Problem 69 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

For a vector function \(A\) possessing continuous second order partial derivatives, \(\nabla \cdot(\nabla \times A)=\) (A) \(A\) (B) \(\nabla \times A\) (C) 0 (D) none of these

Step-by-Step Solution

Answer

The answer is (C) 0.

Step 1: Understand Curl of A

The curl of a vector field \( A \) is represented by \( abla \times A \). This operation results in a new vector field.

Step 2: Understand Divergence of a Vector Field

The divergence of a vector field gives a scalar and is denoted by \( abla \cdot( B ) \), where \( B \) is any vector field. This operation measures the magnitude of a source or sink at a given point in a vector field.

Step 3: Apply Divergence Theorem to Curl

According to vector calculus identities, the divergence of the curl of any vector field, \( A \), is always zero. Mathematically, this identity is represented as: \( abla \cdot (abla \times A) = 0 \).

Step 4: Determine the Correct Answer

Since \( abla \cdot(abla \times A) = 0 \), the correct choice for this problem is option (C) which indicates \( 0 \).

Key Concepts

Curl of a Vector Field

The curl of a vector field is a fundamental concept in vector calculus, especially in the study of fluid dynamics and electromagnetism. In simple terms, the curl measures the rotation or the tendency of particles in the field to rotate around a point.
To calculate the curl of a vector field \( \mathbf{F} \), use the expression \( abla \times \mathbf{F} \). This operation produces a new vector field that represents the rotational effects present in the original field.

• In 3D, if \( \mathbf{F} = (F_1, F_2, F_3) \), the curl is given by a determinant:

\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_1 & F_2 & F_3 \end{vmatrix} \]
This equation captures how the components of the vector field change as we move through space.
In physical contexts, the curl is crucial in formulating the equations of motion for rotating bodies and in understanding how fluids tend to circulate.

Divergence of a Vector Field

The divergence of a vector field is another critical operation. Unlike curl, it measures how much a vector field diverges from a point, or spreads out, like water from a spring or vermicelli in a hot pan.
Mathematically, for a vector field \( \mathbf{F} = (F_1, F_2, F_3) \), the divergence is calculated as:

• \( abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \)

The result is a scalar field representing the density of sources or sinks in the space.
Divergence is notably used in electromagnetism, where the divergence of an electric field helps describe charge density.

Vector Calculus Identities

Vector calculus identities are essential tools that simplify complex calculations and reinforce our understanding of vector fields. These identities are rooted in fundamental principles of mathematics and physics.
One key identity involves the interplay between curl and divergence: the divergence of the curl of a vector field is always zero, expressed as \( abla \cdot (abla \times \mathbf{F}) = 0 \). This means no matter how a field rotates, it doesn't have sources or sinks in the sense of divergence.
These identities not only streamline calculations but also intuitively highlight relationships between different vector operations.

• Remember, these identities are only valid for fields with continuous second-order derivatives, ensuring that the mathematical operations are well-defined and applicable across the space of interest.

They are frequently applied in engineering and physics, where vector fields describe phenomena like fluid flow, gravitational forces, and electromagnetic fields.