Problem 30 - Advanced Engineering Mathematics

Question

Verify the given identity. Assume continuity of all partial derivatives. \(\operatorname{div}(\operatorname{curl} \mathbf{F})=0\)

Step-by-Step Solution

Answer

The identity \( \operatorname{div}(\operatorname{curl} \mathbf{F}) = 0 \) is verified.

Step 1: Understand Vector Operations

The problem involves two vector calculus operations: curl and divergence. The operator \( \operatorname{curl}(\mathbf{F}) \) yields a vector where \( \mathbf{F} \) is a vector field, and \( \operatorname{div}(\mathbf{A}) \) provides a scalar when applied to a vector field \( \mathbf{A} \).

Step 2: Recall the Curl of a Vector Field

If \( \mathbf{F} = (F_1, F_2, F_3) \) is a vector field, then the curl of \( \mathbf{F} \) is given by:\[ \operatorname{curl}(\mathbf{F}) = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \]

Step 3: Express the Curl Result as a Vector Field

Use the result from Step 2, the vector field becomes\[ \mathbf{A} = \left( A_1, A_2, A_3 \right) \] where:\[ A_1 = \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \]\[ A_2 = \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \]\[ A_3 = \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \]

Step 4: Compute the Divergence of the Curl

To find \( \operatorname{div}(\operatorname{curl}(\mathbf{F})) \), calculate the divergence of \( \mathbf{A} \): \[ \operatorname{div}(\mathbf{A}) = \frac{\partial A_1}{\partial x} + \frac{\partial A_2}{\partial y} + \frac{\partial A_3}{\partial z} \]

Step 5: Simplify using Continuous Partial Derivatives

Substitute \( A_1, A_2, A_3 \) in the divergence expression:\[ \operatorname{div}(\mathbf{A}) = \frac{\partial}{\partial x}\left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}\right) + \frac{\partial}{\partial y}\left(\frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}\right) + \frac{\partial}{\partial z}\left(\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right) \]

Step 6: Apply Mixed Partial Derivatives

By the assumption of continuity of partial derivatives, the mixed partial derivatives can be exchanged:\[ \frac{\partial^2 F_3}{\partial x \partial y} = \frac{\partial^2 F_3}{\partial y \partial x}, \frac{\partial^2 F_2}{\partial z \partial x} = \frac{\partial^2 F_2}{\partial x \partial z}, \text{ and so forth}. \] Applying this property, each term in the sum cancels out, resulting in zero.

Step 7: Conclude the Verification

Since all terms in the resulting expression from Step 5 cancel out to zero, we verify the identity:\[ \operatorname{div}(\operatorname{curl}(\mathbf{F})) = 0 \].

Key Concepts

Curl of a Vector Field

In vector calculus, the curl of a vector field is an important operator that measures the rotation of the field at a point. Imagine a small paddle wheel being placed at a point within the vector field—if the wheel starts to spin, then there is a non-zero curl at that point.
The curl is a vector itself and can be computed using partial derivatives.

For a vector field \( \mathbf{F} = (F_1, F_2, F_3) \), the curl is given by:
\[\operatorname{curl}(\mathbf{F}) = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right)\]
This calculation involves taking differences of partial derivatives of the components of \( \mathbf{F} \). Each component of the curl vector represents the rotational effect around its respective axis.
The curl is widely used in electromagnetics and fluid mechanics, often to describe how a fluid or magnetic field rotates around points in space.

Divergence of a Vector Field

Divergence is another foundational concept in vector calculus. It measures the magnitude of a vector field's source or sink at a given point. If you picture air flowing out of a balloon, the divergence would tell you how much air is flowing out at that point.
For a vector field \( \mathbf{A} = (A_1, A_2, A_3) \), the divergence is calculated as:
\[\operatorname{div}(\mathbf{A}) = \frac{\partial A_1}{\partial x} + \frac{\partial A_2}{\partial y} + \frac{\partial A_3}{\partial z}\]
This results in a scalar value, quantifying how a vector field expands at any point.

In the context of the problem, when we find the divergence of the curl, the result is always zero: \( \operatorname{div}(\operatorname{curl}(\mathbf{F})) = 0 \). This important identity states that a curl of a layer can never be a source or sink, which aligns well with the physical intuition of rotation without gaining or losing material or energy.

Partial Derivatives

Partial derivatives are a key concept in understanding how vector calculus operations are performed. They allow us to study the variation of multivariable functions when only one variable changes and the others remain constant.
For example, the partial derivative of \( F_1 \) with respect to \( x \), written as \( \frac{\partial F_1}{\partial x} \), tells us how \( F_1 \) changes as \( x \) changes, keeping all other variables the same.

In the problem, the curl and divergence both employ partial derivatives to calculate how the vector field's components interact and vary along different axes.
One crucial property used in this exercise is that partial derivatives can often be swapped when derivatives are continuous, known as Clairaut's theorem or Schwarz's theorem.

• Mixed partials: \( \frac{\partial^2 F}{\partial x \partial y} = \frac{\partial^2 F}{\partial y \partial x} \)

This property simplifies many calculations in vector calculus, permitting us to reach results like \( \operatorname{div}(\operatorname{curl}(\mathbf{F})) = 0 \). Partial derivatives are invaluable for constructing the mathematical framework to describe physical phenomena.