Problem 30 - Calculus Early Transcendentals

Question

Prove the identity, assuming that F, ?, and G satisfy the hypotheses of the Divergence Theorem and that all necessary differentiability requirements for the functions f(x, y, z) and g(x, y, z) are met. $$ \begin{array}{l}{\iint_{\sigma}(f \nabla g-g \nabla f) \cdot \mathbf{n} d S=\iiint_{G}\left(f \nabla^{2} g-g \nabla^{2} f\right) d V} \\ {[\text {Hint: Interchange } f \text { and } g \text { in } 29 .]}\end{array} $$

Step-by-Step Solution

Answer

The identity is proven by transforming the surface integral into a divergence and applying the Divergence Theorem.

Step 1: Understand the Divergence Theorem

The Divergence Theorem states that for a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the flux of \( \mathbf{F} \) through a closed surface \( \Sigma \) is equal to the volume integral of the divergence of \( \mathbf{F} \) over the region \( G \) enclosed by \( \Sigma \). Mathematically, this is \( \iint_{\Sigma} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{G} abla \cdot \mathbf{F} \, dV \). This concept will be used to prove the given identity.

Step 2: Set Up the Problem

We need to prove that \( \iint_{\Sigma}(f abla g - g abla f) \cdot \mathbf{n} \, dS = \iiint_{G} (f abla^{2} g - g abla^{2} f) \, dV \). Here, \( \mathbf{n} \) is the outward unit normal vector on the closed surface \( \Sigma \). Our goal is to express the left-hand side in terms of a divergence, which we can then relate to a volume integral using the Divergence Theorem.

Step 3: Express the Surface Integral as a Divergence

Consider the vector fields \( \mathbf{F} = f abla g \) and \( \mathbf{H} = g abla f \). The surface integral becomes \( \iint_{\Sigma} (\mathbf{F} - \mathbf{H}) \cdot \mathbf{n} \, dS \), which according to the Divergence Theorem is equivalent to \( \iiint_{G} abla \cdot (\mathbf{F} - \mathbf{H}) \, dV \). Next, simplify \( abla \cdot (\mathbf{F} - \mathbf{H}) \).

Step 4: Compute the Divergence

To find \( abla \cdot \mathbf{F} = abla \cdot (f abla g) \), we use the product rule for divergence: \( abla \cdot (f abla g) = f abla^2 g + abla f \cdot abla g \). Similarly for \( \mathbf{H} \), \( abla \cdot (g abla f) = g abla^2 f + abla g \cdot abla f \). Thus, \( abla \cdot (\mathbf{F} - \mathbf{H}) = f abla^2 g + abla f \cdot abla g - g abla^2 f - abla g \cdot abla f \), which simplifies to \( f abla^2 g - g abla^2 f \).

Step 5: State the Final Result

Substituting this result into the volume integral gives \( \iiint_{G} (f abla^2 g - g abla^2 f) \, dV \), which matches the right-hand side of the identity we want to prove. Therefore, by applying the Divergence Theorem correctly, the identity is proven.

Key Concepts

Surface Integrals

Surface integrals play a significant role in the context of vector calculus, especially when working with the Divergence Theorem. A surface integral allows us to calculate the flow of a vector field across a surface. In simple terms, it is like measuring how much a fluid flows through a surface in a vector field.

For a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the surface integral over a closed surface \( \Sigma \) is given by \( \iint_{\Sigma} \mathbf{F} \cdot \mathbf{n} \, dS \), where \( \mathbf{n} \) is the unit normal vector pointing outward from the surface. This integral represents the total flux of the vector field through \( \Sigma \), accounting for the geometry and the orientation of the surface.

• The surface integral depends on both the magnitude and direction of the vector field.
• The concept of outward normal vector is crucial, as it provides the direction in which flux is being calculated.
• In contexts like electromagnetism and fluid dynamics, surface integrals provide insights into how electric fields and fluids interact with surfaces, respectively.

Understanding and visualizing surface integrals are key to mastering vector calculus and its applications to physics and engineering.

Volume Integrals

Volume integrals extend the idea of integrating over an area to integrating over a three-dimensional volume. It helps us sum up a quantity distributed over a space. In the Divergence Theorem, it relates the flow through a closed surface to the behavior of a vector field within the volume it encloses.

For a scalar or vector field defined over a region \( G \), the volume integral is denoted as \( \iiint_{G} \phi \, dV \), where \( \phi \) could be a scalar field or a divergence of a vector field. This integral is essential for calculating properties like mass, charge, or volume of a region when density varies across space.

• Volume integrals are vital when applying the Divergence Theorem, providing a complete picture connecting the surface's behavior to that within the volume.
• These integrals simplify calculations in complex systems by reducing problems from three dimensions to two via known theorems like the Divergence Theorem.
• In practical applications, volume integrals are used to calculate quantities like heat, fluid pressure, or electric charge distributed within a solid object or space.

Grasping volume integrals allows you to analyze physical properties distributed throughout three-dimensional regions effectively.

Vector Fields

A vector field assigns a vector to each point in space. Think of it as a field of arrows that define the direction and magnitude of a quantity at each point. These arrows could represent anything from the velocity of a flowing fluid to the force exerted by a magnetic field.

Mathematically, a vector field in three-dimensional space is defined as \( \mathbf{F}(x, y, z) = P(x, y, z)\mathbf{i} + Q(x, y, z)\mathbf{j} + R(x, y, z)\mathbf{k} \). Exploring vector fields is crucial for understanding physical phenomena, as they provide a framework for describing how a physical quantity varies over space.

• Vector fields can be visualized graphically using a grid of arrows, where each arrow's length and direction correspond to the vector's magnitude and orientation at that point.
• Common examples include gravitational fields, magnetic fields, and velocity fields in fluid dynamics.
• Vector fields are foundational in applying various calculus theorems, including the Divergence and Stoke's Theorems, which allow conversion between multiple types of integrals.

Learning about vector fields is essential for effectively describing and analyzing how different quantities interact in space.

Product Rule for Divergence

The product rule for divergence is a powerful tool, especially in the context of the Divergence Theorem. This rule helps in finding the divergence of a product of a scalar and a vector field. It is similar to the product rule for differentiation but applied to vector calculus.

The rule states that for scalar functions \( f(x, y, z) \) and \( g(x, y, z) \), the divergence of their product \( f abla g \) is given by:\[abla \cdot (f abla g) = f abla^2 g + abla f \cdot abla g\]This formula is useful in simplifying expressions involving divergence in vector calculus.

• The product rule for divergence allows us to break down complex expressions into simpler, more manageable parts.
• It enables conversion between surface and volume integrals, laying a pathway to apply the Divergence Theorem effectively.
• The rule is fundamental in solving many physics and engineering problems where dynamics and changes within a field are analyzed.

Understanding and applying the product rule for divergence can help significantly in tackling comprehensive integrals and proving identities, as demonstrated with the Divergence Theorem.