Problem 67 - 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 \(r=x i+y j+z k\), then \(\nabla \cdot\left(\frac{r}{r^{3}}\right)=\) (A) 0 (B) 1 (C) \(-1\) (D) none of these

Step-by-Step Solution

Answer

The divergence \( \nabla \cdot \frac{r}{r^3} = 0 \), answer is (A).

Step 1: Understand the Vector Field

The vector field in question is \( \frac{r}{r^3} \) where \( r = \sqrt{x^2 + y^2 + z^2} \). This can be rewritten as \( \frac{x}{(x^2 + y^2 + z^2)^{3/2}} i + \frac{y}{(x^2 + y^2 + z^2)^{3/2}} j + \frac{z}{(x^2 + y^2 + z^2)^{3/2}} k \).

Step 2: Set Up the Divergence Formula

Recall the divergence formula: \( abla \cdot V = \frac{\partial V_1}{\partial x} + \frac{\partial V_2}{\partial y} + \frac{\partial V_3}{\partial z} \). Our task is to find each partial derivative of the components of the vector field \( V \).

Step 3: Compute Partial Derivative with Respect to x

Calculate \( \frac{\partial}{\partial x} \frac{x}{(x^2 + y^2 + z^2)^{3/2}} \). Use the quotient rule and chain rule to find that after simplification, it equals \( \frac{(y^2+z^2)^{3/2} - 3x^2}{(x^2+y^2+z^2)^{5/2}} \).

Step 4: Compute Partial Derivative with Respect to y

Calculate \( \frac{\partial}{\partial y} \frac{y}{(x^2 + y^2 + z^2)^{3/2}} \). By symmetry and using similar calculations, this is given by \( \frac{(x^2+z^2)^{3/2} - 3y^2}{(x^2+y^2+z^2)^{5/2}} \).

Step 5: Compute Partial Derivative with Respect to z

Calculate \( \frac{\partial}{\partial z} \frac{z}{(x^2 + y^2 + z^2)^{3/2}} \). Again, using similar symmetry, this is given by \( \frac{(x^2+y^2)^{3/2} - 3z^2}{(x^2+y^2+z^2)^{5/2}} \).

Step 6: Sum the Partial Derivatives

Sum up the partial derivatives: \[ \frac{\partial}{\partial x} \frac{x}{(x^2 + y^2 + z^2)^{3/2}} + \frac{\partial}{\partial y} \frac{y}{(x^2 + y^2 + z^2)^{3/2}} + \frac{\partial}{\partial z} \frac{z}{(x^2 + y^2 + z^2)^{3/2}} \]. After computation and simplification, the result is \( 0 \).

Step 7: Conclusion

Since the final result of the divergence is zero, the correct choice is (A) \( 0 \).

Key Concepts

del operator

The del operator, represented by \( abla \), is a crucial tool in vector calculus. It combines differentiation with vector operations, making it vital for analyzing vector fields.
In three-dimensional space, the del operator is given by:

• \( abla = \dfrac{\partial}{\partial x} \mathbf{i} + \dfrac{\partial}{\partial y} \mathbf{j} + \dfrac{\partial}{\partial z} \mathbf{k} \)

This represents how the rate of change occurs along each coordinate axis. Think of it as a vector of partial derivatives, capturing how a scalar or vector field changes in space. The del operator can be used to perform several operations, like gradient, divergence, and curl, making it essential in physics and engineering for analyzing phenomena such as fluid flow and electromagnetic fields.
In our exercise, we focused on divergence, which is a scalar result obtained by taking the dot product of \( abla \) with a vector field.

vector calculus

Vector calculus is a branch of mathematics that deals with vector fields and operations on them, using vector operations like gradient, divergence, and curl.
These operations help describe the behavior of vector fields, such as those found in physics and engineering. For example:

• The gradient gives the direction and rate of the fastest increase of a scalar field.
• The divergence measures the "outflowing-ness" of a vector field from a point.
• The curl describes the rotation of a vector field around a point.

Each of these operations relies heavily on the use of the del operator to manipulate vectors.
In vector calculus, a primary objective is to better understand how fields behave spatially. This can be applied to numerous scientific fields, from analyzing electromagnetic fields to understanding fluid dynamics.
The exercise involves finding the divergence of a vector field, which helps us assess if the field is a source, sink, or neither.

partial derivatives

Partial derivatives are used when dealing with functions of multiple variables. They measure how a function changes as one of the variables changes, while keeping the other variables constant.
For example, if we have a function \( f(x, y, z) \), then the partial derivative with respect to \( x \) is denoted as \( \dfrac{\partial f}{\partial x} \). This tells us how \( f \) changes as \( x \) changes, with \( y \) and \( z \) held constant.
Partial derivatives are essential in vector calculus because they allow us to compute gradients, divergences, and curls. In our exercise, we computed the partial derivatives of the components of a vector field to find its divergence.

• This involved calculating partial derivatives with respect to each component \( x, y, \) and \( z \).
• By summing these derivatives, we determined whether the vector field was divergent.

The use of partial derivatives allows for a more precise manipulation and understanding of multi-variable functions in physical contexts.