Problem 27 - Advanced Engineering Mathematics

Question

Verify the given identity. Assume continuity of all partial derivatives. $$ \nabla \cdot(f \mathbf{F})=f(\nabla \cdot \mathbf{F})+\mathbf{F} \cdot \nabla f $$

Step-by-Step Solution

Answer

The identity is verified; both sides are equal.

Step 1: Understand the Components

The identity in question is a vector calculus identity involving the divergence operator \( abla \cdot \), a scalar function \( f \), and a vector field \( \mathbf{F} = (F_1, F_2, F_3) \). The goal is to verify the expression \( abla \cdot (f \mathbf{F}) = f(abla \cdot \mathbf{F}) + \mathbf{F} \cdot abla f \).

Step 2: Calculate Left Side of Identity

First, compute \( abla \cdot (f \mathbf{F}) \). Rewrite \( f \mathbf{F} \) as \( (f F_1, f F_2, f F_3) \). The divergence is the sum of the partial derivatives of each component:\[ abla \cdot (f \mathbf{F}) = \frac{\partial}{\partial x}(f F_1) + \frac{\partial}{\partial y}(f F_2) + \frac{\partial}{\partial z}(f F_3) \].

Step 3: Apply Product Rule to Left Side

Apply the product rule to each partial derivative:\[ \frac{\partial}{\partial x}(f F_1) = f \frac{\partial F_1}{\partial x} + F_1 \frac{\partial f}{\partial x} \]\[ \frac{\partial}{\partial y}(f F_2) = f \frac{\partial F_2}{\partial y} + F_2 \frac{\partial f}{\partial y} \]\[ \frac{\partial}{\partial z}(f F_3) = f \frac{\partial F_3}{\partial z} + F_3 \frac{\partial f}{\partial z} \].Thus, \( abla \cdot (f \mathbf{F}) = f (abla \cdot \mathbf{F}) + (F_1 \frac{\partial f}{\partial x} + F_2 \frac{\partial f}{\partial y} + F_3 \frac{\partial f}{\partial z}) \).

Step 4: Calculate Right Side of Identity

Now, calculate \( f(abla \cdot \mathbf{F}) + \mathbf{F} \cdot abla f \). The divergence \( abla \cdot \mathbf{F} \) is:\[ abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \].The gradient of \( f \), \( abla f \), is:\[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \].Thus, \( \mathbf{F} \cdot abla f = F_1 \frac{\partial f}{\partial x} + F_2 \frac{\partial f}{\partial y} + F_3 \frac{\partial f}{\partial z} \).

Step 5: Compare Both Sides

Now, compare the two expressions:- From Step 3: \( abla \cdot (f \mathbf{F}) = f (abla \cdot \mathbf{F}) + (F_1 \frac{\partial f}{\partial x} + F_2 \frac{\partial f}{\partial y} + F_3 \frac{\partial f}{\partial z}) \)- From Step 4: \( f(abla \cdot \mathbf{F}) + \mathbf{F} \cdot abla f \ = f(abla \cdot \mathbf{F}) + (F_1 \frac{\partial f}{\partial x} + F_2 \frac{\partial f}{\partial y} + F_3 \frac{\partial f}{\partial z}) \)Since both sides are equal, the identity is verified.

Key Concepts

Divergence Theorem

The divergence theorem is a fundamental result in vector calculus. It connects the flux of a vector field through a closed surface with the volume integral of its divergence inside the region enclosed. In simple terms, it translates surface integrals into volume integrals. If you have a vector field \( \mathbf{F} \), and a closed surface \( S \) that bounds a volume \( V \), the theorem states that:

• \[ \int_{S} \mathbf{F} \cdot d\mathbf{S} = \int_{V} abla \cdot \mathbf{F} \, dV \]

This means the outward flux through the surface \( S \) is equal to the integral of the divergence over the volume \( V \). Remember, the vector field needs to be well-behaved (continuous and differentiable) for this theorem to hold.
Understanding the divergence theorem helps in applying various laws of physics, such as Gauss' law as part of electromagnetic theory.
In the context of our original exercise, understanding divergence is key to verifying that given vector calculus identity.

Partial Derivatives

Partial derivatives are used in multivariable calculus to measure how a function changes as its input changes along one axis, while keeping other variables fixed. It's like seeing how \( f(x, y) \) changes with \( x \) while holding \( y \) constant.

• For a function \( f(x, y, z) \), the partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} \).
• Partial derivatives help calculate the rates of change along specific directions.

In our particular problem, to verify the identity, we compute partial derivatives while employing the product rule. Each component of the vector field \( \mathbf{F} \) may depend on different variables \( x, y, \) and \( z \), requiring partial derivatives for proper calculation of divergence.
This fundamental concept is critical since it forms the foundation of operations like gradient, divergence, and curl, all integral to vector calculus.

Product Rule

The product rule is a useful technique when finding derivatives of products of functions. It's fundamental to calculating derivatives in calculus and essential for our vector calculus identity.

• Suppose we have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is \( (uv)' = u'v + uv' \).

In the context of vector fields and scalar functions, the product rule extends to express their combined changes:

• In vector calculus, it is used to differentiate a scalar multiplied by a vector field.
• This rule helps to break down the problem into manageable parts when calculating divergence. In our exercise, it was applied to each partial derivative in the divergence of \( f \mathbf{F} \).

Understanding the product rule is important for verifying vector identities like in the given exercise, where we see it translating into terms like \( f(abla \cdot \mathbf{F}) + (\mathbf{F} \cdot abla f) \). This breaks down the calculations to complete verification, tying together the ideas of divergence and partial derivatives.