Problem 64 - 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 \(r=x i+y j+z k\) \(\nabla r^{n}=\) (A) \(n r n-{ }^{1} r\) (B) \(n m-{ }^{2} r\) (C) \(\bar{n} \overline{r n} r\) (D) none of these

Step-by-Step Solution

Answer

A: \(n r^{n-1} r\).

Step 1 - Calculate Partial Derivative with Respect to x

The expression for calculating gradient \(abla \phi\) is \(\frac{\partial \phi}{\partial x} i + \frac{\partial \phi}{\partial y} j + \frac{\partial \phi}{\partial z} k\). Begin by calculating \(\frac{\partial r^n}{\partial x}\). Since \(r = x i + y j + z k\), we have \(r^n = (x i + y j + z k)^n\). Differentiate \((x i + y j + z k)^n\) with respect to \(x\), treating \(y\) and \(z\) as constants. Thus, \(\frac{\partial r^n}{\partial x} = n(x i + y j + z k)^{n-1}\cdot i = n r^{n-1} i\).

Step 2 - Calculate Partial Derivative with Respect to y

Next, calculate the partial derivative \(\frac{\partial r^n}{\partial y} j\). Using a similar differentiation method as in Step 1, we find that \(\frac{\partial r^n}{\partial y} = n(x i + y j + z k)^{n-1}\cdot j = n r^{n-1} j\).

Step 3 - Calculate Partial Derivative with Respect to z

Proceed with the calculation of \(\frac{\partial r^n}{\partial z} k\) using the same differentiation process. This yields \(\frac{\partial r^n}{\partial z} = n(x i + y j + z k)^{n-1}\cdot k = n r^{n-1} k\).

Step 4 - Combine the Components to Find the Gradient

Combine the results from Steps 1, 2, and 3 to form the gradient expression of \(r^n\). Thus, \(abla r^n = \frac{\partial r^n}{\partial x} i + \frac{\partial r^n}{\partial y} j + \frac{\partial r^n}{\partial z} k = n r^{n-1} (i + j + k)\).

Step 5 - Interpret the Result and Choose the Correct Option

The expression \(abla r^n = n r^{n-1} (i + j + k)\) simplifies to \(n r^{n-1} r\) because \(r = x i + y j + z k\). Therefore, the answer is option (A): \(n r^{n-1} r\).

Key Concepts

Gradient

In the context of vector calculus, the gradient is a vector operator that operates on a scalar field, which is simply a function that assigns a single value to every point in a space. This operator indicates how much a function changes as you move in different directions from a given point—notably, the direction of the steepest ascent. When we calculate the gradient of a function \( \phi(x, y, z) \), we use partial derivatives, represented commonly as \( abla \phi \) or \( \text{grad} \phi \).

This gradient is expressed as the vector \( \left( \frac{\partial \phi}{\partial x} i + \frac{\partial \phi}{\partial y} j + \frac{\partial \phi}{\partial z} k \right) \), where each component is the rate of change of the function along one of the coordinate axes. This vector points in the direction of the greatest increase of the function \( \phi \). Its magnitude tells us how fast the function increases.

If you visualize a temperature map over some land, the gradient at any point on this map gives the direction in which you would feel the most increase in temperature.

Partial Derivatives

Partial derivatives are fundamental to understanding how multivariable functions behave. In a function of multiple variables, each variable might change the outcome of the function in different ways. A partial derivative represents the derivative of that function concerning one variable alone, treating all other variables as constants.

For instance, if you have a function \( \phi(x, y, z) \), and you want to examine the behavior of \( \phi \) with respect to \( x \) alone, you would compute the partial derivative \( \frac{\partial \phi}{\partial x} \). Essentially, you're slicing through the space along the \( x \)-axis and observing how \( \phi \) changes along this path, while \( y \) and \( z \) remain fixed.

Partial derivatives are key to finding the gradient and are used extensively in optimization, where one seeks to find the turning points (maximums and minimums) of multivariable functions. They are the building blocks for tools like the gradient vector which indicates how functions change in multidimensional spaces.

Vector Differential Operator

The vector differential operator \( abla \), also known as "del" or "nabla", is used in vector calculus to calculate several things including the gradient, divergence, and curl of fields. For this exercise, we focus on its role in defining the gradient of a scalar field.

The operator is defined for a three-dimensional Cartesian coordinate as \( abla = \frac{\partial}{\partial x} i + \frac{\partial}{\partial y} j + \frac{\partial}{\partial z} k \). In simpler terms, it embodies taking partial derivatives in each spatial direction and then constructing a vector from these components. Each element of this operator acts to calculate a partial derivative along one directional axis.

In practical terms, when applied to a scalar function \( \phi(x, y, z) \), it provides a vector result, which is the gradient of that function. Additionally, understanding \( abla \) equips one to explore other vector calculus concepts, such as divergence, which measures a vector field's magnitude change per unit volume, and curl, which measures the rotation of a vector field. This versatility makes the \( abla \) operator a cornerstone of vector calculus.