Problem 68 - 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 scalar function \(\phi\), possessing continuous secot order partial derivatives \(\nabla \times(\nabla \phi)=\) (A) \(\phi\) (B) 0 (C) \(\nabla \phi\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (B) 0.

Step 1: Understand the problem

The problem asks us to find the curl of the gradient of a scalar function \( \phi \). Mathematically, this is denoted as \( abla \times (abla \phi) \). The options given suggest that we need to identify which expression matches this curl result.

Step 2: Recall properties of vector calculus

One of the fundamental identities of vector calculus is that the curl of the gradient of any scalar function is always zero. This is stated as \( abla \times (abla \phi) = 0 \). This identity holds provided that \( \phi \) is a sufficiently smooth function, which is ensured by the condition of having continuous second-order partial derivatives.

Step 3: Match the result with the options

Given our understanding from Step 2, the expression \( abla \times (abla \phi) \) equals zero. Therefore, option (B) 0, is the correct answer that matches our derived result.

Key Concepts

Vector Field Differentiation

Vector field differentiation involves my understanding of how we can change vector fields, which are functions that assign a vector to each point in space. A typical vector field is given as \( \vec{V}(x, y, z) = V_1 i + V_2 j + V_3 k \), where \( V_1, V_2, \) and \( V_3 \) are component functions. These components are typically dependent on the spatial coordinates \( x, y, \) and \( z \).
This differentiation is crucial when we deal with operations like the curl, which measures a vector field's rotation or twisting. Here, we take the gradient operator \( abla = \frac{\partial}{\partial x} i + \frac{\partial}{\partial y} j + \frac{\partial}{\partial z} k \), and apply it to our vector field.
Understanding the differentiation helps to better grasp phenomena in physics and engineering, such as fluid flow and electromagnetic fields. It provides insights into how vectors change and interact in varying regions of space.

Determinant Expansion for Vectors

When we talk about determinant expansion for vectors, we're essentially looking at a method to calculate the cross product, which in vector calculus, is an indicator of rotation. To compute the curl using determinants, arrange the base vectors \( i, j, \text{ and } k \), the partial derivatives \( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \), and the vector field components \( V_1, V_2, V_3 \) in a 3x3 matrix. The determinant is then expanded as follows:

• For the \( i \)-component: \( \left(\frac{\partial V_3}{\partial y} - \frac{\partial V_2}{\partial z}\right) i \)
• For the \( j \)-component: \( \left(\frac{\partial V_1}{\partial z} - \frac{\partial V_3}{\partial x}\right) j \)
• For the \( k \)-component: \( \left(\frac{\partial V_2}{\partial x} - \frac{\partial V_1}{\partial y}\right) k \)

This determinant expansion is not only practical for calculating the curl but also essential in physics to describe the rotation at any point in a vector field.

Scalar Function Properties

Scalar functions, unlike vector fields, assign a single value to each point in space. They have intriguing properties, especially when discussing their derivatives. A scalar function \( \phi \) with continuous second-order partial derivatives can allow for interesting applications of vector calculus identities.
The identity \( abla \times (abla \phi) = 0 \) tells us that the curl of a gradient results in a zero vector for any sufficiently smooth scalar function. This results from the path independence of gradient fields and implies that potential functions exist for these fields.
This property forms the foundation of many physical theories and equations. For example, in electrostatics, the electric field is often expressed as the gradient of a potential function. Understanding this identity helps students grasp why certain vector fields behave as they do and how potential energy functions are derived.