Problem 62 - A Complete Resource Book in Mathematics for JEE Main

Question

The vector differential operator DEL, written \(\nabla\), is defined by \(\nabla=\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k=i \frac{\partial}{\partial x}+j \frac{\partial}{\partial y}+k \frac{\partial}{\partial z}\), where \(\frac{\partial}{\partial x}\) rep- resents the derivative w.r.t. \(x\) regarding \(y\) and \(z\) as constant. Similarly, \(\frac{\partial}{\partial y}\) represents the derivative w.r.t. \(y\) regarding \(x\) and \(z\) as constant and \(\frac{\partial}{\partial z}\) represents the derivative w.r.t. \(z\) regarding \(x\) and \(y\) as constant. The operator \(\nabla\) is also known as nabla. Let \(\phi(x, y, z)\) be defined and differentiable at each point \((x, y, z)\) in a certain region of space. Then, the gradient of \(\phi\), written \(\nabla \phi\) or grad \(\phi\), is defined by $$ \nabla \phi=\left(\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k\right) \phi=\frac{\partial \phi}{\partial x} i+\frac{\partial \phi}{\partial y} j+\frac{\partial \phi}{\partial z} k $$ Let \(r\) be any vector such that \(r=x i+y j+z k\) If \(\phi=\ln |r|\) then \(\nabla \phi=\) (A) \(\frac{r}{r^{2}}\) (B) \(\frac{r}{r^{3}}\) (C) \(\frac{r}{r^{4}}\) (D) \(\frac{r}{r}\)

Step-by-Step Solution

Answer

Option (A): \(\frac{r}{r^2}\).

Step 1: Understand the problem

We need to find the gradient of a scalar field \( \phi = \ln |r| \), where \( r = xi + yj + zk \). The options given suggest simplifications using vector notation and magnitudes.

Step 2: Define \( |r| \)

The magnitude of the vector \( r = xi + yj + zk \) is computed as \( |r| = \sqrt{x^2 + y^2 + z^2} \). This is essential to compute the gradient of \( \phi \).

Step 3: Differentiate \( \phi \) with respect to \( x \)

Calculate \( \frac{\partial \phi}{\partial x} = \frac{\partial}{\partial x}(\ln |r|) \). Using the chain rule, we get \( \frac{1}{|r|} \cdot \frac{\partial |r|}{\partial x} \). \( \frac{\partial |r|}{\partial x} = \frac{x}{|r|} \) since \( |r| = \sqrt{x^2 + y^2 + z^2} \). So, \( \frac{\partial \phi}{\partial x} = \frac{x}{x^2+y^2+z^2} \).

Step 4: Differentiate \( \phi \) with respect to \( y \)

Similarly, we find \( \frac{\partial \phi}{\partial y} = \frac{y}{x^2+y^2+z^2} \). This uses the same method as the partial derivative with respect to \( x \).

Step 5: Differentiate \( \phi \) with respect to \( z \)

Finally, \( \frac{\partial \phi}{\partial z} = \frac{z}{x^2+y^2+z^2} \). Again, this applies the same differentiation process focusing on the \( z \)-component of \( r \).

Step 6: Combine results to find \( \nabla \phi \)

Combine the partial derivatives to express \( abla \phi = \frac{x}{x^2 + y^2 + z^2} i + \frac{y}{x^2 + y^2 + z^2} j + \frac{z}{x^2 + y^2 + z^2} k \). This becomes \( \frac{r}{|r|^2} \) since the components align with the vector \( r \).

Step 7: Select the correct option

Now we compare our result \( \frac{r}{r^2} \) with the given options. Option (A) \( \frac{r}{r^2} \) matches our result.

Key Concepts

Gradient

In the world of vector calculus, the gradient is a powerful tool used to measure how a quantity changes in a multi-dimensional space.
It provides both the direction and the rate at which the quantity changes most rapidly. For a scalar field \( \phi(x, y, z) \), the gradient, denoted as \( abla \phi \), essentially forms a vector by taking the partial derivatives with respect to each variable. The formula \( abla \phi = \frac{\partial \phi}{\partial x} i + \frac{\partial \phi}{\partial y} j + \frac{\partial \phi}{\partial z} k \) gives the components of this vector:

• The \( x \)-component \( \frac{\partial \phi}{\partial x} \) represents how \( \phi \) changes with a small change in \( x \) while keeping \( y \) and \( z \) constant.
• Similarly, the \( y \)-component \( \frac{\partial \phi}{\partial y} \) and the \( z \)-component \( \frac{\partial \phi}{\partial z} \) are calculated.

Understanding the gradient helps in various fields such as physics for force fields, or in machine learning for optimizing functions.

Nabla Operator

The nabla operator, represented by the symbol \( abla \), is a vector differential operator.
It is like a compass in vector calculus, helping you navigate through the landscape of a scalar or vector field. The nabla operator takes derivatives with respect to each spatial direction: \( abla = \frac{\partial}{\partial x} i + \frac{\partial}{\partial y} j + \frac{\partial}{\partial z} k \) for three-dimensional space.

Applications of Nabla

- **Gradient**: When applied to a scalar function \( \phi \), as described, \( abla \phi \) gives the gradient.- **Divergence**: Applying \( abla \) to a vector field will provide the divergence, indicating how much a field spreads out from a point.- **Curl**: Applying it in a certain way can also give the curl of a field, often used in fluid dynamics to describe rotation.It smoothly helps perform these operations, simplifying complex multi-dimensional problems.

Partial Derivatives

Partial derivatives are the stepping stones to the gradient and nabla operations.
They help evaluate how a function changes with respect to one variable while keeping others constant. In a function \( f(x, y, z) \), there are three primary partial derivatives:

How Partial Derivatives Work

• \( \frac{\partial f}{\partial x} \): Measures change along the \( x \)-axis, with \( y \) and \( z \) fixed. Imagine slicing through a multi-layered cake along the length.
• \( \frac{\partial f}{\partial y} \): Indicates change along the \( y \)-axis, fixing \( x \) and \( z \). Now it's like slicing it across the width.
• \( \frac{\partial f}{\partial z} \): Focuses on change along the \( z \)-axis with \( x \) and \( y \) constant. This is like going through the layers vertically.

These derivatives simplify the process of analyzing and solving complex functions by breaking them into smaller, more manageable parts. Mastering partial derivatives is crucial for deeper understanding in calculus and various scientific applications.