second Identity for scalar-valued functions \(u\) and \(v\) defined on a region \(D:\) $$\iiint_{D}\left(u \nabla^{2} v-v \nabla^{2} u\right) d V=\iint_{S}(u \nabla v-v \nabla u) \cdot \mathbf{n} d S$$ (Hint: Reverse the roles of \(u\) and \(v\) in Green's First Identity.)

Short answer

Question: Prove Green's Second Identity using Green's First Identity and the hint provided. Answer: Green's Second Identity states that for scalar-valued functions \(u\) and \(v\), the following relationship holds: $$\iiint_D \left(u\nabla^2 v - v\nabla^2 u\right) dV = \iint_S \left[u(\nabla v \cdot \mathbf{n}) - v(\nabla u \cdot \mathbf{n})\right] dS$$ Proof: 1. Start with Green's First Identity: $$\iiint_D (\nabla u \cdot \nabla v)dV = \iint_S (\nabla u \cdot \mathbf{n})v dS - \iiint_D uv \nabla^2 v dV$$ 2. Reverse the roles of \(u\) and \(v\): $$\iiint_D (\nabla v \cdot \nabla u)dV = \iint_S (\nabla v \cdot \mathbf{n})u dS - \iiint_D vu \nabla^2 u dV$$ 3. Subtract the two identities: $$\iiint_D (\nabla u \cdot \nabla v - \nabla v \cdot \nabla u)dV = \iint_S \left[(\nabla u \cdot \mathbf{n})v - (\nabla v \cdot \mathbf{n})u\right] dS + \iiint_D \left(u\nabla^2 v - v\nabla^2 u\right) dV$$ 4. Simplify the result due to the equal dot products: $$0 = \iint_S \left[(\nabla u \cdot \mathbf{n})v - (\nabla v \cdot \mathbf{n})u\right] dS + \iiint_D \left(u\nabla^2 v - v\nabla^2 u\right) dV$$ 5. Rearrange the equation to obtain Green's Second Identity: $$\iiint_D \left(u\nabla^2 v - v\nabla^2 u\right) dV = \iint_S \left[u(\nabla v \cdot \mathbf{n}) - v(\nabla u \cdot \mathbf{n})\right] dS$$

Step-by-step solution

Green's First Identity

Recall Green's First Identity: $$\iiint_D (\nabla u \cdot \nabla v)dV = \iint_S (\nabla u \cdot \mathbf{n})v dS - \iiint_D uv \nabla^2 v dV$$ Where \(\mathbf{n}\) is the outward normal vector to the surface S.

Reverse the roles of \(u\) and \(v\)

Now let's swap \(u\) and \(v\) in Green's First Identity to get a similar expression with reversed roles: $$\iiint_D (\nabla v \cdot \nabla u)dV = \iint_S (\nabla v \cdot \mathbf{n})u dS - \iiint_D vu \nabla^2 u dV$$

Subtract the two identities

Subtract the second identity from the first one: $$\iiint_D (\nabla u \cdot \nabla v - \nabla v \cdot \nabla u)dV = \iint_S \left[(\nabla u \cdot \mathbf{n})v - (\nabla v \cdot \mathbf{n})u\right] dS + \iiint_D \left(u\nabla^2 v - v\nabla^2 u\right) dV$$

Simplify the result

Notice that the dot product \(\nabla u \cdot \nabla v = \nabla v \cdot \nabla u\), which means their difference is zero. This simplifies the above equation to: $$0 = \iint_S \left[(\nabla u \cdot \mathbf{n})v - (\nabla v \cdot \mathbf{n})u\right] dS + \iiint_D \left(u\nabla^2 v - v\nabla^2 u\right) dV$$

Rearrange to obtain Green's Second Identity

Rearrange the above equation to isolate the triple integral on the left. This gives us Green's Second Identity: $$\iiint_D \left(u\nabla^2 v - v\nabla^2 u\right) dV = \iint_S \left[u(\nabla v \cdot \mathbf{n}) - v(\nabla u \cdot \mathbf{n})\right] dS$$ The proof of Green's Second Identity is now complete.

Key concepts

scalar-valued functions

Scalar-valued functions are an important concept in mathematics, particularly in fields like calculus and differential equations. A scalar-valued function assigns a single real number to each point in space. This contrasts with vector-valued functions, which assign a vector (with multiple components) to each point.Understanding scalar-valued functions is essential for applications in physics and engineering. They often represent quantities like temperature, pressure, or potential energy, which vary across a region but are described by one value at each point.
In the context of Green's Second Identity, scalar-valued functions are used to represent potential fields. The identity involves two such functions, often denoted as \(u\) and \(v\), and relates their behavior over a region \(D\) and its boundary \(S\). This relationship is crucial for understanding how these functions interact within a given space, offering insights into collaboration across disciplines such as electromagnetism, fluid dynamics, and more.

• Scalar-valued functions: Assign a single value to each point.
• Examples: Temperature, pressure, potential energy.
• Application: Ideal for analyzing systems in physics and engineering.

Laplace operator

The Laplace operator, often denoted by \(abla^2\) or \(\Delta\), is a second-order differential operator widely used in mathematics and physics. It is a key component in problems involving scalar fields and helps measure the rate at which a field spreads out from a point.The Laplacian of a scalar-valued function \(u\), represented as \(abla^2 u\), can be thought of as the divergence of the gradient of \(u\). In simple terms, it provides information about how the slope of a function behaves, especially whether it is concave or convex. It is instrumental in solving partial differential equations, notably the Laplace equation and the Poisson equation.
In Green's Second Identity, the Laplace operator is applied to the scalar-valued functions \(u\) and \(v\). It helps in understanding how "point sources" of potential influence the field in a region \(D\). The use of the Laplacian thus connects the interior and boundary of the region, bridging local properties with global behavior.

• Laplace operator: Denoted \(abla^2\), measures field spread.
• Useful in: Solving equations like Laplace's and Poisson's.
• Role: Links local and global properties in calculus.

vector calculus

Vector calculus is a branch of mathematics that deals with vector fields and the operation of calculus on them. It involves differentiation and integration of vector functions and finds extensive use in physics and engineering.Key operations in vector calculus include the gradient, divergence, and curl. The gradient (\(abla u\)) gives the direction and magnitude of the steepest ascent of a scalar field, the divergence measures the rate of change of volume for a vector field, and the curl describes the rotation of a field.
In Green's Second Identity, vector calculus plays a pivotal role. The identity itself is a consequence of applying vector calculus operations to scalar-valued functions \(u\) and \(v\). It demonstrates how the changes in these functions within a volume \(D\) relate to their normal components on the surface \(S\).

• Vector calculus: Involves operations like gradient and divergence.
• Applications: Central in physics, engineering, and fluid dynamics.
• Importance: Provides tools to connect changes within volumes to surface effects.