Problem 33 - 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 $$\iint_{S} f \nabla g \cdot \mathbf{n} d \sigma=\iiint_{D}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V \quad \quad\quad(10)$$ 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 proven using the Divergence Theorem.

Step 1: Understand the Formula

We are given a surface integral over a closed surface and need to relate it to a volume integral over the region the surface encloses. Equation (10) is a form of Green's First Formula, which is derived using the Divergence Theorem.

Step 2: Recall Divergence Theorem

The Divergence Theorem relates a surface integral of a vector field across a closed surface to a volume integral over the region it encloses: \(\iint_{S} \mathbf{F} \cdot \mathbf{n} \, d\sigma = \iiint_{D} abla \cdot \mathbf{F} \, dV\). Here, \(\mathbf{F}\) is the vector field, \(\mathbf{n}\) is the outward normal, and \(d\sigma\) is the surface area element.

Step 3: Define the Vector Field \(\mathbf{F}\)

Set \(\mathbf{F} = f abla g\). Our task now is to find \(abla \cdot \mathbf{F}\). The goal is to use the divergence theorem for this vector field \(\mathbf{F}\).

Step 4: Compute \(\nabla \cdot (f \nabla g)\)

Using the product rule for divergence, \(abla \cdot (f abla g) = abla f \cdot abla g + f abla^2 g\). This expression will be used on the right-hand side of the divergence theorem equivalence.

Step 5: Apply the Divergence Theorem

Substitute \(\mathbf{F} = f abla g\) into the Divergence Theorem: \(\iint_{S} f abla g \cdot \mathbf{n} \, d\sigma = \iiint_{D} (abla f \cdot abla g + f abla^2 g) \, dV\). This shows the equivalence with Green's First Formula as given in the exercise.

Key Concepts

Divergence Theorem

The Divergence Theorem is an essential tool in vector calculus that connects the flow of a vector field's divergence through a volume, to the flow across the boundary of that volume. Intuitively, the theorem provides a way to transform a surface integral into an easier-to-compute volume integral. This transformation is invaluable when the surface itself is challenging to integrate directly.
The theorem is formally expressed as:

• If \(\mathbf{F}\) is a continuously differentiable vector field defined on a closed surface \(S\) surrounding a region \(D\), 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 vector on surface \(S\).
• \(d\sigma\) represents an infinitesimal area element.
The theorem simplifies calculations as it transforms a potentially complex surface integral into a volume integral, which may be easier to evaluate.

Scalar Functions

Scalar functions play a fundamental role in multivariable calculus. They are mappings that assign a single real number to each point in the domain, typically representing quantities such as temperature or pressure that do not have direction.
In vector calculus, scalar functions are often denoted by symbols like \(f\) or \(g\), and can have various properties:

• Continuity: These functions do not have any abrupt changes. This means they can be graphed as smooth surfaces in three-dimensional space.
• Differentiability: If their partial derivatives exist and are continuous, they can be considered smooth, allowing for the calculation of gradients.

Gradients of scalar functions are particularly important as they yield vector fields indicating the direction of greatest increase of the function.

Surface Integrals

Surface integrals extend the concept of multiple integrals to functions over curved surfaces, rather than plane areas. In the context of vector calculus, these integrals often involve vector fields and measure the "flux" through the surface.
Calculating a surface integral involves:

• Parameterizing the surface, that is, expressing it in terms of two parameters to facilitate integration.
• Evaluating the vector field at various points on the surface.
• Computing the dot product with the surface's normal vector \(\mathbf{n}\), capturing the component of the field crossing the surface.
• Integrating this quantity over the entire surface.

Surface integrals are used in situations like fluid flow where the interest lies in knowing how much substance flows through a surface over time.

Vector Fields

Vector fields are a crucial concept in understanding the environment within mathematics and physics. They consist of vectors assigned to each point in a subset of space, representing quantities that have both a magnitude and a direction at every point.

• Common examples include velocity fields (describing fluid flow), electromagnetic fields, and force fields.
• In calculus, vector fields are usually denoted by \(\mathbf{F}\), and various operations such as divergence and curl can be performed on them.

Just like scalar functions, vector fields must also be smooth (differentiable and continuous) for theorems like the Divergence Theorem to apply. These fields can be visualized as arrows pointing in the direction of the function's output while the length represents the strength of the vector at each point. Understanding vector fields aid in interpreting physical phenomena that change direction and magnitude across different areas.