Problem 29 - Thomas Calculus

Question

Green's first formula Suppose that \(f\) and \(g\) are scalar functions with continuous first- and second-order partial derivatives throughout a region \(D\) that is bounded by a closed piecewise smooth surface \(S .\) Show that $$\iint_{S} f \nabla g \cdot \mathbf{n} d \sigma=\iiint_{D}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V$$ Equation \((10)\) is Green's first formula. (Hint: Apply the Divergence Theorem to the field \(\mathbf{F}=f \nabla g . )\)

Step-by-Step Solution

Answer

Green's first formula is verified using the divergence theorem.

Step 1: Understand the Divergence Theorem

The Divergence Theorem states that if \( \mathbf{F} \) is a vector field with continuous partial derivatives on a region \( D \) enclosed by a surface \( S \), then:\[ \iint_{S} \mathbf{F} \cdot \mathbf{n} \, d\sigma = \iiint_{D} (abla \cdot \mathbf{F}) \, dV \]where \( \mathbf{n} \) is the outward unit normal on \( S \). This theorem relates a surface integral over \( S \) to a volume integral over \( D \).

Step 2: Define the Vector Field \( \mathbf{F} \)

Let \( \mathbf{F} = f abla g \) as suggested by the hint. Here, \( f \) is a scalar function and \( abla g \) is the gradient of \( g \), creating a vector field when multiplied.

Step 3: Apply the Divergence Theorem

Use the divergence theorem for the vector field \( \mathbf{F} = f abla g \):\[ \iint_{S} f abla g \cdot \mathbf{n} \, d\sigma = \iiint_{D} abla \cdot (f abla g) \, dV \].

Step 4: Expand the Divergence \( \nabla \cdot (f \nabla g) \)

Apply the product rule for divergence on \( abla \cdot (f abla g) \): \[ abla \cdot (f abla g) = abla f \cdot abla g + f abla^2 g \], where \( abla^2 g \) is the Laplacian of \( g \).

Step 5: Substitute and Simplify

Substitute the expanded divergence into the volume integral:\[ \iiint_{D} abla \cdot (f abla g) \, dV = \iiint_{D} (f abla^2 g + abla f \cdot abla g) \, dV \].This equation matches the statement of Green's first formula.

Key Concepts

Divergence Theorem

The Divergence Theorem is a powerful tool in vector calculus that relates a surface integral over a closed surface to a volume integral over the region it encloses. It is sometimes referred to as Gauss's Theorem. In mathematical terms, if \( \mathbf{F} \) is a vector field with continuous partial derivatives over a region \( D \) enclosed by a surface \( S \), the theorem is stated as follows:
\[ \iint_{S} \mathbf{F} \cdot \mathbf{n} \, d\sigma = \iiint_{D} (abla \cdot \mathbf{F}) \, dV \]

• \(S\) is the closed surface bounding the region \(D\).
• \(\mathbf{n}\) is the outward unit normal vector on \(S\).
• The operation \( abla \cdot \mathbf{F} \) is the divergence of the vector field \( \mathbf{F} \).

The Divergence Theorem is widely used in physics and engineering because it simplifies the calculation of flux by converting a surface integral to a volume integral. In the context of Green's first formula, it helps transition from a surface integral to a form that shows the relationship between \( f \) and \( g \).

Surface Integrals

Surface integrals allow us to compute quantities spread across a surface in three-dimensional space. Specifically, they measure the flux of a vector field \( \mathbf{F} \) across a surface \( S \). The surface integral is written as:
\[ \iint_{S} \mathbf{F} \cdot \mathbf{n} \, d\sigma \]
To understand this, consider the following components:

• \(\mathbf{F} \cdot \mathbf{n}\) computes the component of the vector field perpendicular to the surface at each point.
• \(d\sigma\) is a small area element on the surface.

In Green's first formula, the surface integral assesses how the scalar function \( f \) multiplied by the gradient of \( g \) behaves over the boundary surface \( S \). This is crucial in connecting the behavior of \( f \) and \( g \) on the boundary with their combined behavior throughout the volume \( D \).

Volume Integrals

Volume integrals are used to calculate quantities within a two or three-dimensional region. These integrals sum up changes in a scalar function or vector fields across the volume \( D \) enclosed by a surface \( S \). For a scalar field \( \, \rho(x, y, z) \, \), the volume integral is represented as:
\[ \iiint_{D} \rho \, dV \]

• \(dV\) is a differential volume element within the region.
• The integral sums up contributions from every infinitesimally small part of the volume.

In Green’s first formula, the volume integral helps transition the relationship within the boundary defined by \( S \) into a volume encompassing form, showing how both the Laplacian of \( g \) and the gradient interaction between \( f \) and \( g \) manifest inside the entire region \( D \).

Laplacian

The Laplacian is a differential operator that plays a pivotal role in mathematics and physics, especially in the context of Green's formulas. It is denoted as \( abla^2 \) and for a scalar function \( g \), is defined as:
\[ abla^2 g = abla \cdot (abla g) \]
This operator effectively measures the "spread" or "dissipation" of a function. It sums up how fast functions change in space.

• A positive Laplacian at a point indicates a local minimum.
• A negative Laplacian indicates a local maximum.

In Green's first formula, the Laplacian \( abla^2 g \) is crucial because it contributes to the decomposition of the divergence \( abla \cdot (f abla g) \). It helps express intricate properties of the function \( g \) inside the region \( D \) and connects surface properties to volume properties.

Vector Calculus

Vector Calculus is the branch of mathematics concerned with differentiation and integration of vector fields. It extends calculus to functions with multiple variables and is fundamental in engineering and physics.
Key concepts include:

• Gradients: Used to find the rate of change of a scalar field, represented as \( abla g \) for a scalar function \( g \).
• Divergence: Measures the extent to which a vector field diverges from a point, symbolized as \( abla \cdot \mathbf{F} \).
• Curl: Represents the circulation of a vector field around a point, noted as \( abla \times \mathbf{F} \).

In the context of Green's First Formula, vector calculus tools such as the gradient and divergence are essential. They help in handling complex fields and connecting the dots between surface and volume integrals effectively. Understanding these vector calculus elements simplifies comprehension of how \( f abla^2 g + abla f \cdot abla g \) sums up within the region \( D \).