Problem 31 - Calculus: Early Transcendental Functions

Question

Prove Green's first identity in three dimensions (see exercise 43 in section 14.5 for Green's first identity in two dimensions): \(\iiint_{Q} f \nabla^{2} g d V=\iint_{\partial Q} f(\nabla g) \cdot \mathbf{n} d S-\iiint_{Q}(\nabla f \cdot \nabla g) d V\) (Hint: Use the Divergence Theorem applied to \(\mathbf{F}=f \nabla g\).)

Step-by-Step Solution

Answer

Green's first identity in three dimensions is shown by defining a vector field with the given function and gradient, taking divergence, applying divergence theorem and rearranging terms accordingly.

Step 1: Express Vector Field Divergence

First, express the divergence of vector field \( \mathbf{F} = f \nabla g\) given using the definition of divergence in terms of partial derivatives. The divergence of vector field \( \mathbf{F}\) can be written as \( \nabla \cdot \mathbf{F} = \nabla f \cdot \nabla g + f \nabla^2 g \).

Step 2: Apply Divergence Theorem

The Divergence Theorem states that the volume integral of the divergence of a vector field \(\mathbf{F}\) over a volume \(Q\) is equal to the surface integral of \(\mathbf{F}\) dotted with the outward unit normal vector \(\mathbf{n}\) over the boundary of volume \(Q\). Apply this theorem on the divergence of vector field \( \mathbf{F}\). That gives, \( \iiint_{Q} \nabla \cdot \mathbf{F} dV = \iint_{\partial Q} \mathbf{F} \cdot \mathbf{n} dS \).

Step 3: Substitute and Rearrange

Substitute the expression for \( \nabla \cdot \mathbf{F}\) from step 1 and \( \mathbf{F}\) from the given hint into the equation from step 2. After rearranging the terms, we can see that it matches the form of Green's first identity in three dimensions, which is \( \iiint_{Q} f \nabla^{2} g dV = \iint_{\partial Q} f(\nabla g) \cdot \mathbf{n} dS - \iiint_{Q}(\nabla f \cdot \nabla g) dV \). This completes the proof.

Key Concepts

Divergence Theorem

The Divergence Theorem, also known as Gauss's theorem, is a powerful tool in vector calculus that relates the flow (flux) of a vector field through a closed surface to the behavior of the vector field inside the volume enclosed by the surface.

Imagine a vector field representing, for example, the velocity of a fluid or the intensity of an electric field. Now, if you have a closed surface such as a sphere or a cube, the Divergence Theorem allows us to link the total amount of whatever the field represents either exiting or entering the enclosed volume with the divergence of the field within the volume.

Mathematically, if we have a vector field \(\mathbf{F}\), the theorem states that the integral of its divergence over volume \(\iiint_Q\) is equal to the surface integral of the field through boundary \(\partial Q\):

• \[\iiint_{Q} abla \cdot \mathbf{F} dV = \iint_{\partial Q} \mathbf{F} \cdot \mathbf{n} dS\]

Here, \(\mathbf{n}\) represents the outward-pointing unit normal vector on \(\partial Q\). This principle is crucial in various applications such as fluid dynamics, electromagnetism, and heat transfer. It allows us to easily calculate what would otherwise be very complex surface integrals by converting them into volume integrals.

Vector Field Divergence

In vector calculus, divergence is a measure of the magnitude of a vector field's source or sink at a given point. It essentially tells us how much the field is 'spreading out' or 'converging' in space.

For a vector field \(\mathbf{F}\), the divergence at any point is a scalar quantity and is defined by the dot product of the del operator with the vector field:

• \[abla \cdot \mathbf{F} \]

If the divergence is positive at a point, the field is acting like a 'source' emitting flow at that location. Conversely, a negative divergence indicates a 'sink' where the flow is 'sucked into'. A divergence of zero means the field is neither created nor destroyed at that point, which is consistent with the physical principle of conservation.

In the Green's first identity problem, the divergence of the vector field \(f abla g\) reflected both the gradient of the scalar field \(g\) and its interaction with \(f\). This concept is an essential precursor to utilizing the Divergence Theorem, allowing us to express complex phenomena in terms of easily calculable quantities.

Surface Integral

A surface integral is a generalization of multiple integrals to integration over surfaces. It can be used to calculate the total amount of a quantity, such as mass or charge, that passes through a surface or to compute the force exerted by a fluid across a surface.

For a vector field \(\mathbf{F}\), the surface integral over a surface \(S\) with an outward-pointing unit normal vector \(\mathbf{n}\) is defined as:

• \[\iint_{S} \mathbf{F} \cdot \mathbf{n} dS\]

This integral sums up the component of \(\mathbf{F}\) that is perpendicular to \(S\) over each infinitesimal piece of the surface. In practical terms, if \(\mathbf{F}\) represents a fluid flow, the surface integral gives the rate at which the fluid is flowing out of the surface.

In the context of Green's first identity, we deal with a special case where the surface integral on the boundary of a volume \(Q\) is connected to the volume integral within \(Q\) through the application of the Divergence Theorem. This connection simplifies many real-world calculations.

Volume Integral

A volume integral extends the idea of integration to three-dimensional spaces and is used to calculate quantities spread out over a volume. For example, it can represent the total mass of a three-dimensional density distribution or the electric charge within a volume.

The volume integral of a function \(f\) over a volume \(Q\) is denoted by:

• \[\iiint_{Q} f dV\]

The volume integral sums up the function's values at each point within the volume. In the case of vector fields, it's often the divergence of the field that we integrate across a volume, as it represents how much field is being 'produced' at each point within the volume.

In proving Green's first identity, the volume integral takes center stage. The identity showcases how the divergence of the product of two scalar fields—one of them being the gradient of the other—can be represented as a balance between the field's behavior on the boundary surface and its properties inside the volume. This balance is what lends Green's identities their analytical power in various applications like physics and engineering.