Problem 65 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\nabla(x, y, z)=V_{1} i+V_{2} j+V_{3} k\) be defined and differentiable at each point \((x, y, z)\) in a certain region of space. Then, the divergence of \(V\), written \(\nabla \cdot V\) or div \(V\) is defined by $$ \begin{aligned} \nabla \cdot V &=\left(\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k\right) \cdot\left(V_{1} i+V_{2} j+V_{3} k\right) \\ &=\frac{\partial V_{1}}{\partial x}+\frac{\partial V_{2}}{\partial y}+\frac{\partial V_{3}}{\partial z} \end{aligned} $$ Here, \(\nabla=\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k\) is the del operator. Note the analogy with \(A \cdot B=A_{1} B_{1}+A_{2} B_{2}+A_{3} B_{3} .\) Also, note that \(\vec{\nabla} \cdot V \neq V \cdot \nabla\). If \(\phi=2 x^{3} y^{2} z^{4}\), then \(\nabla . \nabla \phi=k\left(3 x y^{2} z^{4}+x^{3} z^{4}+6 x^{3} y^{2} z^{2}\right)\), where \(k=\) (A) 2 (B) 3 (C) 4 (D) 6

Step-by-Step Solution

Answer

\( k = 4 \)

Step 1: Find Gradient of \( \phi \)

Start by finding \( abla \phi \) for the function \( \phi = 2x^3 y^2 z^4 \). Compute \( \frac{\partial \phi}{\partial x} \), \( \frac{\partial \phi}{\partial y} \), and \( \frac{\partial \phi}{\partial z} \). This gives:- \( \frac{\partial \phi}{\partial x} = 6x^2 y^2 z^4 \)- \( \frac{\partial \phi}{\partial y} = 4x^3 y z^4 \)- \( \frac{\partial \phi}{\partial z} = 8x^3 y^2 z^3 \).Thus, \( abla \phi = 6x^2 y^2 z^4 i + 4x^3 y z^4 j + 8x^3 y^2 z^3 k \).

Step 2: Calculate the Divergence of the Gradient of \( \phi \)

Use the formula for divergence: \( abla \cdot abla \phi = \frac{\partial}{\partial x}(6x^2 y^2 z^4) + \frac{\partial}{\partial y}(4x^3 y z^4) + \frac{\partial}{\partial z}(8x^3 y^2 z^3) \).- The first term: \( \frac{\partial}{\partial x}(6x^2 y^2 z^4) = 12x y^2 z^4 \) - The second term: \( \frac{\partial}{\partial y}(4x^3 y z^4) = 4x^3 z^4 \) - The third term: \( \frac{\partial}{\partial z}(8x^3 y^2 z^3) = 24x^3 y^2 z^2 \).Combine these results to find the divergence: \( abla \cdot abla \phi = 12x y^2 z^4 + 4x^3 z^4 + 24x^3 y^2 z^2 \).

Step 3: Solve for \( k \)

The given expression is \( abla \cdot abla \phi = k(3xy^2 z^4 + x^3 z^4 + 6x^3 y^2 z^2) \). Compare this to the derived expression: \( 12xy^2 z^4 + 4x^3 z^4 + 24x^3 y^2 z^2 \).Compare the coefficients: - 12 is 4 times 3 (from \(3xy^2 z^4\))- 4 is 4 times 1 (from \(x^3 z^4\))- 24 is 4 times 6 (from \(6x^3 y^2 z^2\))Thus, \( k = 4 \).

Key Concepts

Divergence

The concept of divergence is crucial in vector calculus, often referred to as \( abla \cdot V \). It describes a vector field's tendency to originate from or converge at a certain point. In essence, it quantifies how much a vector field behaves like a source or a sink at a specific location within a given space.
\[ abla \cdot V = \frac{\partial V_1}{\partial x} + \frac{\partial V_2}{\partial y} + \frac{\partial V_3}{\partial z} \]
The divergence is calculated by summing the partial derivatives of each component of a vector field \( V \), which is composed of component vectors \( V_1, V_2, \) and \( V_3 \) along the \( x, y, \text{and } z \) directions, respectively.
This operation helps in understanding how energy, mass, or other quantities impose changes within a field. Noting that the operation involves distinguishing how a field behaves outwardly or inwardly, divergence plays a fundamental role in fields such as fluid dynamics and electromagnetic theory.

• Important in assessing expansion or contraction rates within a field.
• Integral to the understanding of conservation laws.

Gradient

The gradient \( abla \phi \) of a scalar function \( \phi \) is a vector that points in the direction of the greatest rate of increase of the function, and whose magnitude is the rate of increase in that direction. It transforms scalars into vectors, thus providing critical insight into fields such as physics and engineering.
To compute the gradient of a function \( \phi = 2x^3 y^2 z^4 \), one must determine the partial derivatives with respect to \( x, y, \text{and } z \). This results in:

• \( \frac{\partial \phi}{\partial x} = 6x^2 y^2 z^4 \)
• \( \frac{\partial \phi}{\partial y} = 4x^3 y z^4 \)
• \( \frac{\partial \phi}{\partial z} = 8x^3 y^2 z^3 \)

The resulting gradient vector is \( abla \phi = 6x^2 y^2 z^4 \mathbf{i} + 4x^3 y z^4 \mathbf{j} + 8x^3 y^2 z^3 \mathbf{k} \), indicating how \( \phi \) changes most efficiently.
Essentially, the gradient not only reveals the steepest ascent direction but also bodes accuracy in optimization and analytical calculus processes. It aides significantly when discussing potential and scalar fields.

Del Operator

The del operator, represented by \( abla \), is a symbol used in calculus to denote the gradient, divergence, and curl operations. It is not a vector but an operator that acts on scalar and vector fields.
Mathematically, in three-dimensional Cartesian coordinates, it is expressed as:
\[ abla = \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k} \]
Using the del operator allows simplification of notations and calculations within vector calculus. Employing this operator can help efficiently find:

• The gradient of a scalar function \( \phi \), \( abla \phi \).
• The divergence of a vector field \( V \), \( abla \cdot V \).
• The curl of a vector field, \( abla \times V \) , which measures the rotation of the field.

The del operator is pivotal because it allows easy manipulation and transformation of equations in vector calculus. This operator is indispensable in physics, especially when dealing with electromagnetism and fluid dynamics, providing a concise and effective way to harness the power of calculus.