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

Question

Let \(V(x, y, z)=V_{1} i+V_{2} j+V_{3} k\) be defined and differentiable at each point \((x, y, z)\) in a certain region of space. Then, the curl or roation of \(\vec{V}\), written \(\nabla \times \vec{V}\), curl \(\vec{V}\) or rot \(\vec{V}\), is defined by \(\begin{aligned} \nabla \times V &=\left(\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k\right) \times\left(V_{1} i+V_{2} j+V_{3} k\right) \\ &=\left|\begin{array}{lll}i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ V_{1} & V_{2} & V_{3}\end{array}\right| \\\ &=\left(\frac{\partial V_{3}}{\partial y}-\frac{\partial V_{2}}{\partial z_{2}}\right) i+\left(\frac{\partial V_{2}}{\partial z}-\frac{\partial V_{3}}{\partial x}\right) j+\left(\frac{\partial V_{2}}{\partial x}-\frac{\partial V_{1}}{\partial y}\right) k \end{aligned}\) Note that in the expansion of the determinant the operators \(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\) must precede \(V_{1}, V_{2}, V_{3}\). 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 the Expression

The problem asks us for the divergence of the curl of a vector field \( \vec{A} \). We can start by reviewing the mathematical expression, \( abla \cdot (abla \times \vec{A}) \), which involves computing the curl of \( \vec{A} \) and then finding the divergence of that curl.

Step 2: Recall Vector Calculus Identity

According to the vector calculus identity, the divergence of the curl of any vector field is always zero. This is expressed as \( abla \cdot (abla \times \vec{A}) = 0 \). This is true for any vector function \( \vec{A} \) with continuous second order partial derivatives.

Step 3: Confirm Understanding with Options

Review the given options (A) \( A \), (B) \( abla \times A \), (C) 0, and (D) none of these. The correct answer is (C) because the divergence of the curl of any vector is always zero, according to the identity.

Key Concepts

Curl of a Vector Field

In vector calculus, the curl of a vector field provides a way to measure the rotation of a vector field in three-dimensional space. It applies specifically to situations where the field can be differentiated at every point within the region. Given a vector field \(\vec{V} = V_1 \mathbf{i} + V_2 \mathbf{j} + V_3 \mathbf{k}\), the curl is represented by the expression \(abla \times \vec{V}\). This describes how the vector field twists and turns. To calculate it, a determinant can be used involving the unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\), partial derivative operators \(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\), and the components of \(\vec{V}\).

Following the determinant, you'll find the resulting expression which involves the partial derivatives of \(V_1, V_2,\) and \(V_3\). The formula derived: \[\left(\frac{\partial V_3}{\partial y} - \frac{\partial V_2}{\partial z}\right) \mathbf{i} + \left(\frac{\partial V_1}{\partial z} - \frac{\partial V_3}{\partial x}\right) \mathbf{j} + \left(\frac{\partial V_2}{\partial x} - \frac{\partial V_1}{\partial y}\right) \mathbf{k}\] offers rich insight into how the components of the field are changing in relation to each other. In essence, the curl provides details about the vector field's rotational behavior.

Divergence

Divergence in vector calculus is an operation that produces a scalar output field that describes how much a vector field spreads or diminishes at a given point. It is fundamentally linked to the concept of vector fields that scale rather than rotate. For a vector field denoted as \(\vec{A} = A_1 \mathbf{i} + A_2 \mathbf{j} + A_3 \mathbf{k}\), the divergence is calculated through the operation \(abla \cdot \vec{A}\), which results in a scalar field. The general form of the divergence for a three-dimensional vector is given by \[\frac{\partial A_1}{\partial x} + \frac{\partial A_2}{\partial y} + \frac{\partial A_3}{\partial z}.\]

To find the divergence of the curl of a vector field \(abla \cdot (abla \times \vec{A})\), the identity in vector calculus states that it is zero. This underscores a significant property: a field's curl inherently has no divergence. This fact can greatly simplify analyses in fluid dynamics and electromagnetic theory where such concepts are frequently applied.

Mathematical Expression

A mathematical expression is crucial for articulating concepts in vector calculus. It allows precise communication of intricate ideas such as those involving vector fields and operations on them. The expression \(abla \times \vec{V}\) signifies the curl operation, showing the cross product between the vector differential operator and the vector field itself. This is methodologically like how determinants are used for grasping matrix operations.

Expressions in mathematics break down complex operations into manageable steps. For vector calculus, they allow the visualization of the vector field's structure, showing forms of motion and intensity over space. By using partial derivatives assembled into a determinant, the complex interaction of multi-dimensional systems become more approachable. These expressions are essential for anyone working in fields requiring spatial analysis, like physics or engineering, making it possible to predict and understand natural phenomena.

Partial Derivatives

Partial derivatives are used in calculating changes within multi-variable functions. In vector calculus, they serve an integral role in determining properties such as the curl or divergence. By taking the derivative of a function with respect to one variable while keeping the others constant, partial derivatives provide a lens into how each single dimension affects the whole.

Consider a vector field \( \vec{V}(x, y, z) = V_1 \mathbf{i} + V_2 \mathbf{j} + V_3 \mathbf{k} \). Calculating the curl requires computing partial derivatives like \(\frac{\partial V_3}{\partial y}\) or \(\frac{\partial V_2}{\partial z}\). This detailed breakdown offers insights into how the vector field's component functions change individually when x, y, or z are modified. By isolating the effects of single variables in a multivariable context, partial derivatives are key to unlocking the dynamics of complex systems, allowing solutions from climate models to fluid flow in engineering.