Question

The following equations are variously known as Green's first and second identities or formulas or theorems. Derive them, as indicated, from the divergence theorem. $$(1)\int (\varphi {\mathrm{\nabla }}^2\psi +\mathrm{\nabla }\varphi \cdot \mathrm{\nabla }\psi )d\tau =\oint (\varphi \mathrm{\nabla }\psi )\cdot \mathbf{n}d\sigma$$. To prove this, let $\mathbf{V}=\varphi \mathrm{\nabla }\psi$ in the divergence theorem. $(2)\int (\varphi {\mathrm{\nabla }}^2\psi -\psi {\mathrm{\nabla }}^2\varphi )d\tau =\oint (\varphi \mathrm{\nabla }\psi -\psi \mathrm{\nabla }\varphi )\cdot \mathbf{n}d\sigma$. To prove this, copy Theorem 1 above as is and also with $\varphi$ and $\psi$ interchanged; then subtract the two equations.

Short answer

Green's first identity: $$\int (\varphi {\text{abla}}^2\psi +\text{abla}\varphi \cdot \text{abla}\psi )d\tau =\oint (\varphi \text{abla}\psi \cdot \mathbf{n})d\sigma$$. Green's second identity: $$\int (\varphi {\text{abla}}^2\psi -\psi {\text{abla}}^2\varphi )d\tau =\oint (\varphi \text{abla}\psi -\psi \text{abla}\varphi )\cdot \mathbf{n}d\sigma$$.

Step-by-step solution

State the Divergence Theorem

The Divergence Theorem states that for any vector field \mathbf{V}\, $$\int (\text{abla}\cdot \mathbf{V})d\tau =\oint (\mathbf{V}\cdot \mathbf{n})d\sigma$$.

Let \mathbf{V} = \phi \abla \psi

For the first identity, set \mathbf{V} = \phi \abla \psi. The Divergence Theorem then becomes $$\int (\text{abla}\cdot (\varphi \text{abla}\psi ))d\tau =\oint (\varphi \text{abla}\psi \cdot \mathbf{n})d\sigma$$.

Expand the divergence

Expand the divergence in the integral: $$\text{abla}\cdot (\varphi \text{abla}\psi )=(\text{abla}\varphi )\cdot \text{abla}\psi +\varphi {\text{abla}}^2\psi$$.

Substitute back into the integral

Substitute the expanded divergence back into the integral: $$\int (\varphi {\text{abla}}^2\psi +\text{abla}\varphi \cdot \text{abla}\psi )d\tau =\oint (\varphi \text{abla}\psi \cdot \mathbf{n})d\sigma$$. This proves the first identity.

The second identity – interchange \phi and \psi

To derive the second identity, first copy the result of the first identity with \phi and \psi interchanged: $$\int (\psi {\text{abla}}^2\varphi +\text{abla}\psi \cdot \text{abla}\varphi )d\tau =\oint (\psi \text{abla}\varphi \cdot \mathbf{n})d\sigma$$.

Subtract the two equations

Subtract the equation in Step 4 from the one in Step 5: $$\int (\varphi {\text{abla}}^2\psi -\psi {\text{abla}}^2\varphi )d\tau =\oint (\varphi \text{abla}\psi -\psi \text{abla}\varphi )\cdot \mathbf{n}d\sigma$$. This proves the second identity.

Key concepts

Divergence Theorem

The Divergence Theorem is a cornerstone in vector calculus. It connects the flux of a vector field through a surface to the behavior of the field inside the surface.
Simply put, it states that the integral of the divergence of a vector field $abla\cdot \mathbf{V}$ over a volume $\tau$ is equal to the flux of the vector field $abla\cdot \mathbf{V}$ through the surface $\mathbf{\text{S}}$ that bounds the volume:
$${\int }_{\tau }\text{abla}\cdot \mathbf{V}d\tau ={\oint }_S\mathbf{V}\cdot \mathbf{n}d\sigma$$ Here, $\mathbf{\text{n}}$ is the outward-pointing normal of the surface $\mathbf{\text{S}}$, and $d\sigma$ is the surface element.
This theorem is powerful because it allows us to convert a volume integral into a surface integral. This conversion is useful in many fields, including fluid dynamics, electromagnetics, and more.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and operations on them. It's essential for describing physical phenomena like fluid flow, electromagnetism, and more.
Key operations in vector calculus include:

• Gradient ($\text{abla}\varphi$): Measures the rate of change and the direction of fastest increase of a scalar field.
• Divergence ($\text{abla}\cdot \mathbf{V}$): Measures the magnitude of a source or sink at a given point in a vector field.
• Curl ($\text{abla}\times \mathbf{V}$): Measures the rotation of a vector field.

Combining these operations, we derive useful theorems like the Divergence Theorem and Stokes' Theorem. Understanding these operations and theorems helps in solving complex problems in physics and engineering.
For instance, to explain Green's identities, we use the gradient to expand terms and the divergence to connect volume and surface integrals.

Partial Differential Equations

Partial Differential Equations (PDEs) are equations that involve partial derivatives of unknown functions with respect to multiple variables.
PDEs are crucial in fields like physics, engineering, and finance as they describe various phenomena such as heat conduction, wave propagation, fluid dynamics, and quantum mechanics. The second identity in Green's theorem involves PDEs through the terms $\varphi {\text{abla}}^2\psi$ and $\psi {\text{abla}}^2\varphi$.
These terms involve the Laplacian operator ${\text{abla}}^2$, which is a type of differential operator used in solving PDEs.
The Laplacian, defined as ${\text{abla}}^2=\text{abla}\cdot \text{abla}$, measures the rate at which a function's value accumulates around a point. The identities show how these accumulations can be related to flux through boundaries.

Integral Theorems

Integral theorems are essential tools in mathematical analysis, bridging the gap between local differential properties and global integral properties.
Examples include:

• Green's Theorem: Relates the line integral around a simple closed curve to a double integral over the plane region bounded by the curve.
• The Divergence Theorem: Relates the flux of a vector field through a closed surface to the volume integral of the divergence of the field.
• Stokes' Theorem: Relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary of the surface.

These theorems are not just mathematical curiosities but practical tools. They simplify calculations in physics and engineering, making it easier to relate local behaviors to global outcomes.
For instance, in fluid dynamics, integral theorems help describe the conservation of mass and energy across different regions.