Problem 29 - Advanced Engineering Mathematics

Question

Verify the given identity. Assume continuity of all partial derivatives. \(\operatorname{curl}(\operatorname{grad} f)=\mathbf{0}\)

Step-by-Step Solution

Answer

The identity \(\operatorname{curl}(\operatorname{grad} f) = \mathbf{0}\) is verified.

Step 1: Understanding the Terms

The problem involves vector calculus operations. We first need to know what each operation represents. Gradient (\(\operatorname{grad}\)) of a scalar field \(f\) is a vector that represents the rate and direction of change of \(f\). Curl (\(\operatorname{curl}\)) is an operation applied on a vector field, yielding another vector field that describes the rotation.

Step 2: Expressing Gradient

The gradient \(\operatorname{grad} f\) of a scalar field \(f(x, y, z)\), is given by: \[ \operatorname{grad} f = abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \] where \(abla\) (nabla) is the del operator.

Step 3: Applying Curl to Gradient

Now, apply the \(\operatorname{curl}\) to \(\operatorname{grad} f\). The expression is: \[ \operatorname{curl}(\operatorname{grad} f) = abla \times (abla f) \] where \(abla \times\) denotes the curl.

Step 4: Computing Curl in Terms of Components

The \(\operatorname{curl}\) of a vector \( \mathbf{v} = (v_1, v_2, v_3) \) is expressed as \[ \operatorname{curl}(\mathbf{v}) = \left( \frac{\partial v_3}{\partial y} - \frac{\partial v_2}{\partial z}, \frac{\partial v_1}{\partial z} - \frac{\partial v_3}{\partial x}, \frac{\partial v_2}{\partial x} - \frac{\partial v_1}{\partial y} \right) \] Substitute \(abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \) into this formula.

Step 5: Simplifying the Expression

Applying the curl formula to \(abla f\): \[ \operatorname{curl}(abla f) = \left( \frac{\partial^2 f}{\partial y \partial z} - \frac{\partial^2 f}{\partial z \partial y}, \frac{\partial^2 f}{\partial z \partial x} - \frac{\partial^2 f}{\partial x \partial z}, \frac{\partial^2 f}{\partial x \partial y} - \frac{\partial^2 f}{\partial y \partial x} \right) \] Each term is a difference of mixed partial derivatives of \(f\), which are equal due to the continuity and equality of mixed partial derivatives (i.e., Clairaut's theorem). Hence, each component becomes zero.

Step 6: Concluding the Identity

Since each component of \(\operatorname{curl}(abla f)\) is zero, the identity is verified: \[ \operatorname{curl}(abla f) = \mathbf{0} \] This means the vector resulting from \(\operatorname{curl}(\operatorname{grad} f)\) is indeed the zero vector.

Key Concepts

Gradient

In vector calculus, the gradient is a powerful tool that provides us with the vector representation of the rate of change of a function. When we talk about the gradient, we are typically dealing with a scalar field, which is simply a function defined in three-dimensional space. The gradient of a scalar field \( f(x, y, z) \) is denoted as \( abla f \) and is computed as a vector. This vector consists of partial derivatives of \( f \) with respect to each of its variables:

• \( \frac{\partial f}{\partial x} \): Rate of change with respect to the x-axis.
• \( \frac{\partial f}{\partial y} \): Rate of change with respect to the y-axis.
• \( \frac{\partial f}{\partial z} \): Rate of change with respect to the z-axis.

This results in the vector \( (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}) \), which tells us the direction in which the function increases the fastest. This direction is perpendicular to the level surfaces of \( f \). Understanding the gradient helps in visualizing how a scalar field changes across space.

Curl

The concept of curl describes the rotation or the swirling strength of a vector field. If you imagine water flowing in a river, the curl could be seen as how a leaf might spin if dropped into the water. Mathematically, the curl of a vector field \( \mathbf{F} \), represented as \( \operatorname{curl} \mathbf{F} \), results in another vector field.
The formula to compute the curl involves taking the cross product of the del operator (\( abla \)) with the vector field:

• \( \operatorname{curl} \mathbf{F} = abla \times \mathbf{F} \)

This operation yields a vector that represents the axis and magnitude of the maximum rotation of the vector field. In simpler terms, if you're seeking where and how a vector field turns or rotates, computing its curl will provide the answer. It's important to note that when the curl is zero, it indicates the vector field is irrotational and does not exhibit any rotational motion.

Scalar Field

A scalar field is a function that assigns a scalar (a single real number) to each point in space. Unlike vector fields, scalar fields do not have direction; they only have magnitude at each point. This concept is prevalent in physics and engineering, where quantities like temperature, pressure, and electrical potential are modeled as scalar fields.
To visualize a scalar field, think of a temperature map where each point in space corresponds to a specific temperature, represented by a number. The gradient of a scalar field can then be used to determine the direction and rate of the field's greatest increase.
Scalar fields are fundamental to understanding how different quantities change over a particular region and are key in many fields, including electromagnetism and fluid dynamics.

Clairaut's Theorem

Clairaut’s Theorem, a fundamental theorem in vector calculus, addresses the equality of mixed partial derivatives. It states that if a function’s second mixed partial derivatives are continuous, then the order of differentiation does not matter. In simpler terms:

• \( \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} \)

This theorem underpins many computations in calculus, including the verification of identities like \( \operatorname{curl}(\operatorname{grad} f) = \mathbf{0} \), as explored here. Since the mixed partial derivatives of a scalar field \( f \) are equal, applying Clairaut’s theorem allows simplification when computing curls of gradients, leading to zero vectors.
Understanding Clairaut's theorem aids in simplifying complex mathematical expressions, making seemingly daunting problems more manageable.